1. No category

DRAFT CHAPTER 5: Derivatives of Exponential and Trigonometric Functions
Errors will be corrected before printing. Final book will be available August 2008.
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Chapter 5
DERIVATIVES OF EXPONENTIAL AND
TRIGONOMETRIC FUNCTIONS
The world’s population experiences exponential growth—the rate of growth
becomes more rapid as the size of the population increases. But is this explained in
the language of calculus? Well, the rate of growth of the population is described by
an exponential function, and the derivative of the population with respect to time
is a constant multiple of the population. But there are other examples of growth
that require not just exponential functions but compositions of exponential
functions with other functions. These examples include electronic signal
transmission with amplification, the “bell curve” used in statistics, the effects of
shock absorbers on car vibration, or describing population growth in an
environment that has a maximum sustainable population. By combining the
techniques in this chapter with the derivative rules seen earlier, we can find the
derivative of an exponential function that is combined with other functions.
Logarithmic functions and exponential functions are inverses of each other, and in
this chapter, you will see how their graphs and properties are related to each other.
CHAPTER EXPECTATIONS
In this chapter, you will
• define e and the derivative of y 5 e x, Section 5.1
• determine the derivative of the general exponential function y 5 b x, Section 5.2
• compare the graph of the exponential function with the graph of its derivative,
Section 5.1, 5.2
• solve optimization problems using exponential functions, Section 5.3
• investigate and determine the derivatives of sinusoidal functions, Section 5.4
• determine the derivative of the tangent function, Section 5.5
• Solve rate of change problems involving exponential and trigonometric function
models using their derivatives, Sections 5.1 to 5.5
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Review of Prerequisite Skills
In Chapter 5, you will be studying the derivatives of two classes of functions that
occur frequently in calculus problems: exponential functions and trigonometric
functions. To begin, we will review some properties of exponential and
trigonometric functions.
Properties of Exponents
• b mb n 5 b m1n
bm
• n 5 b m2n, bn Þ 0
b
• 1b m 2 n 5 b mn
• b logb1m2 5 m
Properties of the exponential function, y 5 b x
• The base b is positive and b Þ 1.
• The y-intercept is 1.
• The x-axis is a horizontal asymptote.
• The domain is the set of real numbers, R.
• The range is the set of positive real numbers.
• The exponential function is always increasing if b 7 1.
• The exponential function is always decreasing if 0 6 b 6 1.
• The inverse of y 5 b x is x 5 b y.
• The inverse is called the logarithmic function and is written as logb x 5 y.
The Graphs of y 5 logb x and y 5 b x
4
y
4
y = bx
y = bx 2
y
2
x
x
–4 –2 0
y = x –2
–4
2
4
y = logbx
–4 –2 0
2 4
y = x –2 y = log x
b
for b 7 1
• If b m 5 n for b 7 0, then logb n 5 m.
2
REVIEW OF PREREQUISITE SKILLS
–4
for 0 6 b 6 1
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Review of Basic Properties
Radian Measure
A radian is the measure of an angle subtended at the centre of a circle by an arc
equal in length to the radius of the circle.
p radians 5 180°
r
r
u
r
u = 1 radian
The Sine and Cosine Functions
f(x) 5 sin x and f(x) 5 cos x
The Tangent Function
f(x) 5 tan x
ex [ RZx Þ ;
p
3p
5p
,;
,;
,p f
2
2
2
Domain
x[R
Domain
Range
21 # sin x # 1
Range
21 # cos x # 1
5 y [ R6
Periodicity
tan1x 1 p2 5 tan x
Periodicity
sin1x 1 2p2 5 sin x
5
y
y = tanx
cos1x 1 2p2 5 cos x
3
2
y
1
y = sinx
x
1
x
0
–1
–2
p p 3p 2p
2
2
y = cosx
0
–1
p
2
p 3p 2p 5p
2
2
–3
–5
Transformations of Sinusoidal Functions
For y 5 a sin k1x 2 d 2 1 c and y 5 a cos k1x 2 d 2 1 c
• the amplitude is 0a 0
2p
• the period is
ZkZ
• the horizontal shift is d, and
• the vertical translation is c
CHAPTER 5
3
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Trigonometric Identities
Reciprocal Identities
Pythagorean Identities
csc u 5
1
sin u
sin2u 1 cos2u 5 1
sec u 5
1
cos u
tan2u 1 1 5 sec2u
cot u 5
1
tan u
1 1 cot2u 5 csc2u
Quotient Identities
Reflection Identities
Cofunction Identities
tan u 5
sin u
cos u
sin12u2 5 2sin u
cos a
cot u 5
cos u
sin u
cos12u2 5 cos u
sin a
p
2 x b 5 sin x
2
p
2 x b 5 cos x
2
Exercise
1. Evaluate each of the following:
2 22
d. a b
3
2. Express each of the following in the equivalent logarithmic form.
2
a. 322
2
c. 2723
b. 325
a. 54 5 625
b. 422 5
1
16
c. x 3 5 3
e. 38 5 z
d. 10w 5 450
f. a b 5 T
3. Sketch the graph of each function and state its x-intercept.
a. y 5 log10 1x 1 2 2
b. y 5 5x13
4. Refer to the figure below. State the values of
y
P(x, y)
r
x
u
a. sin u
4
REVIEW OF PREREQUISITE SKILLS
b. cos u
c. tan u
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5. Convert the following angles to radian measure.
a. 360°
c. 290°
e. 270°
g. 225°
b. 45°
d. 30°
f. 2120°
h. 330°
6. Refer to the figure below. State the values of
(0,1) y
u
(b,a)
(a,b)
x
u
(1,0)
a. sin u
c. cos u
b. tan u
d. sin a
p
2 ub
2
e. cos a
p
2 ub
2
f. sin12u 2
7. The value of sin u, cos u, or tan u is given. Determine the values of the other
two functions if u lies in the given interval.
a. sin u 5
5 p
, #u#p
13 2
2
3p
b. cos u 5 2 , p # u #
3
2
c. tan u 5 22,
3p
# u # 2p
2
d. sin u 5 1, 0 # u # p
8. State the period and amplitude for each of the following.
2
a. y 5 cos 2x
d. y 5 cos112x2
7
x
p
b. y 5 2 sin
e. y 5 5 sin a u 2 b
2
6
c. y 5 23 sin1px2 1 1
f. y 5 03sin x 0
9. Sketch the graph of each function over two complete periods.
p
a. y 5 sin 2x 1 1
b. y 5 3 cos a x 1 b
2
10. Prove the following identities.
sin x
a. tan x 1 cot x 5 sec x csc x
b.
5 tan x sec x
1 2 sin2x
11. Solve the following equations where x [ 30, 2p 4 . Express your answers to
one decimal place.
a. 3 sin x 5 sin x 1 1
b. cos x 2 1 5 2cos x
CHAPTER 5
5
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Investigate
CHAPTER 5: RATE-OF-CHANGE MODELS IN MICROBIOLOGY
While many real life situations can be modelled fairly well by polynomial
functions, there are also many scenarios that are best modelled by other types of
functions, including exponential, logarithmic, and trigonometric functions.
Because determining the derivative of a polynomial function is simple, finding
the rate of change for models described by polynomial functions is also simple.
Often the rate of change at various times is more important to the person
studying the scenario than is the value of the function. In this chapter, you will
learn how to differntiate exponential and trigonometric functions, increasing the
number of function types you can use to model real world situations and in turn
analyze using rates of change.
Case Study—Microbiologist
Microbiologists contribute their expertise to many fields, including medicine,
environmental science, and biotechnology. Enumerating, the process of counting
bacteria, allows microbiologists to build mathematical models that predict
populations. Once they can predict a population accurately, the model could be
used in medicine, for example, to predict the dose of medication required to kill
a certain bacterial infection. The data set in the table
Time
Population
shown was used by a microbiologist to produce a
(in hours)
polynomial-based mathematical model to predict
0
1000
population p(t), as a function of time t, in hours, for
0.5
1649
the growth of a certain strain of bacteria:
p1t2 5 1000 a 1 1 t 1
1 2 1 3
1 4 1 5
t 1 t 1
t
t b
2
6
24 120
1.0
2718
1.5
4482
2.0
7389
DISCUSSION QUESTIONS
1. How well does the function fit the data set? Use the data, the equation, a
graph, and/or a graphing calculator to comment on the “goodness of fit.”
2. What is the population after 0.5 h? How fast is the population growing at
this time? (Use calculus to determine this.) Complete these calculations for
the 1.0 h point.
3. What pattern did you notice in your calculations? Explain this pattern by
examining the terms of this equation to find the reason why.
The polynomial function in this case is an approximation of a special function in
mathematics, natural science, and economics, f 1x2 5 e x, where e has a value of
2.718 28…. At the end of this chapter, you will complete a task on rates of
change of exponential growth in a biotechnology case study.
6
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Section 5.1—Derivatives of Exponential
Functions, y 5 e x
Many mathematical relations in the world are nonlinear. We have already studied
various polynomial and rational functions and their rates of change. Another type of
nonlinear model is the exponential function. Exponential functions are often used to
model rapid change. Compound interest, population growth, the intensity of an
earthquake, and radioactive decay are just a few examples of exponential change.
In this section, we will study the exponential function y 5 e x and its derivative.
The number e is a special irrational number, like the number p. It is called
Euler’s (or natural) number in honour of the Swiss mathematician Leonhard Euler
(pronounced “oiler”), 1707–1783. We use a rational approximation for e of about
2.718. The rules developed thus far have been applied to polynomial functions
and rational functions; we are going to show how the derivative of an exponential
function can be found.
INVESTIGATION
Tech
Support
To evaluate
powers of e, such
as e22, press
LN
2ND
2
.
S
In this investigation, you will
a. graph the exponential function f 1x2 5 e x and its derivative
b. determine the relationship between the exponential function and its derivative
A. Consider the function f 1x2 5 e x. Create a table similar to the one shown below.
Complete the f 1x2 column using a graphing calculator by calculating
the values of e x for the values of x provided.
2
x
f (x)
22
0.135
f´(x)
21
0
1
2
3
B. Graph the function f 1x 2 5 e x.
C. Use a graphing calculator to calculate the value of the derivative f ¿1x2 at each
of the given points as follows:
MAscroll down to 8:nDeriv( under the
To calculate f ¿1x2 press MATH
and
MATH menu.
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C
Press ENTER and the display on the screen will be nDeriv(.
To find the derivative, key in the expression e x, the variable x, and the x value
at which you want the derivative.
d x
For example, to determine dx
e at x 5 22 the display will be nDeriv( e x, X, 22).
Press ENTER and the approximate value of f ¿1222 will be returned.
D. What do you notice about the values of f 1x 2 and f ¿1x2 ?
E. Draw the graph of the derivative function f ¿1x2 on the same set of axes as f 1x 2.
How do the two graphs compare?
F. Try a few other values of x to see if the pattern continues.
G. What conclusion can you make about the function f 1x 2 5 e x and its derivative?
PROPERTIES OF y 5 e x
Since y 5 e x is an exponential function, it has the same properties as other
exponential functions you have studied.
Recall that the logarithm function is the inverse of the exponential function. For
example, y 5 log2 x is the inverse of y 5 2x. The function y 5 e x also has an
inverse, y 5 loge x. Their graphs are reflections in the line y 5 x. The function
y 5 loge x can be written as y 5 ln x and is called the natural logarithm function.
12
y
y=x
8
4
y = ex
x
–12 –8 –4 0
–4
4 8
y = In x
12
–8
–12
All the properties of exponential functions and logarithmic functions you are
familiar with also apply to y 5 e x and y 5 ln x.
y 5 ex
• the domain is 5x [ R6
• the domain is 5x [ R 0 x 7 06
• passes through 10, 12
• passes through 11, 02
• the range is 5 y [ R 0 y 7 06
• e ln x 5 x
• y 5 0 is the horizontal asymptote
8
y 5 ln x
5 . 1 D E R I VAT I V E S O F E X P O N E N T I A L F U N C T I O N S, y 5 ex
• the range is 5 y [ R6
• ln e x 5 x
• x 5 0 is the vertical asymptote
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From the investigation, you should have noticed that all of the values of the
derivative f ¿1x 2 were exactly the same as those of the original function f 1x 2 5 e x.
This is a very significant result, since this function is its own derivative, that is,
f 1x2 5 f ¿1x2. Since the derivative also represents the slope of the tangent at any
given point, the function f 1x 2 5 e x has the special property that the slope of the
tangent at a point is the value of the function at that point.
12
8
f (x) = ex
(1, e1) 4
–12 –8 –4 0
–4
The slope of the
tagent at (1, e1)
is e1.
y
(2, e2)
The slope of the
tagent at (2, e2) is
e2 , and so on. x
4
8
12
–8
–12
The Derivative of f(x) 5 e x
For the function f 1x 2 5 e x, f ¿ 1x2 5 e x.
EXAMPLE 1
Selecting a strategy to differentiate a composite function involving e x
Determine the derivative of f 1x 2 5 e 3x.
Solution
To find the derivative, use the Chain Rule.
df 1x2
dx
5
d1e 3x 2 d13x2
d13x2
5 e 3x # 3
dx
5 3e 3x
The Derivative of a Composite Function Involving ex
In general, if f 1x 2 5 eg1x2, then f ¿1x 2 5 eg1x2 g¿1x 2 by the Chain Rule.
CHAPTER 5
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Derivatives of exponential functions involving e x
Determine the derivative of each function.
2
a. g 1x 2 5 e x 2x
b. f 1x 2 5 x 2e x
Solution
2
a. To find the derivative of g 1x 2 5 e x 2x, we use the Chain rule.
dg1x 2
dx
5
5
d1e x
2
2x
dx
d1e x
2
2
2x
2
# d1x
d1x 2 2 x2
5 ex
2
2x
2
2 x2
(Chain rule)
dx
12x 2 1 2
b. Using the Product Rule
f ¿ 1x2 5
d1x 2 2
dx
# e x 1 x 2 de
x
(Product rule)
dx
5 2xe x 1 x 2e x
(Factor)
5 e 12x 1 x 2
x
EXAMPLE 3
2
Selecting a strategy to determine the value of the derivative
Given f 1x 2 5 3e x , determine f ¿121 2.
2
Solution
First, find an expression for the derivative of f ¿1x2 .
d13e x 2 d 13x2
2
f ¿1x2 5
d1x2 2
5 3e x 12x 2
dx
(Chain rule)
2
5 6xe x
2
Then f ¿1212 5 26e.
Answers are usually left as exact values in this form. If desired, numeric approximations can be obtained from a calculator. Here, using e 8 2.718, we obtain the
answer 216.308.
10
5 . 1 D E R I VAT I V E S O F E X P O N E N T I A L F U N C T I O N S, y 5 ex
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Connecting the derivative of an exponential function to the slope of
a tangent
x
Determine the equation of the line tangent to y 5 ex 2 where x 5 2.
Solution
Use the derivative to determine the slope of the required tangent.
y5
ex
(Rewrite as a product)
x2
5 x 22e x
dy
5 122x 23 2e x 1 x 22e x
dx
22e x
ex
5 3 1 2
x
x
5
5
5
22e x
x3
1
xe x
(Product rule)
(Determine a common denominator)
(Simplify)
x3
22e x 1 xe x
(Factor)
x3
122 1 x2e x
x3
2
When x 5 2, y 5 e4 . When x 5 2, dy
5 0, and the tangent is horizontal.
dx
2
Therefore, the equation of the required tangent is y 5 e4 . A calculator yields the
x
following graph for y 5 xe 2, and we see the horizontal tangent at x 5 2.
How does the derivative of the general exponential function g 1x 2 5 b x compare
to the derivative of f 1x 2 5 e x? We will answer this question in section 5.2.
CHAPTER 5
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IN SUMMARY
Key Ideas
d x
• For f1x2 5 e x, f ¿1x2 5 e x. In Leibniz notation, dx
1e 2 5 ex.
1e g1x2 2 1g1x 22
• For f1x2 5 e g1x2, f ¿1x2 5 e g1x2 ? g¿ 1x2. In Leibniz notation, dd 1g1x 22 d dx
• The slope of the tangent at a point on the graph of y 5 e x equals the value
of the function at that point.
Need to Know
• The rules for differentiating functions, such as the Product, Quotient, and
Chain rules, also apply to combinations involving exponential functions of the
form f1x2 5 e g1x2.
• e is called Euler's or the natural number, where e 8 2.718
Exercise 5.1
PART A
1. Why can you not use the Power Rule for derivatives to differentiate y 5 e x?
2. Differentiate each of the following.
K
a. y 5 e 3x
c. y 5 2e 10t
e. y 5 e 526x1x
b. s 5 e 3t25
d. y 5 e 23x
f. y 5 e Vx
2
3. Detemine the derivative of each of the following.
a. y 5 2e
e 2x
c. f 1x 2 5
x
x3
3
d. f 1x 2 5 Vxe x
b. y 5 xe 3x
e. h 1t 2 5 et 2 1 3e 2t
f. g 1t 2 5
e 2t
1 1 e 2t
1
4. a. If f 1x2 5 1e 3x 1 e 23x 2, calculate f ¿112 .
3
b. If f 1x2 5 e 21 x 1 1 2, calculate f ¿102 .
1
c. If h1z 2 5 z 2 11 1 e 2z 2, calculate h¿121 2 .
x
5. a. Determine the equation of the tangent to the curve defined by y 5 1 2e
at
1 ex
the point 10, 1 2 .
b. Use graphing technology to graph the function in 5. a. and draw the
tangent at 10, 12.
c. Compare the equation in 5. a. to the equation generated by graphing
technology.
12
5 . 1 D E R I VAT I V E S O F E X P O N E N T I A L F U N C T I O N S, y 5 ex
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PART B
6. Determine the equation of the tangent to the curve y 5 e 2x at the point where
x 5 21. Graph the original curve and the tangent.
7. Determine the equation of the tangent to the curve defined by y 5 xe 2x at the
point A11, e 21 2 .
8. Determine the coordinates of all points at which the tangent to the curve
defined by y 5 x 2e 2x is horizontal.
9. If y 5 52 1e5 1 e 2 5 2 , prove that y– 5 25.
x
x
y
dy d 2y
10. a. For the function y 5 e 23x, determine dx, dx2 and
d 3y
.
dx3
dny
b. From the pattern in 10. a., state the value for dxn
11. Determine the first and second derivatives for each function.
a. y 5 23e x
A
c. y 5 e x 14 2 x 2
b. y 5 xe 2x
12. The number, N, of bacteria
in a culture at time t in hours is
t
N1t2 5 1000330 1 e 2 304
a. What is the initial number of bacteria in the culture?
b. Determine the rate of change in the number of bacteria at time t.
c. How fast is the number of bacteria changing when t 5 20 h?
d. Determine the largest number of bacteria in the culture during the interval
0 # t # 50.
13. The distance s (in metres) fallen by a skydiver t seconds after jumping (and
before the parachute opens) is s 5 160Q4t 2 1 1 e 2 4R.
1
t
a. Determine the velocity, v, at time t.
1
b. Show that acceleration is given by a 5 10 2 4 v.
c. Determine vT 5 lim v. This is the “terminal” velocity, the constant velocity
tSq
attained when the air resistance balances the force of gravity.
d. At what time is the velocity 95% of the terminal velocity? How far has the
skydiver fallen at that time?
C
14. Use a table of values and successive approximation to evaluate each of the
following:
a. lim a 1 1
xSq
1
b
x
b. lim 11 1 x2
1
x
xS0
c. Discuss the results.
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PART C
15. Use the definition of the derivative to evaluate each limit.
eh 2 1
hS0
h
e 21h 2 e 2
hS0
h
a. lim
b. lim
16. For what values of m does the function y 5 Ae mt satisfy the following
equation?
d 2y
dx
2
1
dy
2 6y 5 0
dx
17. The hyperbolic functions are defined as sinh x 5 12 1e x 2 e 2x 2 and
cosh x 5 12 1e x 1 e 2x 2 .
a. Prove
b. Prove
c. Prove
d1sinh x2
dx
d1cosh x2
dx
d1tanh x2
dx
5 cosh x.
5 sinh x.
5
1
1cosh x2
2
if tanh x 5
sinh x
.
cosh x
1
1
1
1
1
18. a. Another expression for e is e 5 1 1 1!
1 2!
1 3!
1 4!
1 5!
1 p.
Evaluate this expression using four, five, six, and seven consecutive terms
of this expression. (Note: 2! Is read “two factorial” 2! 5 2 3 1 and
5! 5 5 3 4 3 3 3 2 3 1 2
b. Explain why the expression for e in (a) is a special case of
1
2
3
4
x
x
e x 5 1 1 1!
1 2!
1 x3! 1 x4! 1 p . What is the value of x?
Extension: Graphing the Hyperbolic Function
1. Use graphing technology to graph y 5 cosh x by using the definition
cosh x 5 12 1e x 1 e 2x 2 .
CATALOG
2ND
MODE
DELof CATALOG items and select cosh( to
0 for the list
2. Press
investigate if cosh is a built-in function.
A
3. On the same window, graph y 5 1.25x 2 1 1 and y 5 1.05x 2 1 1. Investigate
changes in the coefficient a in the equation y 5 ax 2 1 1 to see if you can
create a parabola that will approximate the hyperbolic cosine function.
14
5 . 1 D E R I VAT I V E S O F E X P O N E N T I A L F U N C T I O N S, y 5 ex
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Section 5.2—Derivative of the General
Exponential Function, y 5 b x
In the previous section, we investigated the exponential function y 5 e x and its
derivative. That function has the special property that the function is its own
derivative. The graph of the derivative function is the same as the graph of
y 5 e x. In this section, we will look at the general exponential function y 5 b x
and its derivative.
10
y
y = bx
8
6
4
0<b<1 2
b>1
x
–6 –4 –2 0
–2
INVESTIGATION
2
4
6
In this investigation, you will
a. graph and compare the general exponential function and its derivative using
the slopes of the tangents at various points and with different bases
b. determine the relationship between the general exponential function and its
derivative by means of a special ratio
A. Consider the function f 1x2 5 2x. Create a table with the headings shown
below. Use the equation of the function to complete the f 1x2 column.
x
f (x)
f 9(x)
f 9(x)
f (x)
22
21
0
1
2
3
B. Graph the function f 1x2 5 2x.
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C. Calculate the value of the derivative f ¿1x2 at each of the given points. To
calculate f ¿1x2 use the nDeriv( function. (See the investigation in section 5.1
for detailed instructions.)
D. Draw the graph of the derivative function on the same set of axes as f 1x2
using the given x-values and the corresponding values of f ¿1x2.
E. Compare the graph of the derivative with the graph of f 1x2.
f ¿ 1x 2
F. a. Calculate the ratio f 1x 2 and record these values in the last column of your table.
b. What do you notice about this ratio for the different values of x?
c. Is the ratio greater or less than 1?
G. Repeat parts A to F for the function f 1x2 5 3x.
H. Compare the ratio
f ¿ 1x 2
for the function f 1x2 5 2x and f 1x2 5 3x.
f 1x 2
I. Repeat parts A to F for the function f 1x2 5 b x using different values of b.
Does the pattern you found for f 1x2 5 2x and f 1x2 5 3x continue?
J. What conclusions can you make about the general exponential function and
its derivative?
Properties of y 5 b x
In this investigation, you worked with the functions f 1x2 5 2x, f 1x2 5 3x, and
their derivatives. You should have made the following conclusions:
f ¿ 1x 2
• For the function f 1x2 5 2x, the ratio f 1x 2 is approximately equal to 0.69.
• The derivative of f 1x2 5 2x is approximately equal to 0.69 3 2x.
f ¿ 1x 2
• For the function f 1x2 5 3x, the ratio f 1x 2 is approximately equal to 1.10.
• The derivative of f 1x2 5 3x is approximately equal to 1.10 3 3x.
y
10
f(x) = 2x
8
10
6
6
4
4
f '(x) = 1.10 3 3x
2
2
–6 –4 –2 0
–2
8
f '(x) = 0.69 3 2x
x
2
4
6
The derivative of f 1x2 5 2x is an
exponential function.
The graph of f ¿1x2 is a vertical
compression of the graph of f 1x2.
16
5.2
D E R I VAT I V E O F T H E G E N E R A L E X P O N E N T I A L F U N C T I O N, y 5 b x
y
f(x) = 3x
x
–6 –4 –2 0
–2
2
4
6
The derivative of f 1x2 5 3x is
an exponential function.
The graph of f ¿1x2 is a vertical stretch
of the graph of f 1x2.
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In general, for the exponential function f 1x2 5 b x we can conclude that
• f 1x2 and f ¿1x2 are both exponential functions
• the slope of the tangent at a point on the curve is proportional to the value of
the function at that point
• f ¿1x2 is a vertical stretch or compression of f 1x2 dependent on the value of b
f ¿ 1x 2
• the ratio f 1x 2 is a constant and is equivalent to the stretch/compression factor
We can use the definition of the derivative to determine the derivative of the
exponential function f 1x2 5 b x.
f ¿1x2 5 lim
hS0
f 1x 1 h2 2 f 1x2
h
b x1h 2 b x
hS0
h
5 lim
bx # bh 2 bx
hS0
h
5 lim
5 lim
hS0
(Substitution)
(Properties of the exponential function)
b x 1b h 2 12
h
(Common factor)
The factor b x is constant as h S 0 and does not depend on h. Therefore,
bh 2 1
f ¿1x2 5 b x lim
hS0
h
Consider the functions from our investigation:
• For f 1x2 5 2x, we determined that f ¿ 1x2 8 0.69 3 2x and so
2h 2 1
8 0.69.
hS0
h
lim
• For f 1x2 5 3x, we determined that f ¿ 1x2 8 1.10 3 3x and so
3h 2 1
8 1.10.
hS0
h
lim
h
In the previous section, for f 1x2 5 e x, f ¿1x2 5 e x and so lim e h2 1 5 1.
hS0
Can we find a way of determining this constant of proportionality without using a
table of values?
The derivative of f 1x2 5 e x might give us a hint at the answer to this question.
From the previous section, we know that
f ¿1x2 5 1 3 e x.
We also know that loge e 5 1, or ln e 5 1.
Now consider ln 2 and ln 3.
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ln 2 8 0.693 147 and
ln 3 8 1.098 612
f ¿ 1x 2
These match the constants f 1x 2 that we determined in our investigation.
This leads to the following conclusion:
The Derivative of f(x) 5 bx
bh 2 1
5 ln b and
hS0
h
lim
if f 1x2 5 b x, then f ¿1x2 5 1ln b2 3 b x
EXAMPLE 1
Selecting a strategy to determine derivatives involving b x
Determine the derivative of
a. f 1x2 5 5x
b. f 1x2 5 53x22
Solution
a. f 1x2 5 5x Use the derivative of f 1x2 5 b x.
f ¿1x2 5 1ln 52 3 5x
b. To differentiate f 1x2 5 53x22, use the Chain Rule and the derivative of
f 1x2 5 b x.
f 1x2 5 53x22
We have f 1x2 5 5g1x2 with g1x2 5 3x 2 2.
Then g¿ 1x2 5 3.
Now, f ¿1x2 5 53x22 3 1ln 52 3 3
5 3153x22 2 ln 5
The Derivative of f(x) = bg(x)
For f 1x2 5 b g1x2, f ¿1x2 5 b g1x2 ln b1g¿1x22 .
EXAMPLE 2
Solving a problem involving an exponential model
On January 1, 1850, the population of Goldrushtown was 50 000. Since then, the
size of the population has been modelled by the function P1t2 5 50 00010.98 2 t,
where t is the number of years since January 1, 1850.
a. What was the population of Goldrushtown on January 1, 1900?
b. At what rate was the population of Goldrushtown changing on January 1,
1900? Was it increasing or decreasing at that time?
18
5.2
D E R I VAT I V E O F T H E G E N E R A L E X P O N E N T I A L F U N C T I O N, y 5 b x
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Solution
a. January 1, 1900, is exactly 50 years after January 1, 1850, so we let t 5 50.
P1502 5 50 00010.98 2 50
5 18 208.484
The population on January 1, 1900 was approximately 18 208.
b. To determine the rate of change in the population, we require the derivative
of P.
P¿1t2 5 50 00010.982 tln10.98 2
P¿1502 5 50 00010.98 2 50ln10.98 2
8 2367.861
Hence, after 50 years, the population was decreasing at a rate of approximately
368 people per year. (We expected the rate of change to be negative, because the
original population function was a decaying exponential function since the base
was less than 1.)
60 000 P(t)
50 000
Population
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40 000
P(t) is a decreasing
function. P'(t) < 0
30 000
20 000
10 000
t
0
20 40 60 80 100
time (years)
IN SUMMARY
Key Ideas
d x
1b 2 5 bx # In b
dx
? lnb ? g¿1x2. In Leibniz notation,
• If f1x2 5 b x, then f ¿1x2 5 b x ? lnb. In Leibniz notation,
• If f1x2 5 b g1x2, then f ¿1x2 5 b g1x2
d 1 b g1x22 d1g1x22
d
1 b g1x2 2 5
dx
d1g1x22
dx
Need to Know
bh 2 1
5 lnb
hS0
h
• lim
• When you are differentiating a function that involves an exponential function,
use the above rules, along with the Sum, Difference, Product, Quotient, and
Chain rules as required.
CHAPTER 5
19
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Exercise 5.2
K
PART A
1. Differentiate each of the following functions.
d. w 5 101526n1n
a. y 5 23x
b. y 5 3.1x 1 x 3
e. y 5 3x
2
2
2
12
f. y 5 40012 2 x13
c. s 5 103t25
2. Determine the derivative of each of the following.
2t
a. y 5 x 5 3 152 x
c. v 5
t
b. y 5 x13 2 x
3. If f 1t2 5 10
2
3t25
d. f 1x2 5
V3x
x2
# e , determine the values of t so that f ¿1t2 5 0.
2t 2
PART B
4. Determine the equation of the tangent to the curve y 5 312x 2 at the point
where x 5 3.
5. Determine the equation of the tangent to the curve y 5 10x at the point
11, 10 2.
A
6. A certain radioactive material decays exponentially. The percent, P, of the
material left after t years is given by P1t2 5 10011.2 2 2t.
a. Determine the half-life of the substance.
b. How fast is the substance decaying at the point where the half-life is reached?
T
7. Historical data show that the amount of money sent out of Canada for interest
and dividend payments during the period 1967–1979 can be approximated by
the model P 5 15 3 108 2 e 0.200 15 t, where t is measured in years 1t 5 0 in
1967 2 and P is the total payment in Canadian dollars.
a. Determine and compare the rates of increase for the years 1968 and 1978.
b. Compare the rate of increase for 1988 to the rate of increase for 1998.
c. Check the Statistics Canada website to see if the rates of increase
predicted by this model were accurate for 1988 and 1998.
2
8. Determine the equation of the tangent to the curve y 5 22x at the point on
the curve where x 5 0. Graph the curve and the tangent at that point.
C
20
5.2
PART C
9. The velocity of a car is given by v1t2 5 12011 2 0.85t 2 . Graph the function.
Describe the acceleration of the car.
D E R I VAT I V E O F T H E G E N E R A L E X P O N E N T I A L F U N C T I O N, y 5 b x
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Section 5.3—Optimization Problems Involving
Exponential Functions
In earlier chapters, you considered numerous situations in which you were asked
to optimize a given situation. Recall that to optimize means to determine values
of variables so that a function that represents quantities such as cost, area,
number of objects, or distance can be minimized or maximized.
Here we will consider further optimization problems, using exponential function
models.
EXAMPLE 1
Solving an optimization problem involving an exponential model
The effectiveness of studying for an exam depends on how many hours a student
studies. Some experiments show that, if the effectiveness, E, is put on a scale of
t
0 to 10, then E1t2 5 0.53 10 1 te220 4, where t is the number of hours spent studying
for an examination. If a student has up to 30 h for studying, how many hours are
needed for maximum effectiveness?
Solution
t
We wish to find the maximum value for the function E1t2 5 0.5 310 1 te 220 4, on
the interval 0 # t # 30.
First find critical numbers by determining E¿1t2 .
E¿1t2 5 0.5 a e 220 1 t a 2
t
5 0.5e 220 a 1 2
t
1 220t
e bb
20
(Product and Chain rules)
t
b
20
t
E¿ is defined for t [ R, and e 220 7 0 for all values of t. So, E¿1t2 5 0 when
t
1 2 20 5 0.
Therefore, t 5 20 is the critical number.
To determine the maximum effectiveness, we use the algorithm for finding
extreme values.
E102 5 0.5110 1 0e 0 2 5 5
E1202 5 0.5110 1 20e 21 2 8 8.7
E1302 5 0.5110 1 30e 21.5 2 8 8.3
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21
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Therefore, the maximum effectiveness measure of 8.7 is achieved when a student
studies 20 h for the exam.
t
Examining the graph of the function E1t2 5 0.5 3 10 1 te 220 4 confirms our result.
10 E (t) Maximum point (20, 8.7)
Effectiveness
8
6
4
2
t
0
5
10 15 20 25 30
time (h)
EXAMPLE 2
Using calculus techniques to analyze an exponential business model
A mathematical consultant determines that the proportion of people who will have
responded to the advertisement of a new product after it has been marketed for
t days is given by f 1t2 5 0.711 2 e 20.2t 2. The area covered by the advertisement
contains 10 million potential customers, and each response to the advertisement
results in revenue to the company of $0.70 (on average), excluding the cost of
advertising. The advertising costs $30 000 to produce and a further $5000 per day
to run.
a. Determine lim f 1t2 and interpret the result.
tSq
b. What percentage of potential customers have responded after 7 days of
advertising?
c. Write the function P(t) that represents the profit after t days of advertising.
What is the profit after 7 days?
d. For how many full days should the advertising campaign be run in order to
maximize the profit? Assume an advertising budget of $200 000.
Solution
a. As t S q, e 20.2t S 0, so lim f 1t2 5 lim 0.711 2 e 20.2t 2 5 0.7. This result
tSq
tSq
means that, if the advertising is left in place indefinitely (forever), 70% of the
population will respond.
b. f 172 5 0.711 2 e 20.2172 2 8 0.53
After seven days of advertising, about 53% of the population has responded.
c. The profit is the difference between the revenue received from all customers
responding to the ad and the advertising costs. Since the area covered by the ad
contains 10 million potential customers, the number of customers responding to
the ad after t days is 107 3 0.711 2 e 20.2t 2 4 5 7 3 106 11 2 e 20.2t 2 .
22
5.3 O P T I M I Z AT I O N P R O B L E M S
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The revenue to the company from these respondents is
R1t2 5 0.737 3 106 11 2 e 20.2t 2 4 5 4.9 3 106 11 2 e 20.2t 2 .
The advertising costs for t days are C1t2 5 30 000 1 5000t.
Therefore, the profit earned after t days of advertising is given by
P1t2 5 R1t2 2 C1t2
5 4.9 3 106 11 2 e 20.2t 2 2 30 000 2 5000t.
After seven days of advertising, the profit is
P172 5 4.9 3 106 11 2 e 20.2172 2 2 30 000 2 5000172
8 3 627 000.
d. If the total advertising budget is $200 000, then we require that
30 000 1 5000t # 200 000
5000t # 170 000
t # 34.
We wish to maximize the profit function P 1t2 on the interval 0 # t # 34 .
For critical numbers, determine P ¿1t2.
P ¿1t2 5 4.9 3 106 10.2e 2 0.2t 2 2 5000
5 9.8 3 105e 2 0.2t 2 5000
P¿ 1t2 is defined for t [ R . Let P¿ 1t2 5 0.
9.8 3 105e 20.2t 2 5000 5 0
e 20.2t 5
5000
(Isolate e 20.2t )
9.8 3 105
e 20.2t 8 0.005 102 04
20.2t 5 ln10.005 102 04 2
(Take the ln of both sides)
(Solve)
t 8 26
To determine the maximum profit, we evaluate
P1262 5 4.9 3 106 11 2 e 20.21262 2 2 30 000 2 5000126 2
8 4 713 000
P102 5 4.9 3 106 11 2 e 0 2 2 30 000 2 0
5 230 000 (they’re losing money!)
P1342 5 4.9 3 106 11 2 e 20.21342 2 2 30 000 2 5000134 2
8 4 695 000
The maximum profit of $4 713 000 occurs when the ad campaign runs for
26 days.
CHAPTER 5
23
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Examining the graph of the function P 1t2 confirms our result.
5 000 000 P(t)
Profit ($)
4 000 000
Maximum point
(26, 4 713 000)
3 000 000
2 000 000
1 000 000
t
–20 –10 0
10 20 30 40
time (days)
IN SUMMARY
Key Ideas
• Optimizing means to determine the values of the independent variable so that
the functions values that models a situation can be minimized or maximized.
• The techniques used to optimize an exponential function model are the same
as those used on polynomial and rational functions.
Need to Know
• Apply the algorithm for solving optimization problems introduced in Chapter 3
to solve optimization problems:
1. Understand the problem and identify quantities that can vary. Determine a
function in one variable that represents the quantity to be optimized.
2. Determine the domain of the function to be optimized, using the
information given in the problem.
3. Use the algorithm for finding extreme values from Chapter 3 to find the
absolute maximum or minimum function value on the domain.
4. Use the result of step 4 to answer the original problem.
5. Graph the original function using technology to confirm your results.
24
5.3 O P T I M I Z AT I O N P R O B L E M S
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Exercise 5.3
PART A
1. Use graphing technology to graph each of the following functions. From the
graph, find the absolute maximum and absolute minimum values of the given
functions on the indicated intervals.
a. f 1x2 5 e 2x 2 e 23x on 0 # x # 10
b. m1x2 5 1x 1 2 2 e 22x on x [ 3 24, 4 4
2. a. Use the algorithm for finding extreme values to determine the absolute
maximum and minimum values for the functions in question 1.
b. Explain which approach is easier to use for the functions in question 1.
3. A small self-contained forest was studied for squirrel population by a
biologist. It was found that the squirrel population, P, is a function of time, t,
20
where t is measured in weeks. The function is P1t2 5 1 1 3e20.02t .
a. Determine the population at the start of the study when t 5 0.
b. The largest population the forest can sustain is represented mathematically
by the limit as t S q. Determine this limit.
c. Determine the point of inflection.
d. Graph the function.
e. Explain the meaning of the point of inflection in terms of squirrel
population growth.
PART B
4. The net monthly profit from the sale of a certain product is given (in dollars)
by the formula P1x2 5 106 3 1 1 1x 2 1 2e 20.001x 4 , where x is the number of
items sold.
a. Determine the number of items that yield the maximum profit. At full
capacity, the factory can produce 2000 items per month.
b. Repeat 4. a. assuming that, at most, 500 items can be produced per month.
K
5. Suppose the revenue (in thousands of dollars) for sales of x hundred units of
an electronic item is given by the function R1x2 5 40x 2e 20.4x 1 30, where
the maximum capacity of the plant is 800 units. Determine the number of
units to produce to maximize revenue.
6. A rumour spreads through a population in such a way that, t hours after the
rumour starts, the percentage of people involved in passing it on is given by
P1t2 5 1001e 21 2 e 24t 2. What is the highest percentage of people involved
in spreading the rumour within the first 3 h? When does this occur?
CHAPTER 5
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7. Small countries trying to rapidly develop an industrial economy often try to
achieve their objectives by importing foreign capital and technology. Statistics
Canada data show that, when Canada attempted this strategy from 1867 to
1967, the amount of U.S. investment in Canada increased from about
$15 3 106 to $280 305 3 106. This increase in foreign investment can be
represented by the simple mathematical model C1t2 5 0.015 3 109e 0.075 33t,
where t represents the number of years (starting with 1867 as zero), and C
represents the total capital investment from U.S. sources.
a. Graph the curve for the 100-year period.
b. Compare the growth rate of U.S. investment in 1947 to the rate in 1967.
c. Determine the growth rate of investment in 1967 as a percentage of the
amount invested.
d. If this model is used up to 1977, calculate the total U.S. investment and the
growth rate.
e. Use the Internet to determine the actual total U.S. investment in 1977, and
calculate the error in the model.
f. If the model is used up to 2007, calculate the expected U.S. investment
and the expected growth rate.
t
A
8. A colony of bacteria in a culture grows at a rate given by N1t2 5 25, where N is
the number of bacteria t minutes from the beginning. The colony is allowed to
grow for 60 min, at which time a drug is introduced that kills the bacteria. The
t
number of bacteria killed is given by K1t2 5 e3, where K bacteria are killed at
time t minutes.
a. Determine the maximum number of bacteria present and the time at which
this occurs.
b. Determine the time at which the bacteria colony is obliterated.
9. Lorianne is studying for two different exams. Because of the nature of
the courses, the measure of study effectiveness for the first course is
t
E1 5 0.619 1 te 220 2 , while the measure for the second course is
E2 5 0.5110 1 te 210 2 . She is prepared to spend 30 h total in preparing for the
exams. The total effectiveness is given by f 1t2 5 E1 1 E2. How should this
time be allocated to maximize total effectiveness?
t
C
T
26
10. Explain the steps you would use to determine the absolute extrema of
f 1x2 5 x 2 e2x on the interval x # 3 22, 24 .
11. a) For f 1x2 5 x 2e x, determine the intervals of increase and decrease.
5.3 O P T I M I Z AT I O N P R O B L E M S
b) Determine the absolute minimum value of f 1x2.
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12. Find the maximum and minimum values of each function. Graph each function.
a. y 5 ex 1 2
c. y 5 2xe2x
b. y 5 xex 1 3
d. y 5 3xe2x 1 x
2
13. The profit function of a commodity is P1x2 5 xe20.5x , where x 7 0. Find the
maximum value of the function.
14. You have just walked out the front door of your home. You notice that it closes
quickly at first and then closes more slowly. In fact, a model of the movement
of a closing door is given by d1t2 5 200t12 2 2t, where d is the number of
degrees between the door frame and the door at t seconds.
a. Graph this relation.
b. Determine when the speed of the closing door is increasing and decreasing.
c. Determine the maximum speed of the closing door.
d. At what point would you consider the door closed?
PART C
15. Suppose that, in question 9, Lorianne has only 25 h to study for the two exams.
Is it possible to determine the time to be allocated to each exam? If so, how?
16. Although it is true that many animal populations grow exponentially for a
period of time, it must be remembered that, eventually, the food available to
sustain the population will run out, and at that point, the population will
decrease through starvation. Over a period of time, the population will level
out to the maximum attainable value, L. One mathematical model to describe
a population that grows exponentially at the beginning and then levels off to
a limiting value, L, is the logistic model. The equation for this model is
aL
P 5 a 1 1L 2 a 2 e2kLt, where the independent variable t represents the time,
and P represents the size of the population. The constant a is the size of the
population at t 5 0, L is the limiting value of the population, and k is a
mathematical constant.
a. Suppose a biologist starts a cell colony with 100 cells and finds that the
limiting size of the colony is 10 000 cells. If the constant k 5 0.0001,
draw a graph to illustrate this population, where t is in days.
b. At what point in time does the cell colony stop growing exponentially?
How large is the colony at this point?
c. Compare the growth rate of the colony at the end of day 3 to the end of
day 8. Explain what is happening.
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Mid-Chapter Review
1. Determine the derivative of each function.
a. y 5 5e 23x
1
b. y 5 7e 7x
c. y 5 x 3e 22x
d. y 5 1x 2 12 2e x
e. y 5 1x 2 e 2x 2 2
f. y 5
e x 2 e 2x
e x 1 e 2x
2. A certain radioactive substance decays exponentially over time. The amount
of a sample of the substance that remains, P, after t years is given by
P1t2 5 100e 25t, where P is expressed as a percentage.
a. Determine the rate of change of the function, dP
.
dt
b. What is the rate of decay when 50% of the original sample has decayed?
3. Determine the equation of the tangent to the curve y 5 2 2 xe x at the point
where x 5 0.
4. Determine the first and second derivatives for each function.
a. y 5 23e x
b. y 5 xe 2x
5. Determine the derivative of each function.
a. y 5 82x15
b. y 5 3.21102 0.2x
c. f 1x2 5 x 22x
d. H1x2 5 30015 2 3x21
c. y 5 e x 14 2 x2
e. q1x2 5 1.9x 1 x 1.9
f. f 1x2 5 1x 2 2 2 2 3 4x
6. The population of rabbits in a forest at time t in months is
t
R1t2 5 500310 1 e 210 4 .
a. What is the initial number of rabbits in the forest?
b. Determine the rate of change of the number of rabbits at time t.
c. How fast is the number of rabbits changing after 1 year?
d. Determine the largest number of rabbits in the forest during the first 3 years.
e. Use graphing technology to graph R versus t and give physical reasons
why the population of rabbits might behave this way.
7. A drug is injected into the body in such a way that the concentration, C, in
the blood at time t hours is given by the function C1t2 5 101e 22t 2 e 23t 2. At
what time does the highest concentration occur within the first 5 h?
8. Given y 5 c1e2 kx, for what values of k will the function represent growth?
For what values of k will the function represent decay?
28
MID-CHAPTER REVIEW
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9. The rapid growth in the population of a type of insect is given by
P1t2 5 5000e0.02t, where t is the number of days.
a. What is the initial population 1t 5 0 2 ?
b. How many insects will there be after a week?
c. How many insects will there be after a month (30 days)?
10. You have probably noticed that the air pressure in a plane varies. The atmospheric pressure, y, varies with the altitude, x kilometers, above Earth. For altitudes up to 10 km, the pressure in millimeters of mercury (mm Hg) is given
by y 5 760e 20.125x. What is the atmospheric pressure
a. 5 km above Earth?
c. 9 km above Earth?
b. 7 km above Earth?
11. A radioactive substance decays in such a way that the amount left after t years
is given by A 5 100e 20.3t. The amount, A, is expressed as a percent. Find the
function, A¿ , that describes the rate of decay. What is the rate of decay when
50% of the substance is gone?
12. Given f 1x2 5 xe x, find all x-values for which f ¿1x2 7 0. What is the significance of this?
2
13. Find the equation of the tangent to the curve y 5 2 2x at the point on the
curve where x 5 0. Graph the curve and the tangent at that point.
14. $1000 is invested in a savings account that pays 6%/a, compounded annually.
a. Find the derivative A¿1t2 of the function.
b. At what rate is the amount growing at the end of 2 years? 5 years?
10 years?
c. Is the rate constant?
A¿ 1t 2
d. Determine the ratio of A1t 2 for each value that you determined for A¿1t2 .
e. What do you notice?
15. The function y 5 e x is its own derivative. But this function is not the only one
that has this property. Show that, for every value of c, y 5 c 1e2 x has the same
property.
CHAPTER 5
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Section 5.4—Derivatives of y 5 sin x and
y 5 cos x
In this section, we will investigate to determine the derivatives of y 5 sin x and
y 5 cos x.
INVESTIGATION 1 A. Using your graphing calculator, graph y 5 sin x where x is measured in
radians.
ZO
Use the following WINDOW settings:
Xmin 5 0, Xmax 5 9.4, Xscl 5 p 4 2
Ymin 5 23.1, Ymax 5 3.1, Yscl 5 1
Enter y 5 sin x into Y1 and graph the function.
dy
Tech
Support
dy
To calculate dx at a
point press
2nd
MODE
A
GRAPH
5
and
enter the desired x
coordinate of your
point, then press
enter
TRACE
,
dy
B. Use the CALC function (with dx selected) to compute y and dx, respectively, for
y 5 sin x, and record these values in a table such as the following (correct to
four decimal places):
DEL
x
sin x
d
( sin x)
dx
0
0.5
1.0
:
:
:
6.5
d
C. Create another column to the right of the dx
1sin x2 column with cos x as the
heading. Using your graphing calculator, graph y 5 cos x with the same
window settings as above.
Tech
Support
For help calculating
a value of a
function using a
graphing calculator
see Technical
Appendix X–XX
30
D. Compute the values of cos x for x 5 0, 0.5, 1.0, p , 6.5, correct to four
decimal places, and record them in the cos x column.
d
E. Compare the values in the dx
1sin x2 column with those in the cos x column and
write a concluding equation.
5 . 4 D E R I VAT I V E S O F y 5 sin x A N D y 5 cos x
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INVESTIGATION 2 1. Using your graphing calculator, graph y 5 cos x, where x is measured in
radians.
ZO
Use the following WINDOW settings:
Xmin 5 0, Xmax 5 9.4, Xscl 5 p 4 2
Ymin 5 23.1, Ymax 5 3.1, Yscl 5 1
Enter y 5 cos x into Y1 and graph the function.
2. Use the CALC function (with dy
selected) to compute y and dy
, respectively, for
dx
dx
y 5 cos x, and record these values, correct to four decimal places, in a table
such as the following:
x
cos x
d
(cos x)
dx
0
0.5
1.0
:
:
:
6.5
d
3. Create another column to the right of the dx
1cos x2 column with 2sin x as the
heading. Using your graphing calculator, graph y 5 2sin x with the same
window settings as above.
4. Compute the values of 2sin x for x 5 0, 0.5, 1.0, p , 6.5, correct to four
decimal places, and record them in the 2sin x column.
d
5. Compare the values in the dx
1cos x2 column with those in the 2sin x column
and write a concluding equation.
The investigations lead to the following conclusions:
The Derivatives of Sinusoidal Functions
d
1sin x2 5 cos x
dx
EXAMPLE 1
d
1cos x2 5 2sin x
dx
Selecting a strategy to determine the derivative of a sinusoidal function
dy
Determine dx for each function.
a. y 5 cos 3x
b. y 5 x sin x
CHAPTER 5
31
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Solution
a. To differentiate this function, use the Chain Rule.
y 5 cos 3x
d1cos 3x2 d13x2
dy
#
5
dx
d13x2
dx
(Chain rule)
5 2sin 3x # 13 2
5 23 sin 3x
b. To find the derivative in this case, use the Product Rule.
y 5 x sin x
d1sin x2
dy
dx #
5
sin x 1 x
dx
dx
dx
(Product rule)
5 11 2 # sin x 1 x cos x
5 sin x 1 x cos x
EXAMPLE 2
Reasoning about the derivatives of sinusoidal functions
dy
Determine dx for each function.
a. y 5 sin x 2
b. y 5 sin2x
Solution
a. To differentiate this composite function, use the Chain Rule and change of
variable.
Here the inner function is u 5 x 2, and the outer function is y 5 sin u.
dy du
dy
(Chain rule)
Then
5
dx
du dx
5 1cos u 2 12x2
5 2 x cos x
(Substitute)
2
b. Since y 5 sin2x 5 1sin x2 2, we use the Chain Rule with y 5 u 2, where
u 5 sin x.
dy
dy du
(Chain rule)
Then
5
dx
du dx
5 12u 2 1cos x2
5 2 sin x cos x
32
5 . 4 D E R I VAT I V E S O F y 5 sin x A N D y 5 cos x
(Substitute)
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With practice, you will learn to apply the Chain rule without the intermediate step
of introducing the variable u. For y 5 sin x 2, for example, you can skip this
step and immediately write dy
5 1cos x 2 2 12x2.
dx
Derivatives of Composite Sinusoidal Functions
dy
If y 5 sin f 1x2, then dx 5 cos f 1x2 # f ¿1x2 . In Leibniz notation,
d1sin f 1x22 d 1f 1x22
d
1sin f 1x22 5
dx
d 1f 1x22
dx
dy
If y 5 cos f 1x2, then dx 5 2sin f 1x2 # f ¿1x2. In Leibniz notation,
d1cos f 1x22 d 1f 1x22
d
1cos f 1x22 5
dx
d 1f 1x22
dx
EXAMPLE 3
Differentiating a composite cosine function
Determine dx for y 5 cos11 1 x 3 2.
dy
Solution
y 5 cos11 1 x 3 2
d3cos11 1 x 3 2 4 d11 1 x 3 2
dy
#
5
dx
dx
d11 1 x 3 2
(Chain rule)
5 2sin11 1 x 3 2 13x 2 2
5 23x 2sin11 1 x 3 2
EXAMPLE 4
Differentiating a combination of functions
Determine y¿ for y 5 e sin x1cos x.
Solution
y 5 e sin x1cos x
y¿ 5
d1e sin x1cos x 2
d1sin x 1 cos x2
# d1sin x 1 cos x2
5 e sin x1cos x 1cos x 2 sin x2
(Chain rule)
dx
CHAPTER 5
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Connecting the derivative of a sinusoidal function to the slope
of a tangent
p
Determine the equation of the tangent to the graph of y 5 x cos 2x at x 5 2 .
Solution
When x 5 p
,y5p
cos p 5 2p
.
2
2
2
p
The point of tangency is Qp
, 2 2 R.
2
The slope of the tangent at any point on the graph is given by
d1cos 2x2
dy
dx #
5
cos 2x 1 x #
dx
dx
dx
5 11 2 1cos 2x2 1 x12sin 2x2 12 2
5 cos 2x 2 2x sin 2x
p dy
At x 5 ,
5 cos p 2 p1sin p2
2 dx
(Product and Chain Rules)
(Simplify)
(Evaluate)
5 21
The equation of the tangent is
y1
EXAMPLE 6
p
p
5 2 a x 2 b or y 5 2x.
2
2
Connecting the derivative of a sinusoidal function to its extreme values
Determine the maximum and minimum values of the function f 1x2 5 cos2x on
the interval x [ 30, 2p4
Solution
By the algorithm for finding extreme values, the maximum and minimum values
occur at points on the graph where f ¿1x2 5 0, or at an end point of the interval.
The derivative of f 1x2 is
f ¿1x2 5 21cos x2 12sin x2
(Chain rule)
5 22 sin x cos x
5 2sin 2x
Solving f ¿1x2 5 0
2sin 2x 5 0
sin 2x 5 0
2x 5 0, p, 2p, 3p, or 4p
p
3p
and x 5 0, , p,
, or 2p.
2
2
34
5 . 4 D E R I VAT I V E S O F y 5 sin x A N D y 5 cos x
(Using the double angle identity)
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We evaluate f 1x2 at the critical numbers. (In this case the endpoints of the interval
are included.)
x
0
p
2
p
3p
2
2p
f 1x2 5 cos2x
1
0
1
0
1
The maximum value is 1 when x 5 0, p, 2p and the minimum value is 0 when
or 3p
x5p
.
2
2
The above solution is verified by our knowledge of the cosine function. For the
function y 5 cos x
• Domain x [ R
• Range 21 # cos x # 1
For the given function y 5 cos2x
• Domain x [ R
• Range 0 # cos2x # 1
Therefore, the maximum value is 1 and the minimum value is 0.
2
y
y = cos2 x
1
p
2
0
–1
x
p 3p 2p
2
IN SUMMARY
Key Ideas
• The derivatives of sinusoidal functions are found as follows:
•
•
•
d sin x
d cos x
5 cos x and
5 2sin x
dx
dx
dy
5 cos f 1x2 # f ¿1x2.
If y 5 sin f 1x2, then
dx
dy
5 2sin f 1x2 # f ¿1x2.
If y 5 cos f 1x2, then
dx
Need to Know
• When you are differentiating a function that involves sinusoidal functions, use
the above rules, along with the Sum, Difference, Product, Quotient, and
Chain rules as required.
CHAPTER 5
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Exercise 5.4
K
PART A
dy
1. Determine dx for each of the following.
f. y 5 2x 1 2 sin x 2 2 cos x
a. y 5 sin 2x
g. y 5 sin1e x 2
b. y 5 2 cos 3x
c. y 5 sin1x 3 2 2x 1 4 2
h. y 5 3 sin13x 1 2p2
d. y 5 2 cos124x2
i. y 5 x 2 1 cos x 1 sin
e. y 5 sin 3x 2 cos 4x
j. y 5 sin
p
4
1
x
2. Differentiate the following functions.
sin x
1 1 cos x
a. y 5 2 sin x cos x
d. y 5
cos 2x
x
c. y 5 cos1sin 2x2
e. y 5 ex 1cos x 1 sin x2
b. y 5
f. y 5 2x 3sin x 2 3x cos x
PART B
3. Determine an equation for the tangent at the point with the given x-coordinate
for each of the following functions.
a. f 1x2 5 sin x, x 5
C
d. f 1x2 5 sin 2x 1 cos x, x 5
p
3
b. f 1x2 5 x 1 sin x, x 5 0
p
2
p
p
e. f 1x2 5 cos a 2x 1 b , x 5
4
3
c. f 1x2 5 cos14x2 , x 5
f. f 1x2 5 2 sin x cos x, x 5
p
4
4. a. If f 1x2 5 sin2x and g 1x2 5 1 2 cos2x, explain why f ¿1x 2 5 g¿1x 2 .
b. If f 1x2 5 sin2x and g 1x2 5 1 1 cos2x, how are f ¿1x2 and g¿1x2 related?
5. Differentiate each function.
a. v1t2 5 sin2 1 Ït2
c. h1x2 5 sin x sin 2x sin 3x
b. v1t2 5 Ï1 1 cos t 1 sin t
36
p
2
5 . 4 D E R I VAT I V E S O F y 5 sin x A N D y 5 cos y
2
d. m1x2 5 1x 2 1 cos2x2 3
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6. Determine the absolute extreme values of each function on the given interval.
(Verify your results with graphing technology.)
a. y 5 cos x 1 sin x, 0 # x # 2p
b. y 5 x 1 2 cos x, 2p # x # p
c. y 5 sin x 2 cos x, x [ 3 0, 2p4
d. y 5 3 sin x 1 4 cos x, x [ 30, 2p 4
A
7. A particle moves along a line so that, at time t, its position is s1t2 5 8 sin 2t.
a. For what values of t does the particle change direction?
b. What is the particle’s maximum velocity?
8. a. Graph the function f 1x2 5 cos x 1 sin x.
b. Find the coordinates of the point where the tangent to the curve of f 1x2 is
horizontal, where 0 # x # p.
9. Determine expressions for the derivatives of csc x and sec x.
x 1
10. Determine the slope of the tangent to the curve y 5 cos 2x at point a , b .
6 2
11. A particle moves along a line so that at time t its position is s 5 4 sin 4t.
a. When does the particle change direction?
b. What is the particle’s maximum velocity?
c. What is the particle’s minimum distance from the origin? maximum
distance?
T
12. An irrigation channel is constructed by bending a sheet of metal that is 3 m
wide, as shown in the diagram. What angle u will maximize the cross-sectional
area (and thus the capacity) of the channel?
1m
u
1m
u
1m
13. An isosceles triangle is inscribed in a circle of radius R. Find the value of u
that maximizes the area of the triangle.
2u
PART C
14. If y 5 A cos kt 1 B sin kt, where A, B, and k are constants, show that
y– 1 k 2y 5 0.
CHAPTER 5
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Section 5.5—Derivative of y 5 tan x
In this section, we will study the derivative of the remaining primary trigonometric
function—tangent.
Since this function can be expressed in terms of sine and cosine, we can find its
derivative using the product rule.
EXAMPLE 1
Reasoning about the derivative of the tangent function
Determine dy
for y 5 tan x.
dx
Solution
y 5 tan x
sin x
cos x
5 1sin x2 1cos x2 21
5
d1cos x2 21
dy
d sin x #
1cos x2 21 1 sin x
5
dx
dx
dx
(Product rule)
5 1cos x2 1cos x2 21 1 sin x1212 1cos x2 22 12sin x2
(Chain rule)
2
511
sin x
cos2 x
5 1 1 tan2 x
(Using the Pythagorean Identity)
2
5 sec x
Therefore,
EXAMPLE 2
d1tan x2
dx
5 sec2 x.
Selecting a strategy to determine the derivative of a composite
tangent function
dy
Determine dx for y 5 tan1x 2 1 3x2 .
Solution
y 5 tan1x 2 1 3x2
d tan1x 2 1 3x2 d1x 2 1 3x2
dy
#
5
dx
dx
d1x 2 1 3x2
5 sec2 1x 2 1 3x2 # 12x 1 32
38
5 . 5 D E R I VAT I V E O F y 5 tan x
5 12x 1 32sec2 12x2 1 3x2
(Chain rule)
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Derivatives of Composite Functions Involving y 5 tan x
If y 5 tan f 1x2, then
dy
5 sec2f 1x2 # f ¿1x2. In Leibniz notation,
dx
d1tan f 1x22 df1x22
d
1tan f 1x22 5
dx
d1f 1x22
EXAMPLE 3
dx
Determining the derivative of a combination of functions
Determine dx for y 5 1sin x 1 tan x2 4.
dy
Solution
y 5 1sin x 1 tan x2 4
dy
5 41sin x 1 tan x2 3 1cos x 1 sec2x2
dx
EXAMPLE 4
(Chain rule)
Determining the derivative of a product involving the tangent function
Determine dx for y 5 x tan12x 2 1 2 .
dy
Solution
y 5 x tan12x 2 12
d12x 2 1 2
dy
5 11 2tan12x 2 1 2 1 1x2sec2 12x 2 1 2
dx
dx
(Product rule)
5 tan12x 2 1 2 1 2x sec2 12x 2 1 2
IN SUMMARY
Key Ideas
• The derivatives of functions involving the tangent function are found as follows:
d1tan x2
•
•
dx
5 sec2x
d
1tan f 1x2 2 5 sec2 f 1x2 # f ¿1x2
dx
Need to Know
• Trigonometric identities can be used to write one expression as an equivalent
expression and then used to differentiate. In some cases, the new function
will be easier to work with.
CHAPTER 5
39
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Page 40
Exercise 5.5
K
PART A
1. Determine dy
for each of the following.
dx
d. y 5
b. y 5 2 tan x 2 tan 2x
e. y 5 tan1x 2 2 2tan2 x
c. y 5 tan2 1x 3 2
A
x2
tanpx
a. y 5 tan 3x
f. y 5 3 sin 5x tan 5x
2. Determine an equation for the tangent to each function at the point with the
given x-coordinate.
a. f 1x2 5 tan x, x 5
p
4
b. f 1x2 5 6 tan x 2 tan 2x, x 5 0
PART B
3. Determine y¿ for each of the following.
a. y 5 tan1sin x2
b. y 5 3tan1x 2 2 12 4 22
c. y 5 tan2 1cos x2
d. y 5 1tan x 1 cos x2 2
e. y 5 sin3x tan x
f. y 5 e tanVx
d 2y
4. Determine dx2 for each of the following.
a. y 5 sin x tan x
b. y 5 tan2 x
5. Determine all the values of x, 0 # x # 2p, for which the slope of the tangent
to f 1x2 5 sin x tan x is zero.
6. Determine the local maximum point on the curve y 5 2x 2 tan x in the
interval 2p
6 x 6 p
.
2
2
T
7. Prove that y 5 sec x 1 tan x is always increasing in the interval
2p
6 x 6 p
.
2
2
8. Determine the equation of the line tangent to y 5 2 tan x, where x 5 p
.
4
PART C
9. Determine the derivative of cot x.
10. Determine f –1x2 where f 1x2 5 cot 4x
40
5 . 5 D E R I VAT I V E O F y 5 tan x
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Key Concepts Review
In this chapter, we introduced a new base for exponential functions, namely
the number e, where e 8 2.718281. We studied the derivatives of the exponential
functions along with the primary trigonometric functions. You should be able to
apply all the rules of differentiation that you learned in Chapter 2 to expressions
that involve the exponential, sine, cosine, and tangent functions combined with
polynomial and rational functions.
We have also considered some applications of exponential and trigonometric
functions. The calculus techniques used to determine instantaneous rate of
change, equations of tangent lines, and absolute extrema used earlier on
polynomial and rational functions, also apply to exponential and trigonometric
functions.
Derivative Rules for Exponential Functions
d x
d
1e 2 5 e x and 1e g1x2 2 5 e g1x2 # g¿1x2
dx
dx
d x
d
•
1b 2 5 b x ln b and 1b g1x2 2 5 b g1x2 1ln b 2 g¿1x2
dx
dx
•
Derivative Rules for Primary Trigonometric Functions
d
d
1sin x2 5 cos x and 1sin f 1x2 2 5 cos f 1x2 # f ¿1x2
dx
dx
d
d
•
1cos x2 5 2sin x and 1cos f 1x2 2 5 2sin f 1x2 # f ¿1x2
dx
dx
d
d
•
1tan x2 5 sec2 x and 1tan f 1x2 2 5 sec2 f 1x2 # f ¿1x2
dx
dx
•
CHAPTER 5
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CAREER LINK WRAP-UP Investigate and Apply
CHAPTER 5: RATE-OF-CHANGE MODELS IN MICROBIOLOGY
A simplified model for bacterial growth is P1t2 5 P0 e rt, where P(t) is the population
of the bacteria colony after t hours, P0 is the initial population of the colony (the
population at t 5 0), and r determines the growth rate of the colony. This model is
simple in that it does not account for limited resources, such as space and
nutrients. In this model, as time increases, so does the population, but there is no
bound on the population. While a model like this can describe the population for a
short period of time or can be made to describe the population for a longer period
of time by adjusting conditions in a laboratory experiment, in general, populations
are better described by more complex models.
To determine how the population of a particular type of bacteria will grow over
time under controlled conditions, a microbiologist observes the initial population
and the population every half hour for 8 hours. (The microbiologist also controls
the environment in which the colony is growing to make sure that temperature
and light conditions remain constant and ensures that the amount of nutrients
available to the colony as it grows is sufficient for the increasing population.)
After analyzing the population data, the microbiologist determines that the
population of the bacteria colony can be modelled by the equation:
P1t2 5 500 e0.1t
a. What is the initial population of the bacteria colony?
b. What function describes the instantaneous rate of change in the bacteria
population after t hours?
c. What is the instantaneous rate of change of the population after 1 h? What is
the instantaneous rate of change after 8 h?
d. How do your answers to c. help you in making a prediction about how long it
will take for the bacteria colony to double in size? Make a prediction for the
number of hours it will take the population to double, using the answers to c.
and/or other information.
e. Determine the actual doubling time, the time that it takes the colony to grow to
twice its initial population. (Hint: solve for t when P1t2 5 1000.)
f. Compare your prediction for the doubling time to the calculated value. If your
prediction was not close to the actual value, what factors do you think might
account for the difference?
g. When is the instantaneous rate of change equal to 500 bacteria/h?
42
CAREER LINK WRAP-UP
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Review Exercise
1. Differentiate each of the following.
a. y 5 6 2 e x
d. y 5 e 23x
b. y 5 2x 1 3e x
e. y 5 xe x
c. y 5 e 2x13
f. s 5
2
15x
et 2 1
et 1 1
2. Determine dy
for each of the following.
dx
a. y 5 10x
b. y 5 43x
2
c. y 5 15x2 15x 2
e. y 5
4x
4x
d. y 5 1x 4 2 2x
f. y 5
5Vx
x
3. Differentiate each of the following.
a. y 5 3 sin 2x 2 4 cos 2x
d. y 5 x tan 2x
e. y 5 1sin 2x2 e 3x
b. y 5 tan 3x
c. y 5
1
2 2 cos x
f. y 5 cos2 2x
4. a. Given the function f 1x2 5 e , solve the equation f ¿ 1x2 5 0.
x
b. Discuss the significance of the solution found in 4a.
x
5. a. If f 1x2 5 xe 22x, find f ¿Q12R.
b. Explain what this number represents.
6. Determine the second derivative for each of the following.
a. y 5 xe x 2 e x
b. y 5 xe 10x
2x
1
7. If y 5 ee 2x 2
, prove that dy
5 1 2 y 2.
11
dx
8. Find the equation of the tangent to the curve defined by y 5 x 2 e 2x that is
parallel to the line represented by 3x 2 y 2 9 5 0.
9. Determine the equation of the tangent to the curve y 5 x sin x at the point
where x 5 p
.
2
t
10. An object moves along a line so that, at time t, its position is s 5 3 1sincos
,
2t
where s is the displacement in metres. Calculate the object’s velocity at t 5 p
.
4
CHAPTER 5
43
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Page 44
11. The number of bacteria in a culture, N, at time t is given by
t
N1t2 5 2000330 1 te 220 4.
a. When is the rate of change of the number of bacteria equal to zero?
b. If the bacterial culture is placed into a colony of mice, the number of mice
that become infected, M, is related to the number of bacteria present by the
3
equation M1t2 5 Ï
N 1 1000. After 10 days, how many mice are infected
per day?
12. The concentrations of two medicines in the blood stream t hours after
injection are c1 1t2 5 te 2t and c2 1t2 5 t 2e 2t.
a. Which medicine has the larger maximum concentration?
b. Within the first half-hour, which medicine has the larger maximum
concentration?
13. Differentiate.
a. y 5 12 1 3e2x 2 3
x
c. y 5 ee
d. y 5 11 2 e5x 2 5
b. y 5 xe
14. Differentiate.
c. y 5 152 2 2x
e. y 5 41e2 x
a. y 5 sin 2x
c. y 5 sin a
e. y 5 cos2x
b. y 5 x 2 sin x
d. y 5 cos x sin x
a. y 5 5x
b. y 5 10.472 x
d. y 5 512 2 x
15. Determine y¿ .
f. y 5 22110 2 3x
p
2 xb
2
f. y 5 cos x sin2x
16. Determine the equation of the tangent to the curve y 5 cos x at Qp
, 0R.
2
17. An object is suspended from the end of a spring. Its displacement from the
equilibrium position is s 5 8 sin110pt2 . Calculate the velocity and acceleration
2
of the object at any time t, and show that ddt 2s 1 100p2s 5 0.
18. The motion of a particle is given by s 5 5 cos12t 1 p
2 . What are the maximum
4
values of the displacement, the velocity, and the acceleration?
19. The hypotenuse of a right-angled triangle is 12 cm in length. Calculate the
measures of the remaining angles in the triangle that maximize its perimeter.
12
y
u
x
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20. A fence is 1.5 m high and is 1 m from a wall. A ladder must start from the
ground, touch the top of the fense, and rest somewhere on the wall. Calculate
the minimum length of such a ladder.
21. A thin rigid pole is to be carried horizontally around a corner joining two
corridors of width 1 m and 0.8 m, respectively. Calculate the length of the
longest pole that can be transported in this manner.
y
0.8
u
x
u
1
22. When the rules of hockey were developed. Canada didn’t use the metric
system. Thus the distance between the goal posts was designated to be six feet
(slightly less than two metres). If Paul Kariya is on the goal line, three feet
outside one of the goal posts, how far should he go out (perpendicular to the
goal line) in order to maximize the angle in which he can shoot at the goal?
Hint: Determine the values of x that maximize u.
6
3
x
u
CHAPTER 5
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Chapter 5 Test
1. Determine the derivative dy
for each of the following.
dx
a. y 5 e 22x
2
x 2 13x
b. y 5 3
c. y 5
e 3x 1 e 23x
2
d. y 5 2 sin x 2 3 cos 5x
e. y 5 sin3 1x 2 2
f. y 5 tanV1 2 x
2. Determine the equation of the tangent to the curve defined by y 5 2e 3x that is
parallel to the line defined by 26x 1 y 5 2.
3. Determine the equation of the tangent to y 5 e x 1 sin x at 10, 1 2 .
4. The velocity of a particular particle that moves in a straight line under the
influence of forces is given by v1t2 5 10e 2kt, where k is a positive constant
and v1t2 is in cm> s.
a. Show that the acceleration of this particle is proportional to (a constant
multiple of) its velocity. Explain what is happening to this particle.
b. What is the initial velocity of the particle?
c. At what time is the velocity equal to half of the initial velocity? What is
the acceleration at this time?
5. Determine f –1x2 .
a. f 1x2 5 cos2 x
b. f 1x2 5 cos x cot x
6. Determine the absolute extreme values of f 1x2 5 sin2 x where x [ 30, p 4
7. Calculate the slope of the tangent line that passes through y 5 5x where
x 5 2. Express your answer to two decimal places.
8. Determine all the maximum and minimum values of y 5 xe x 1 3e x.
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CHAPTER 5 TEST