Math 1220: Calculus 1
Last Homework
Due Monday, April 27
Course Summary
1. Basic Theory of Calculus
• know the meaning of the slope of a curve and understand the
difficulty in precisely defining the tangent line to a curve
• be able to sketch slope and area graphs for a given curve
• how to recover a sketch of f from given sketches or sign data for
f 0 (and f 00 )
• the relationship between slope and area: what the Fundamental
Theorem of Calculus is, and why it is true.
2. Computing Derivatives and Antiderivatives
• know the derivatives of basic functions such as: powers of x;
trigonometric functions; exponential functions; logarithmic functions; inverse trigonometric functions
• be able to apply the basic rules to compute derivatives of sums,
products and compositions of the basic functions
• be able to ‘reason backwards’ and use your knowledge of derivatives to find antiderivatives
3. Evaluating Definite Integrals
• know how to use your power to find antiderivatives to evaluate
definite integrals
• use the geometric meaning of definite integrals to evaluate or estimate their values
4. Linear Approximation
• be able to apply the linear approximation technique; this includes
being able to make a good choice of reference point
• be able to give a rock-solid, clearly justified upper bound for the
error of your approximation
5. Using Calculus to Solve Actual Problems
(a) Related Rates Problems
• be able to take a problem described in English and convert
it to a problem of calculus; this includes providing enough
explanation that the conversion can be understood and verified by an independent reader; it also includes being able to
take the situation described in the problem and extract one
or more equations from it
• be able to solve the resulting calculus problem
(b) Optimization Problems
• be able to take a problem described in English and convert
it to a problem of calculus; this includes providing enough
explanation that the conversion can be understood and verified by an independent reader; it also includes being able to
take the situation described in the problem and extract one
or more equations from it
• be able to solve the resulting calculus problem
(c) Integration
• integration allows you to solve difficult problems by cutting
them into tiny easily solved pieces, and then adding the results; be able to apply this technique to a convert (with explanation) difficult problems into integral problems.
• be able to solve the resulting calculus problem (if the finding
the antiderivative is not beyond what we’ve learned)
Plan
1. Each day, come to class with two problems (they don’t need to be from
the same section) already worked on.1 I’ll mark down that you did this
(and which problems you worked on) as long as there is evidence2 that
you actually spent time thinking about the problems; you’ll get full
homework credit for having done this.
2. At the end of class, I’ll have you sign out showing that you worked on
problems in class. You’ll get homework credit for that, too.
3. On Monday, April 27, the first day of Finals Week, you can turn in as
many finished solutions as you like. If you want copies for studying,
then make copies for studying; you won’t get the ones you turn in to me
back in time. Finished solutions will replace earlier homework grades
with 4s if they are perfect. Perfect means:
•
•
•
•
One problem per piece of lined paper.
The problem is neatly written on the top of paper.
The solution is clearly labeled and neatly written.
Your solution only attempts one substantial step (or a few very
simple steps) at a time, and there is a blank line between each
step and the next.
• Where appropriate, you have provided justification for your steps.
(i.e., don’t just write down ‘ xy = z2 ’, say instead ‘because the big
triangle and the little triangle are similar, xy = z2 ’.)
• Any diagrams you use are clearly labeled, neatly drawn and their
connection to the problem is clearly explained.
• Your solution is correct and easy to understand.
At any time in the next two weeks, I’ll be happy to look at your work
problem and tell you if it is acceptable, and what you’d have to do to
make it acceptable. I’m happy to give hints and suggestions on finding
your solutions, too.
1
This work does not need to be ‘perfect’ in the sense I’ve described in part (3).
Substantial progress on a solution is one kind of evidence; a well thought out question
clearly explaining what is keeping you from making progress on the problem is another
kind of evidence.
2
1
1.1
Basic Theory of Calculus
Theory
Problem 1
(a) Explain carefully but efficiently what the Fundamental Theorem
of Calculus says.
(b) Explain how you can use the Fundamental Theorem of Calculus
to evaluate a definite integral.
Problem 2
(a) Briefly describe some of the problems that arise when you attempt
to give unambiguous instructions on how to draw a tangent line
to a curve.
(b) How did we end up defining tangent lines?
(c) Explain how to get from our definition of tangent line to the limit
formula
f (x) − f (t)
.
f 0 (x) = lim
t→x
x−t
1.2
Practice
Problem 3
(a) Here is the graph of f .
Sketch a slope graph and an area graph for f . Make sure to mark
all the important points on your graphs.
(b) Here are the graphs of g 0 and g 00
First, decide which graph is which. Then use the graphs to sketch
the graph of g . (You can do this by extracting the sign data from
the graphs. If you are more ambitious, you can incorporate even
more detail.) Make sure to mark all the important points on your
graph.
Problem 4 Let f (x) = x5 − 5x3 + x.
(a) Use your calculus superpowers to sketch the graph f for all real
numbers.
(b) Find the maximum and minimum values of f (x) for x between
− 21 and 1.
2
Computing Derivatives and Antiderivatives
Problem 5
2
(a) Find the derivative of h(x) = ex cos( x12 ).
(b) Find the derivative of i(x) = sin(cos(sin2 (cos3 (x4 )))).
Problem 6 Let n be a positive integer.
(a) Find a formula (probably involving n) for the derivative of the
function k(x) = sinn (x) (which means (sin(x))n ).
Rπ
(b) Find a formula (probably involving n) for 02 sinn (x) cos(x) dx.
3
Evaluating Definite Integrals
Problem 7
R1 √
4 − x2 dx.
R2 1
(b) Explain why 1 x4 +1 dx can be no greater than 2. Can you find a
lower bound for this integral?
(c) Consider the two integrals
Z 2b Z b
x
f (x) dx
and
f
dx.
2
2a
a
Draw a representative picture of regions whose area they might
compute. By analyzing these regions, determine how these integrals are related: that is, suppose I tell you the first integral is,
say, 49; then what can you deduce about the second integral?
(a) Evaluate
−1
Problem 8
(a) Evaluate
Rπ
(b) Evaluate
R1
0
sin(x) dx.
1
0 x2 +1
dx.
4
Linear Approximation
Problem 9
(a) Use a linear approximation to estimate
√
4
2.
(b) Use your answer to (a) as the reference point for a second linear
√
approximation and get, hopefully, an even better estimate of 4 2.
(c) Find an upper bound for the error of your approximation in (b).
Problem 10
(a) Find the linear function ` whose graph is the tangent line to the
graph of f (x) = sin(x) at reference point a = π6 .
√
√
(b) You know that sin( π4 ) = 22 , so `( π4 ) ≈ 22 . Use this approximation
to derive an estimate for the number π.
(c) Can you find an upper bound on the error of your approximation?
5
Using Calculus to Solve Actual Problems
5.1
Related Rates Problems
Problem 11 You are a calculus instructor, trying to think of an interesting
related rates problem for your students. You start by imagining a light rising
up a pole while a person walks away, then you start assigning numbers: at
the special time in question
• the person (who is 5 feet tall) is 10 feet horizontally from the pole and
walking at 3 feet per second
• the light is 10 feet off the ground and moving up at ...?
“This is boring,” you think, “It’s the same old problem!” But then you have
an idea: if you carefully choose the motion of the light, then the length
shadow of the person won’t change at all! Cackling diabolically, you set
about determining the required velocity for the light. What is it?
Problem 12 A big conical tank, 12 feet tall with a top radius of 4 feet is full
to the top of maple syrup to dump onto the world’s biggest pancake, which
is being maneuvered into place by a gang of four flatbead tractor trailors.
At the chosen moment, after a suspenseful countdown of course, you pull
the chain that opens the nozzle at the bottom and the syrup begins to flow.
Some time later, when the level of the tank is down to 8 feet, you observe
that the tank is emptying at a rate of about 1 cubic foot of syrup per minute.
How quickly is the level of the tank decreasing at that time?
5.2
Optimization Problems
Problem 13 You want to fit a rectangular solid box inside a circular cone
with height 10 feet and base radius 3 feet. It will sit on the base and the
upper corners will all touch the diagonal surface of the cone. What are the
dimensions of the box with the largest possible volume?
Problem 14 You are in the business of designing posters. Typically your
customer says something like:
“I need a printed area of A with right side margin r, left side
margin l, top side margin t and bottom side margin b, and I need
it to be as cheap as possible.”
Every time, you dust off your calculus skills and figure out the dimensions of
the poster with the smallest total area, and your customers are always pleased
with the results. Finally, though, you get tired of solving these problems and
decide to come up with a formula that will give the perfect dimensions every
time, without needing to do the same old problems over and over. Find the
required formula!
5.3
Integration
Problem 15 You bought a strange vegetable at the market and commenced
slicing it thinly as soon as you got home. The slices are pentagons and the
length of the side of the pentagonal slice x inches from the left end of the
vegetable is given by the expression (x2 +1)(x3 −16x) (it was four inches long
before you started cutting it). What was the total volume of the vegetable?
Problem 16 An airplane is flying through the air leaving behind a curved
1 5
t + cos(t). If
trail of smoke. At time t its speed is v(t) = 1 + t − 13 t3 + 120
the plane starts flying at t = 0 and ends at time t = 8π, then how long is
the smoke trail at the end?