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MATH 130
HW Problems Key
Exam 2
Homework Key for Section 1
1. Find the first and second derivative of the following functions:
(a)
f 0 (x) = 8x − 2 f 00 (x) = 8
(b)
f 0 (x) = 6x2 − 6x f 00 (x) = 12x − 6
(c)
f 0 (t) = 4t3 − 6t2 + 12t − 3 f 00 (t) = 12t2 − 12t + 12
(d)
f 0 (x) = 20x(2x2 + 3)4
f 00 (x) = 320x2 (2x2 + 3)3 + 20(2x2 + 3)4
(e)
f 0 (x) =
−x2 − 4
(x2 − 4)2
f 00 (x) =
2x3 + 24x
(x2 − 4)3
2. Find the third derivative of the following functions:
(a)
f 000 (x) = 72x + 24
(b)
f 000 (x) = −
6
x4
f 000 (x) = −
48
x5
(c)
3. An object is dropped from a height of 256 feet. The distance it falls in t seconds is given by
s(t) = 16t2 . Find BOTH the objects velocity AND acceleration when it strikes the ground.
16t2 = 256 when t = 4 , v(t) = 32t . a(t) = 32 =⇒ v(4) = 128 and a(4) = 32
Homework Key for Section 2
1. For the following functions find dy/dx by implicit differentiation:
(a)
dy
x
=
dx
2y
(b)
x−y
dy
=
dx
x
(c)
dy
y
=−
dx
x
2. Find an EQUATION of the tangent line to 4x2 + 9y 2 = 36 at the point (0, 2)
y=2
3. A spectator watches a rowing race from the edge of the river. The lead boat is moving in a
straight line that is 120 feet from the river bank. If the boat is moving at a constant speed
of 24 ft/sec, how fast is the boat moving away from the spectator when it is 50 feet past her?
9.23 f t/sec
4. The base of a 13 foot ladder leaning against a wall begins to slide away from the wall. At
the instant of time when the base is 12 feet from the wall, the base is moving at a rate of
8 ft/sec. How fast is the top of the ladder sliding down the wall at that time?
19.2 f t/sec
Homework Key for Section 3
1. Determine where the following functions are increasing and decreasing:
(a) increasing on (−∞, ∞)
(b) decreasing on (−∞, 23 ) and increasing on ( 32 , ∞)
√
(c) decreasing on (−∞, −
3
)
3
∪ (
√
3
, ∞)
3
and increasing on (−
√
√
3
3
,
)
3
3
(d) increasing on (−∞, −2) ∪ (0, ∞) and decreasing on (−2, 0)
2. Find the relative max and relative min for the following:
(a) min at x = 2
(b) max at x = 3
(c) none
Homework Key for Section 4
1. Determine where the following functions are concave up and concave down:
(a) CU on (−∞, ∞)
(b) CD on (−∞, 0) and CU on (0, ∞)
(c) CD on (−∞, 4)
(d) CD on (−∞, 2) and CU on (2, ∞)
2. Find the inflection points, if any:
(a) IP at x = 0
(b) IP at x = 1
3. Use the SECOND derivative test to find the relative extrema of f (x) = 2x3 + 1
no extrema
4. Sketch a graph with the given properties
answers vary
Homework Key for Section 5
1. Find the absolute mins and absolute maxs (if any) of the following functions:
(a)
absolute min is −
41
8
(b)
no absolute extrema
(c)
absolute max is 1
(d)
absolute max is 5 absolute min is − 4
(e)
absolute max is 10 absolute min is 1
2. An open box is made from a tin sheet that is 8 inches square by cutting out identical
squares from each corner and bending up the resulting flaps. Determine the dimensions
of the largest box that can be made.
16
4
16
in by
in by in
3
3
3
3. A rectangular box is to have a square base and a volume of 20 f t3 . If the material for the
base costs 30 cents per square foot, the sides cost 10 cents per square foot and the top
costs 20 cents per square foot, find the dimensions of the box that can be constructed at
minimum cost.
2 by 2 by 5
4. An aluminum can is to be constructed so as to have a volume of 36 in3 . Find the dimensions that uses the least amount of material.
r
36
3 18
h =
r =
2/3
π
π 18
π
Homework Key for Section 6
1. Evaluate the following:
(a) 16
(b) 27
(c) 3
(d) −3
(e) 25
2. Simplify the expression:
(a) 4x3
√
(b) 5xy 2 x
(c)
2
a2
3. Solve the following for x:
(a) x = 2
(b) x = 3
(c) x =
5
4
4. Express the following equations in log form:
(a) log2 64 = 6
(b) log3
1
9
= −2
5. Given that log 3 = 0.4771 and log 4 = 0.6021 find the value of the following:
(a) 1.2042
(b) 1.0792
6. Write the given expressions as one log, that is combine them as much as you can.
(a) ln a2 b3
(b) ln
√
3 xy
√
3z
7. Use the log laws to simplify the following:
(a) log x + 4 log (x + 1)
(b)
1
2
log (x + 1) − log (x2 + 1)
(c) ln x − x2
8. Solve the following:
(a) 5.1986
(b) −0.0912
(c) −8.0472
Homework Key for Section 7
1. Differentiate:
(a) f 0 (x) = 3e3x
(b) f 0 (x) = −e−x
(c) f 0 (x) = ex + 1
(d) f 0 (x) = x3 ex + 3x2 ex
(e) f 0 (x) = e
√
x 1 −1/2
x
2
3x+2
(f) f 0 (x) = (x − 1)e
(g) f 0 (x) =
(3) + e3x+2
2ex
(ex +1)2
2. Find an EQUATION of the tangent line to f (x) = e2x−3 at ( 23 , 1).
y = 2x − 2
2
3. Find the absolute extrema of f (x) = e−x on [−1, 1].
abs max at 1 and abs min at e−1
4. Find the second derivative of f (x) = e−4x + 2e3x
2(8e−4x + 9e3x )
5. Find f 0 (x) for the following functions:
(a)
f 0 (x) =
5
x
(b)
1
x+1
f 0 (x) =
(c)
f 0 (x) =
8
x
f 0 (x) =
1
2x
f 0 (x) =
−2
x
(d)
(e)
(f)
f 0 (x) =
8x − 6
− 6x + 3
4x2
(g)
f 0 (x) =
1
√
2x ln x
f 0 (x) =
3(ln x)2
x
(h)
(i)
f 0 (x) =
6. Use LD to find the derivative of the following.
(a) y 0 = (x + 1)(5x + 7)(x + 2)2
(b) y 0 = 3x ln 3
2
2
(c) y 0 = (x2 + 1)x [ x2x
2 +1 + ln(x + 1)]
3x2
x3 + 1
Homework Key for Section 8
1. A bacteria culture initially contains 100 cells and grows at a rate proportional to its size.
After one hour the population is 420.
(a) Find an expression for the number of bacteria after t hours.
Q(t) = 100e1.435t
(b) Find the number of bacteria after 3 hours.
≈ 7409
(c) When will the population reach 10,000?
≈ 3.2 hours
2. The half-life of Cesium-137 is 30 years. Suppose we have a 100 mg sample
(a) Find an expression for the mass after t years.
Q(t) = 100e−0.0231t
(b) How much remains after 100 years?
≈ 9.29 mg
(c) When will the mass be 1 mg?
≈ 199.3 years
3. Skeletal remains at a dig have lost 82 % of the carbon 14 that they had originally. Determine its approximate age.
≈ 14, 290 years old