MATH 165
Name:
Exam 3 Practice Questions,
Sec:
1
3.10 Related Rates
1. A spherical balloon is expanding from the sun’s heat at a constant rate of 10 cubic meters
per hour. How fast is its radius increasing when the radius is 5 meters.
2. A balloon leaving the ground 1200ft from an observer rises at the rate of 200 ft/min.
How fast is the angle θ of elevation of the observer’s line of sight increasing when the
balloon is at an altitude of 1600ft? Be sure to the units in your answer.
2
3.11 Linearization/Differentials
1. Find the linearization of
centered at x =
√
x
y = sin−1 ( )
2
3 and use it to approximate f (1.7).
2. Construct a useful linearization of f (x) = ln|x| and use it to estimate f (1.997).
3. Find the linearization L(x) of the function f (x) = cosx at π6 .
4. Find the linearization of f (x) = ln(x2 +3x+1) at x = 0. Use it to find an approximation
for f (0.002).
3
4.1 Extreme Values
1. Find the local minima and maxima of the function f (x) = 2x3 − 54x and find the
intervals on which the functon is increasing or decreasing.
2. Identify all critical points, and the maximum and minimum values on the interval provided:
1
f (x) =
,
1 + x2
on the interval [−3, 1].
3. A function f has derivative
f 0 (x) = (x + 1)4 (x − 4)3 .
a. Find and classify all critical point(s) for f . Justify you answers.
b. Find all inflection point(s) for f . Justify your answer.
4. Determine the values of the constants a, b, c, and d so that f (x) = ax3 + bx2 + cx + d
has a local maximum at the point (0, 0) and a local minimum at the point (1,-1).
4
4.3 and 4.4 Increasing/Decreasing/Concavity/Curve
Sketching
1. For each of the following, either sketch a function with the given properties or explain
why no such function exists.
a. f (x) defined on (−∞, ∞), f 0 (x) < 0 and f 0 (x) decreasing for all x.
b. f (x) defined on (−∞, ∞), f 00 (x) > 0 and f (x) < 0 for all x.
c. f (x) defined on (−∞, ∞), f 0 (x) > 0 and f (x) < 0 for all x.
2. Using our process for graph sketching, carefully sketch the graph of f (x) = (2 − x2 )2/3 .
3. Using our process for graph sketching, carefully sketch the graph of f (x) =
credit will be given for simply using the calculator, you must show all steps).
3
.
x−4
(No
4. Find the intervals on which f is both increasing and concave down for
f (x) = x4 − 4x5 .
5. For a function f defined on −∞ < x < ∞ we are given the derivative
f 0 (x) = (2x + 7)3 (3x − 6)4 .
a. Give the intervals on which f is increasing and intervals on which f is decreasing.
b. List all of the critical values of f , and determine (with reason) whether it is a
relative min, relative max, or neither.
c. Find the x values of all inflection points of f . Justify your answer.
5
4.6 Applied Optimization
1. A farmer has 600 meters of fence. She wants to use the fence to enclose a yard divided
into three identical pens, as shown in the figure. Find the dimensions for the pen that
result in the largest enclosed area.
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2. Find the point on the parabola f (x) = x2 closest to the point (3,0). (Recall that the
distance formula between any two points (x1 , y1 ) and (x2 , y2 ) is
d2 = (x2 − x1 )2 + (y2 − y1 )2 .
3. We are constructing a 200 square foot garden next to a brick wall by fencing in 3 sides
(the side along the wall does not need to be fenced). Calculate the dimensions of the
side that will minimize the amount of fencing used.
4. You are planning to make an open rectangular box from an 8-in by 15-in piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are
the dimensions of the box of the largest volume you can make this way, and what is it
volume?
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4.7 Newton’s Method
1. Use Newton’s method to find a positive fourth root of 2 by solving the equation x4 −2 = 0.
Start with x0 = 1 and find x2 .
2. For what value of x does cosx = 2x?
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4.8 Antiderivatives
1. Evaluate
Z
1
+ 3e−x )dx
(x4 − 1 + √
7
x
.
2. Evaluate
3. Evaluate
Z
12
.
1 + x2
Z
cot2 xdx.
(Hint: 1 + cot2 x = csc2 x)
4. Solve the initial value problem:
dv
3
= √2
, v(2) = 0.
dt
t t −1
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