The Chain Rule

Transcription

The Chain Rule

Exam II Review - Worksheet
Math 1131
Name:
Section:
The Chain Rule
1. Let y =
Class #31
√
2x + 1. Find
dy
dx .
2. Let y = (1 − x)200 . Find y 0 (x).
2
3. Let y = e1−x . Find
dy
dx .
4. Let f (x) = ln(x2 ). Find f 0 (x).
5. Let f (x) =
2
ln(x) . Find f 0 (x).
1
Math 1131
Name:
6. Let f (x) =
Section:
p
ex + sin(x) + 5x3 . Find f 0 (x).
7. Let f (x) = tan(ex · x2 ). Find f 0 (x).
8. Compute the derivative of f (x) = cos(x2 ) · sin(x2 )
9. Let f (x) =
Class #31
p
ln(x2 + x + 1). Find f 0 (x).
2
Math 1131
Name:
Class #31
Section:
10. Let f (x) = sin(cos(tan(x))). Find f 0 (x).
11. Let f (x) = cot(x · csc(2x)). Find f 0 (x).
12. Let f (x) = sec(cos(sin(x))). Find f 0 (x).
3
Math 1131
Name:
Section:
Implicit Differentiation
1. Suppose that xy = x2 . Find
Class #31
dy
.
dx
2. Suppose that y 2 − 4xy + x2 = 1. Find
3. Suppose that ey = cos(x) + y. Find
dy
.
dx
dy
.
dx
4. Suppose that ey = y cos(y) + x. Find
dy
.
dx
4
Math 1131
Name:
Section:
Logarithms and Differentiation
1. Let f (x) = ln
Class #31
p
ex · cos(x) · (x2 − 1) . Find f 0 (x)
2. Let f (x) = ln xsin(x) . Find f 0 (x)
5
Math 1131
Name:
Section:
ex
3. Let f (x) = x
. Find f 0 (x).
4. Let f (x) =
Class #31
sin(x)
ln(x)
. Find f 0 (x).
6
Math 1131
Name:
Class #31
Section:
Related Rates
1. An airplane flies directly over a radar station, at a constant altitude of 3 mi above the ground.
A little while later, the radar station measures that (a) the distance between the plane and
radar station equals 5 mi and (b) that the distance between the plane and radar station is
increasing at a rate of 500 mi/hr. What is the ground speed of the airplane at the time of
the second measurement?
2. An ice cube melts, with its surface area decreasing at a rate of 3 in2 /s. How fast is the side
length decreasing when the side length is 1 in?
7
Math 1131
Name:
Class #31
Section:
3. A streetlight is mounted at the top of a 6 meter pole, and a 2 meter tall person is walking
toward it at 2 meters per second. How fast is the length of their shadow changing when they
are 4 meters from the streetlight? What about when they are 1 meter from the light?
4. A police officer is walking down a city street, when they spot a wanted felon standing 200 ft
away at the corner of the next block. The police officer takes off after the felon at 12 ft/s,
and the felon immediately cuts around the corner and runs away at 9 ft/s. What is the rate
of change of the distance between the officer and the felon after 10 seconds have passed?
8
Math 1131
Name:
Class #31
Section:
5. Suppose there is a 100 cm long water trough shaped as a triangular prism whose cross-section
is an inverted triangle ∇ which is 20 cm across the top, and is 10 cm tall. If the tank is being
being filled with water at a constant rate of 400 cm3 /s, how fast is the height changing when
the water is half way up the tank?
6. Suppose the water trough above leaks (100 cm long, cross section is a ∇, top = 20 cm, and
height = 1 cm). If water is being added to the tank at a rate of 400 cm3 /s, and is leaking
out of the tank at 100 cm3 /s, how fast is the height changing when the water is half way up
the tank?
9
Math 1131
Name:
Class #31
Section:
L’Hospital’s Rule
Use L’Hospital’s Rule to evaluate the following limits. Remember to show all work.
This is the best way to learn to do these problems correctly!
lim
ln(x3 )
x3 + 3
2. Compute the limit lim
ln(x3 )
x3 + 3
1. Compute the limit
x→0+
x→∞
3. Compute the limit lim 2x sin
x→∞
1
2x
10
Math 1131
Name:
Section:
4. Compute the limit lim
x→∞
5. Compute the limit
Class #31
lim
x→0−
8
1+
x
x
1 + 3 sin(x)
2/x
11
Math 1131
Name:
Class #31
Section:
Linear Approximations
1. Approximate e−0.1 using the method of linear approximation. You must give your answer as
a decimal without using a calculator.
√
2. Approximate 8 using a linear approximation. You must give your answer as a decimal
without using a calculator.
12
Math 1131
Name:
Class #31
Section:
3. Find the linear approximation to f (x) = x1 + x at a = 2. Use your answer to approximate
f (1.9). You must give your answer as a decimal without using a calculator.
4. Find the linearization of f (x) = x ln(x2 ) at a = 1. Use your answer to approximate f (1.2).
You must give your answer as a decimal without using a calculator.
13
Math 1131
Name:
Class #31
Section:
5. Find a linearization of the function f (x) = sin(x) at a = π4 .
Use your answer to approximate sin 5π
16 .
6. Find a linear approximation for the function f (x) = cos(x) at a =
Use your answer to approximate cos 5π
16 .
7. Find a linearization of the function f (x) = tan(x) at a =
Use your answer to approximate tan 5π
16 .
14
π
4
π
4
Math 1131
Name:
Class #31
Section:
Maxima, Minima, and Graph Shape
1. Let f (x) = x3 − 6x2 + 9x + 1
Find the following. If a requested quantity doesn’t exist, answer “DNE”.
(a) Find the intervals where f is increasing and decreasing
(b) Find the intervals where f is concave up and concave down
(c) Find the numbers x where the are local maxima and minima
(d) Find the numbers x where the is a point of inflection
15
Math 1131
Name:
Class #31
Section:
2. Let f (x) = x4 − 8x2 + 16.
Find the following. If a requested quantity doesn’t exist, answer “DNE”.
(a) Find the intervals where f is increasing and decreasing
(b) Find the intervals where f is concave up and concave down
(c) Find the numbers x where the are local maxima and minima
(d) Find the numbers x where the is a point of inflection
16
Math 1131
Name:
Class #31
Section:
Optimization
1. In optimization problems, it is important to verify that you have the correct answer.
(a) Find two positive numbers whose sum is 25 and whose product is as large as possible.
(b) Find two positive numbers whose sum is 25 and whose product is as small as possible.
17
Math 1131
Name:
Class #31
Section:
2. Find two numbers x and y such that xy 2 = 54 and which minimizes F = x2 + y 2 .
18
Math 1131
Name:
Class #31
Section:
3. You are building a rectangular garden. You are tired of your neighbors stealing your produce
and you have looked into several types of electric fence. The material for the side of the fence
near your house is $1 per foot. The other three sides will cost $2 per foot. If you have $100,
what is the maximum amount of area you can protect?
You must show all work, including verifying that area is maximized.
19
Math 1131
Name:
Class #31
Section:
4. You want to turn part of your yard into a dog park. For zoning reasons, you want the park
to be exactly 100 ft2 . Suppose that your house will form the west boundary of the yard, that
the fence for the north boundary will cost $3 per foot and that the other two sides will cost
$1 per foot. What are the dimensions of the yard with the cheapest surrounding fence?
You must show all work, including verifying that cost is minimized.
20
Math 1131
Name:
Class #31
Section:
5. You are designing a rectangular display box with an open top. The box should have a volume
of 2 m3 , and the length of the base should equal twice the width of the base. The material
for the base costs $5 per square meter, and the material for the sides costs $10 per square
meter. Find the cost of the materials for the cheapest container.
You must show all work, including verifying that cost is minimized.
21
Math 1131
Name:
Class #31
Section:
6. You are designing a rectangular display box with an open top. The width of the base must
equal twice its length. The material for the base costs $6/in2 and the material for the sides
costs $2/in2 . What is the largest volume box that you can make for $120?
You must show all work, including verifying that volume is maximized.
22
Math 1131
Name:
Class #31
Section:
7. You are crossing a 2 mile wide river. Your camp is on the other side, 5 miles downstream.
You are hungry, and you want to get there as quickly as possible. If you can swim at 3 mi/hr
and hike at 5 mi/hr, where should you land to get there the fastest?
You must show all work, but this time you do not need to verify that time is minimized.
23
Math 1131
Name:
Class #31
Section:
Anti-Derivatives
1. Find the (general) anti-derivative of f (x) = sin(x) + π cos(x) +
2. Find the (general) anti-derivative of f (x) =
3π
2
sec2 (x).
1
x2 + x + 1
+√ .
x3
x
3. Suppose f 0 (x) = x2 + 2x + 5 and that f (0) = 3. Find f (x).
4. Suppose f 0 (x) =
1
x
+ x + ex and that f (1) = 0. Find f (x).
5. Suppose f 00 (x) = 12x2 − ex , that f (0) = 1, and that f 0 (0) = 2. Find f (x).
24

The Chain Rule

Transcription

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