03 Derivatives and Their Applications

[Pages:20]AP Calculus Derivatives and Their Applications

Student Handout

2017-2018 EDITION

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Derivatives and Their Applications

Students should be able to compute derivatives of the following: Polynomials Rational functions (quotients) Radical functions Trigonometric functions Exponential and logarithmic functions

Students should be able to apply various techniques and rules including:

The power rule for integer, rational (fractional) exponents, expressions with radicals.

Derivatives of sum, differences, products, and quotients

The chain rule for composite functions

Implicit differentiation

Fundamental Theorems of Calculus (FTC)

The derivative of expression in functional form

h(

x)

f (g(x)),

q(x)

f g

(x) (x)

,

etc.

where the functions are not given but values are taken from table or graph.

Students should be able to apply the concept of derivative to

Solve related rate problems Solve optimization problems Determine the slope of tangent line to a curve at a point Determine the equations of tangent lines Approximate a value on a function using a tangent line and determine if the estimate is an

over- or under- approximation based on concavity of the function

Students need to be able to recognize different ways that a tangent line approximation can appear on the AP exam:

Tangent line approximation

Linear approximation

Linearization

Euler's method (BC)

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1

Multiple Choice

1. (calculator not allowed)

2

If f (x) (x2 2x 1) 3 , then f '(0) is

(A)

4 3

(B) 0

(C)

2 3

(D)

4 3

(E) 2

2. (calculator not allowed)

If f (x) 2x 14 , then the 4th derivative of f (x) at x 0 is

(A) 0 (B) 24 (C) 48 (D) 240 (E) 384

For questions 3-5: The table below shows some of the values of two differentiable functions f

and g and their derivatives.

x

f (x)

g(x)

f '(x)

g '(x)

3

-3

6

-5

2

4

0

3

-3

9

5

3

-2

4

5

3. (calculator allowed)

If

h(x)

f

(x)g(x)

f (x) g(x)

,

then

find

the

value

of

h '(5)

.

(A)

5 4

(B)

35 4

(C)

84 5

(D)

37 2

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4. (calculator not allowed) If h(x) f (g(2x)) , then find the value of h '(2) . (A) 90 (B) 45 (C) 5 (D) 3

5. (calculator not allowed)

If h(x) g(x)2 , then find the value of h '(3) .

(A) 4 (B) 12 (C) 24 (D) 36

6. (calculator not allowed)

If

y

2

cos

x 2

,

then

d2y dx2

is

(A)

8

cos

x 2

(B)

2

cos

x 2

(C)

sin

x 2

(D)

cos

x 2

(E)

1 2

cos

x 2

7. (calculator not allowed)

The cost, in dollars, to shred the confidential documents of a company is modeled by C, a

differentiable function of the weight of documents in pounds. Of the following, which is the

best interpretation of C '(500) 80 ?

(A) The cost to shred 500 pounds of documents is $80.

(B) The

average

cost

to

shred

documents

is

80 500

dollar

per

pound.

(C) Increasing the weight of the documents by 500 pounds will increase the cost to shred the

documents by approximately $80.

(D) The cost to shred documents is increasing at a rate of $80 per pound when the weight of

the documents is 500 pounds.

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8. (calculator not allowed)

What is the slope of the tangent line to the curve 3y2 2x2 6 2xy at the point (3, 2) ?

(A) 0

(B)

4 9

(C)

7 9

(D)

6 7

(E)

5 3

9. (calculator allowed)

The temperature, in degrees Fahrenheit O F , of water in a pond is modeled by the function

H

given

by

H

(t)

55

9

cos

2 365

(t

10)

,

where

t

is

the

number

of

days

since

January

1

t 0 . What is the instantaneous rate of change of the temperature of the water at time

t 90 days? (A) 0.114 OF / day

(B) 0.153OF / day

(C) 50.252 OF / day

(D) 56.350 OF / day

10. (calculator not allowed)

If

sin(xy)

x, then

dy dx

is

(A)

1 cos( xy)

(B)

1 x cos(xy)

(C)

1 cos(xy) cos(xy)

(D) 1 y cos(xy) x cos(xy)

(E) y(1 cos(xy)) x

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11. (calculator not allowed) If f and g are twice differentiable and if h(x) f (g(x)) then h(x)

(A) f g(x)g(x)2 f g(x) g(x)

(B) f (g(x))g(x) f (g(x))g(x)

(C) f ''(g(x))g '(x)2

(D) f ''(g(x))g ''(x) (E) f ''(g(x))

12. (calculator not allowed)

If f (x) tan1 x, then f ' 3

(A)

1 4

(B)

1 2

(C)

6

(D)

3

13. (calculator not allowed)

Let f (x) ex 3 . What is the approximation for f (0.5) found by using the line tangent to

the graph of f at x 0 ? (A) 1.25 (B) 1.5 (C) 2.125 (D) 2.25

14. (calculator not allowed)

If f (x) ln x 4 e3x , then f '(0) is

(A)

2 5

(B)

1 5

(C)

1 4

(D)

2 5

(E) nonexistent

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15. (calculator not allowed)

If f (x) x 1 x2 2 3 , then f '(x) (A) 6x x2 2 2 (B) 6x x 1 x2 22 (C) x2 22 x2 3x 1 (D) x2 22 7x2 6x 2 (E) 3 x 1 x2 22

16. (calculator not allowed)

Let f be a differentiable function such that f (3) 15, f (6) 3, f '(3) 8 , and f '(6) 2 . The function g is differentiable and g(x) f 1(x) for all x. What is the value of g '(3) ?

(A)

1 2

(B)

1 8

(C)

1 6

(D)

1 3

(E) The value of g '(3) cannot be determined from the information given.

17. (calculator not allowed)

The function h(x) x3 kx2 where k is a constant. If the tangent line to the graph of h at

x 2 is parallel to the line that contains the points 1, 3 and 4,15 , what is the value of

k?

(A) 4

(B)

49 16

(C)

47 16

(D) 2

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