Chapter 3 Worksheet Packet AP Calculus AB Name, Assignments of Calculus

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Chapter 3 Worksheet Packet

AP Calculus AB

Name

Calculus Practice: Derivatives

Find the derivative and give the domain of the derivative for each of the following

functions. If the derivative does not exist at any point, explain why and justify your

answer.

1) f (x) = x

4

2) f (x)= x'

3) f (

Find the equation of the tangent line to the graph of f (x) at the point P.

5) f (x) P (1,2)

2

6) f (x) = x 3 +2x P(8,20)

7) Find the velocity of the particle at time t = 3 if the position function for the particle is

given by s (t) — 2t.

8) Find the x values of all points on the graph of y = x 4 — 2x 2 where the tangent line is

horizontal.

tanx'

2. Given f (x)

a) d) 4^ e) undefined

AP Calculus

Derivatives Practice

Name

I. If f (x) = sin4 x, then f =

a) - 7 b)

Date

d) e) 4 4

3. Differentiate: f (x) = x 2 + 2 tan x

a) 2x + 2 tan x b) 2x + sec 2 x^ c) 2 + sec2 x^ d) 2x + 2 sec 2 x^ e) 2x + 2 cot x

dy 3x

4. Find the derivative, — dx

, of y = x 2 + 1

3 3 3x 2 — 3 d) 3(1 — x 2 )^ 6x + x a) b) c) e) 1 + x 2 2x^ (1 + x 2 ) 3^ + x2 )2 (x2 + 1) 2

dy x 2 — 1

5. Find the derivative, — dx '

of f (x) = x 2 + 1.

4x 4x 2 — 4x 2 — 4x a)

4x b) 1 c)^ d)^ e) x2 + ) 2 (x 2 + 1) 2 (x2 + 1) 2 (x2 + 1)

6. If f (x) = (x 3 + 4x2 — 12x + 8)(3x 2 — 9x + 7), then find 01).

a) —4 b) 4 c) —3 d) 3 e) 7

7. Find the slope of a line tangent to the graph of f (x) = xx+j at the point (1, 4).

a) — b) - 9 c)

Page 2

8. Find an equation of the tangent line to the curve f(x) - x 2 - 10 passing through the point (5, 1).

a) y - 1 = -10(x - 5) (^) b) y + 5 = -10(x + 1) (^) c) y + 1 =

d) y - 1 = 10(x - 5) (^) e) y - 5 = 10(x - 1)

sin x rr

Find the slope of the tangent to the (^) graph f(x)= where x =

10(x + 5)

e) 3A/j

e) 28x 7

6x 3 + 3 e)

cos 2x 6

a)

b) T'

c) 3 3

F' d f'(x) for (^) f(x) - (2x2 + 5) 7 •

a) 7(4x) 6 b) (4x) 7 (^) c) 28x(2x 2 + 5) 6 (^) d) 7(2x 2 + 5) 6

dy

Find —

dx

for y = x 3 A/2x + 1

x2 (7x + 3) (^) 3x2 8x3 + 3x 2 (^) 8x + 3 a) (^) b) c) (^) d) A/2x + 1 (^) 2A/2x + 1 (^) 2./2x4 + x3 A/2x + 1 (^) A/2x + 1

12. If y =

dv (3x 2 + 5) 5 (x + 2) 4 , then dx

a) 2(x + 2) 3 (3x 5)4 (^) b) 2(21x 2 + 30x + 10)(x + 2) 3 (3x2 + 5)

c) (x + 2) 3 (3x 2 + 5)(21x 2 + 30x + 10) (^) d) 24(x + 2) 3 (3x 2 + 5)4 (21x 2 + 30x + 10)

e) 12(x + 2) 3 (3x 2 + 5) 4 (21x + 30)

13. Find the derivative of y - cos x 3.

a) 3x 2 sinx 3 (^) b) 3 cos x 3 (^) c) -3x 2 sin x 3 (^) d) 3 sinx 3 cos 2 x 3 e) 3x cos x 2

14. Find f' (x) (^) given f(x) = sin3 (4x).

a) 4 cos 3 (4x) (^) b) 3 sin2 4x cos(4x) (^) c) cos 3 4x

d) 12 sin2 4x cos(4x) (^) e) 12 cos 2 (4x)

5. The ends of a water trough 14 ft. long are equilateral triangles whose sides are 5 ft.

long. If the water is being pumped into the trough at a rate of 10 cu. ft. per min., find

the rate at which the water level is rising when the depth is 6 in.

6. A spherical balloon is being inflated with gas. Use differentials to approximate the

increase in surface area of the balloon if the radius changes from 2 ft. to 2.05 ft.

7. Use a linear approximation to estimate the value of V.

8. The radius of a spherical balloon is measured as 8 in. with a maximum error o .1 in.

Approximate the relative error for the calculated value of the volume.

Chapter 3 Test Practice/AP Calculus

The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds).

1. s = 6 sin t - cos t

Find the body's velocity at time t = rt/6 sec.

Find the derivative of the given function.

2. y = 2 sin -1 (4x 3 )

3. y = tan -1. -,/

4. y = + sin 4x

5. y = ln 4x

7

6. y - sin x

7. y = 5 sec2 x

8. y = 4x

9. y = cot (2x - 5)

10. y = 5 sec6x

11. y = log (5x - 4)

12. y = 3xex - 3ex

Solve.

13. Find the tangent line to the graph of x 2 + y2 - 2x + 4y = 8 at (4, 0)

14. Find the normal line to the graph of 4x 2 y -rt cos y = 5n at (

Use implicit differentiation to find d 2y/dx.

15. y2 _ x2 = 9

Solve the problem.

16. The position of a particle moving along a coordinate line is s = (^) ---71t,5, with s in meters and t in seconds. Find

the particle's velocity at t = 1 sec.

17. The profit in dollars from the sale of x thousand compact disc players is P(x) = x 3 - 3x 2 + 4x + 8. Find the

marginal profit when the value of x is 9.

1

Answer Key

Testname: CHAPTER 3 TEST PRACTICE

(-2\ 24x 2

(3 2(1 + 3x)45;

f<N 4 2 cos 4x

-----1" 43 + sin 4x

rz)_ 2

• 7 csc x cot x

10 tan x sec2x

0 4x ln 4

0) -2 csc 2 (2x - 5)

30 tan x sec6x

5

3 (5x - 4) ln 10

3 3xex

3

y=

1 _ 1 +

Tt 27 27r

_41 x d2y _ y2 - x 2

e:9 dx Ydx 2 y

C)

rn/sec 3

a(6) = -6 rn/sec2, a(8) = 6 rn/sec

20. vertical tangent

21. 011011111111101111wommuoium.

22. ,irm..441E-

23. At x = 0, 3, -

24)2<t<3,5<t<

X

/6(5--K)-i- /

(a7-9e-oox +9)

(2)( 3+ (icx ?-1)