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Review for Exam 1

p.93 19, 25, 27;

p.186 Review Exercises: #1,3,4,5,7,8, 15, 17, 19, 20, 27a&b, 35

186

CHAPTER 2 Differentiation: Basic Concepts

2-84

Checkup for Chapter 2

-7

CHAPTER SUMMARY

c. y

1. In each case, find the derivative

dy

dx

5

a. y = 3×4 – 4V8+

x2

b. y = (3x – x + 1)(4 – x)

5×2 – 3x + 2

1 – 2x

d. y = (3 – 4x + 3x)3/2

2. Find the second derivative of the function

f(t) = (21 + 1)?

3. Find an equation for the tangent line to the curve

y = x2 – 2x + 1 at the point where x = -1.

4. Find the rate of change of the function

with respect to x when x = 1.

1 – 5x

5. PROPERTY TAX Records indicate that x years

after the year 2010, the average property tax on a

four-bedroom home in a suburb of a major city

was T(x) = 3x² + 40x + 1,800 dollars.

a. At what rate is the property tax increasing with

respect to time in 2013?

b. At what percentage rate is the property tax

increasing in 2013?

6. MOTION ON A LINE An object moves along

a line in such a way that its position at time t is

given by s(t) = 27 – 3x² + 2 for 1 2 0.

a. Find the velocity v(t) and acceleration aft) of the

object.

b. When is the object stationary? When is it

advancing? Retreating?

c. What is the total distance traveled by the object

for 0 51 522

7. PRODUCTION COST Suppose the cost

of producing x hundred units of a particular

commodity is C(x) = 0.04×2 + 5x + 73 thousand

dollars. Use marginal cost to estimate the cost of

producing the 410th unit. What is the actual cost

of producing the 410th unit?

8. INDUSTRIAL OUTPUT At a certain factory,

the daily output is Q = 50023/4 units, where L

denotes the size of the labor force in worker-

hours. Currently, 2,401 worker-hours of labor are

used each day. Use calculus (increments) to

estimate the effect on output of increasing the size

of the labor force by 200 worker-hours from its

current level.

9. PEDIATRIC MEASUREMENT Pediatricians

use the formula S = 0.20290425 to estimate the

surface area S (m²) of a child 1 meter tall who

weighs w kilograms (kg). A particular child

weighs 30 kg and is gaining weight at the rate

of 0.13 kg per week while remaining 1 meter tall.

At what rate is this child’s surface area changing?

10. GROWTH OF A TUMOR A cancerous tumor

is modeled as a sphere of radius r сm.

a. At what rate is the volume V = for changing

with respect to r when r=0.75 cm?

b. Estimate the percentage error that can be

allowed in the measurement of the radius r to

ensure that there will be no more than an 8%

error in the calculation of volume.

f(x) = x+1

(1 – 3x)

1 – 2x

3x + 2

13. y =

24. Use the chain rule to find for the given value

of x.

dx

a. y = u-u?; u = x – 3; for x = 0

34

b. y =

,u = Vx- 1; for x

U +1

U-12

In Exercises 14 through 17, find an equation for the

tangent line to the graph of the given function at the

specified point.

14. f(x) = x2 – 3x + 2; x = 1

4

15. f(x)

; x = 1

=

x – 3

25. Use the chain rule to find for the given value

dx

of x.

a. y = x – 4u? + 5u + 2; u = x + 1; for x = 1

b. y = Vu, u = x² + 2x – 4; for x = 2

26. Find the second derivative:

a. f(x) = 6x – 4x + 5×2 – 2x + =

1

=

x=0

x +1

X

=

16. f(x)

17. f(x) = x2 + 5; x = -2

18. In each of these cases, find the rate of change of

f(t) with respect to t at the given value of t.

a. f(t) = p – 41+51V1 – 5 at t = 4

212 – 5

b. f(1) =

at t = -1

1 – 31

19. In each of these cases, find the rate of change of

f(t) with respect to t at the given value of t.

a. f(t) = + (1 – 1) at t=0

b. f(t) = (1 – 3+ + 6)1/2 at t = 1

20. In each of these cases, find the percentage rate of

change of the function f(t) with respect to t at the

given value of t.

a. f(t) = 72 – 3t+ Vt at t = 4

b. f(t)

21. In each of these cases, find the percentage rate of

change of the function f(t) with respect to t at the

given value of t.

a. f(0) = f(3 – 21) at 1 = 1

1

b. f(t) =

2

b. z =

1 +r?

c. y = (3x + 2)

27. Find the second derivative:

a. f(x) = 4x – 3x

b. f(x) = 2x(x + 4)

X-1

c. f(x)

(x + 1)2

dy

28. Find by implicit differentiation.

dx

a. 5x + 3y = 12

b. (2x + 3y) = x +1

dy

29. Find

dx by implicit differentiation.

a. xy = 1

b. (1 – 2xy) = x + 4y

=

1

=

att = 4

t – 3

att = 0

t +1

35. POPULATION GROWTH Suppose that a

5-year projection of population trends suggests

that 1 years from now, the population of a certain

community will be P thousand, where

P(t) = -27° + 91° +8t + 200

a. At what rate will the population be growing

3 years from now?

b. At what rate will the rate of population growth be

changing with respect to time 3 years from now?

In Exercises 19 through 32, either find the given limit

or show it does not exist. If the limit is infinite, indicate

whether it is + or -.

x2 + x-2

x2 – 3x

19. lim

20. lim

x-1

1-2 X+1

X-8

1

21. lim

22. lim

1-2 2 – x

x

23. lim 2 –

X

3)

24. lim 2 +

60

(2+3)

(1-3)

25.

lim

26. lim

20

x – 3x + 5

28. lim

2x + 3

+

—r? + 5

*4 + 3x² – 2x + 7

3r2

lim

+x+1

1 1

1+

1+-+

x2

lim

+-x2 + 3x – 1

27.

29.

30.

x(x – 3)

lim

— 7 – r?

1

1

31.

lim x

32.

lim

10

-V:(+3)

x

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