Math 1325 Final Exam Review True or False - Blinn College · Math 1325 – Final Exam Review 1. True or False: a. fa is equivalent to the slope of the tangent line at a f a, b. h

Math 1325 – Final Exam Review

1. True or False:

a. f a is equivalent to the slope of the tangent line at ,a f a

b. h

fhf

h

11lim

0 is equivalent to 1f

c. f g x f x g x

d. f g x f x g x

e. f g x f x g x

f. f xf

xg g x

g. All first-order critical values occur when 0f x .

h. All relative extrema occur when 0f x .

i. If ,c f c is a point of inflection, then 0f c .

j. If 0f c , then ,c f c is a point of inflection.

k. If the slopes of the tangent lines to f x are positive on ,a b , then 0f x on ,a b .

l. If 0f c and 0f c , then f c is a relative minimum.

m. If f x is continuous on the closed interval ,a b , then f x must have an absolute minimum

and an absolute maximum on the interval.

2. Find the limit for each of the following.

a. xxx

2

13lim

b. 22

2lim

6x

x

x x

c. 2

1

2lim

1x

x x

x

d. 2

3 21

2 1lim

2x

x x

x x x

e. 2

3

4 8lim

4x

x x

x

f. 2 3

lim5 7x

x

x

g. 4 3

3lim

12 128x

x x

x

h. 2

2

1lim

( 2)x

x

x x

i. 0

2 3limx

x

x

j. 2

23

5 6lim

2 3x

x x

x x

k. 1

22lim

2

1 x

xx

x

l. 4

2lim

4 x

x

x

m. 3

2lim

5 4x

x

x

n.

9 2lim

1 3x

x

x

o. 4

limx

f x , given 2

4 , 0

2 , 0 4

3 , 4

x x

f x x x x

x x

.

3. Find all values where the given functions are discontinuous.

a. 34

322

2

xx

xxxk

b. 6

8 if 0

7 if 0 3

3 if 3

x

x x

f x e x

x x

c. if 1

2 if 1

x xf x

x x

d. 2

3

4 if 2

if 2

x xf x

x x

e. 0 if 4

4 if 4

xf x

x x

f.

2

1if 0

if 0 3

if 3x

xx

f x x x

e x

g.

4. Find all values of k such that f x is continuous.

a. 2

4 if 5

if 5

kx xf x

kx x

b.

3 if 2

if 2

x k xf x

x k x

5. Find the average rate of change of 21 3f x x between x = –2 and x = 0.

6. Find the instantaneous rate of change for each function at the given value.

a. 2 32 1 3 7f x x x x at 1x

b. 33 17 4g x x x at 2x

7. Find the equation of the line tangent to the graph at the given value.

a. 2 2f x x at 1x

b. 92xxf at 4x

c. 4

f xx

at 1x

8. Find the derivative of the following.

a. t

tttp5

612 4

b. 5 34 7 2 3y x x x x

c. 3

2

3 4 7

2 3

x xf x

x

d. 1

2 24 5 4y x x

e. 323 13

5

xxy

f. 24 156 tttm

g. 223 12 xxy

h. 5

35

83

t

ttr

i. 3 8 1y x

j. 23 410 xy

k. 7 53 4 27 x x xy e

l. 32 42 xxey

m. 2ln xy

n. 25 ln( )xy e x

o. 2

ln xy

p. 12ln xy

q. 32log xy

r. 2

52

2 2log xxy

9. Given the function 3 24 5 8f x x x x , find all values of x where the tangent line is horizontal.

10. For each of the following functions, find all critical values, state the interval(s) on which the function is

increasing, state the interval(s) on which the function is decreasing and state all relative extrema using

appropriate and complete sentences.

a. 3 24 9 30 6f x x x x

b. 1

2

x

xxf

c. xxxf 39 32

11. Find all intervals where each function is concave up or concave down. Also, find all points of

inflection.

a. 2

3f x x x

b.

45

12

1

20

1xxxf

12. Given f x defined for ,x , determine which of the following are true statements and which

are false.

a. 5

lim 5x

f x

b. 5 is undefinedf

c. 4 0f

d. 4 0f

e. 3 0f

f. 2 is undefinedf

g. 1 0f

h. 0 is undefinedf

i. 3 0f

j. 0 for , 5f x x

k. 0f x for 3, 1 3,x

13. Sketch a graph with the following properties.

a. lim 0x

f x

b. The vertical asymptotes are 3x and 3x .

c. Increasing nowhere

d. Concave up on the intervals 3,0 3,

e. Concave down on the intervals , 3 0,3

14. Using curve sketching techniques, sketch a graph of 2 1

3

xf x

x. Show all work including the

horizontal asymptote, vertical asymptotes, x-intercept(s) and y-intercept (if they exist).

15. Given the graph of f x , find the interval(s) on which f x is increasing and the interval(s) on which

f x is decreasing.

16. Given the graph of f x , find the interval(s) on which f x is concave up and the interval(s) on which

f x is concave down.

17. Find the location of all absolute extrema for the given functions with the specified domains.

a. 3 26 9 8f x x x x ; 2,5

b. x

xxf

8

8; 4,6

18. Optimize the following.

a. A fence must be built to enclose a rectangular area of 25000 ft . Fencing material costs $3 per

foot for the two sides facing north and south and $6 per foot for the other two sides. Find the

cost of the least expensive fence.

b. An open box is to be made by cutting a square from each corner of a 3-ft by 8-ft piece of

cardboard and then folding up the sides. What size square should be cut from each corner in

order to produce a box of maximum volume?

19. Given the following equations, find dy dx by implicit differentiation.

a. 2 3 4 2x y xy

b. 3 29 4

3x y

c. 2 2 33 7 4x xy y x

d. 2 2 9 35 7 5x x y y x

e.

1/22

3

x yx

x y

f.

20. Given the following equations and information, evaluate dy dt .

a. 3 2 52 3 7 17x xy y given that 3dx dt , 2x , and 1y

b. 2 23 3 2 7x xy y t given that 2dx dt , 1x , 4y and 1t

21. Determine the following.

a. A spherical snowball is placed in the sun. The sun melts the snowball so that its radius decreases 116 in. per hour. Find the rate of change of the volume with respect to the time at the instant the

radius is 4 in. Recall, the formula for the volume of a sphere is 34

3V r .

b. A sand storage tank used by the highway department for winter storms is leaking. As the sand

leaks out, it forms a conical pile. The radius of the base of the pile increases at the rate of ¼ in.

per minute. The height of the pile is always twice the radius of the base. Find the rate at which

the volume of the pile increasing at the instant the radius of the base is 6 in. Leave your answer

in terms of π. Recall, the formula for the volume of a cone is 21

3V r h .

22. Find the following.

a. 3 42 3 4x x x dx

b. 2

4 3y dy

c. 558x dx

x

d. 3 9 27 15x xe e x dx

e. 1211 31

9

xx e dx

f. 2

3

xedx

x

g. 2

3

12 4

9

tdt

t t

h. 3

5 2 42 3 5 3x x x x dx

i. 29 8x x dx

j. 2 32 xx e dx

k. ln x dx

l. ln 12 x dx

m. 2

ln xdx

x

23. Evaluate each definite integral.

a. 1

2

4

x dx

b. 2

2

1

3 5x dx

c. 1

19

e

x dxx

d. 4

2

1

7x x dx

e.

1

2

0

5 4x x dx

f.

4

0

2 xe dx

g.

24. Find a function g x whose graph has slope 32 7 3g x x x and passes through the point 2, 3 .

25. Without using the Riemann program and given the function 2 1f x x , approximate the area

between 1a and 7b using 3n rectangles based on left endpoints, right endpoints, average and

midpoints.

26. Find the area between the curves 2 1f x x and 2 2g x x x on 0,1 . You must provide a

graph and find any points of intersection.

27. Find the area between the curves 22f x x x and 2 4g x x . You must provide a graph and

find any points of intersection.

28. Find the area completely enclosed by the graphs of 3 2f x x x and

2g x x .

29. Find the area completely by the graphs of 3 3 2f x x x and 2g x x .

30. Find all second-order partial derivatives of the following.

a. 3 2 3 2, 4 3 2f x y x x y y

b. 3 2, 2 8xf x y e x y

Math 1325 – Final Exam Review Solutions

1.

a. True

b. False; It is equivalent to 1f .

c. True

d. True

e. False; f g x f x g x g x f x

f. False; 2

f x g x g x f xfx

g g x

g. False; They also occur when f x does not exist.

h. True

i. True

j. False; f c must equal 0 AND f x must change signs on either side of c .

k. False; If the slopes of the tangent lines to f x are positive on ,a b , then 0f x on ,a b .

l. True; This is the second derivative test.

m. True

2.

a. 2

b. 1

5

c. 3

d. 1 e. 0

f. 2

5

g.

h.

i.

j. 1

4

k. 2

l. 1

4

m. 0

n. 2

3

o. 8

p.

3.

a. Discontinuous at 1,3x

b. Discontinuous at 3x

c. Discontinuous nowhere

d. Discontinuous at 2x

e. Discontinuous nowhere

f. Discontinuous at 0,3x

4.

a. 2

15k

b. 3k

5. AROC 6

6.

a. IROC 10

b. IROC 19

7.

a. 2 1y x

b. 4 9

5 5y x

c. 4 8y x

8.

a. 3 1 2 248 3 5p t t t t

b. 7 5 4 332 48 60 28 28 21dy dx x x x x x

c. 4 2

22

6 35 28 12

2 3

x x xf x

x

d. 1 2

22 10 4 5 4y x x x

e. 4

2 3 215 9 2 3 1dy dx x x x x

f. 4 46 5 1 45 1m t t t

g. 2 31 7 3 8dy dx x x x x

h. 6

3 12 35

5 3

tr t

t

i. 12

8 1y

x

j. 23 46 ln10 10 xy x

k. 7 56 4 3 4 27 7 15 4 x x xdy dx x x e

l. 3

22 426 2 4 4x x

y x x x e

m. 2

yx

n. 1

10 lnxy e xx

o. 2lndy x

dx x

p. 1

2 1

dy

dx x

q. 2

ln10 2 3

dy

dx x

r. 2

5 4 1

2 ln 2 2

xdy

dx x x

9. 5

1,3

x

10.

a. Increasing on 5

, 1 ,2

; Decreasing on 5

1,2

;

There is a relative maximum of 23 at 1x .

There is a relative minimum of 251

4 at

5

2x .

b. Increasing nowhere; Decreasing on , 1 1, ;

There is no relative maximum.

There is no relative minimum.

c. Increasing on 0,8 ; Decreasing on ,0 8, ;

There is a relative maximum of 12 at 8x .

There is a relative minimum of 0 at 0x .

11.

a. Concave up on , 2 ; Concave down on 2, ;

There is a point of inflection at 2, 2 .

b. Concave up on 1, ; Concave down on ,0 0,1 ;

There is a point of inflection at 1

1,30

.

12. a. False

b. True

c. False

d. True

e. True

f. False

g. False

h. False

i. False

j. True

k. True

13. This is an example. There are many correct answers.

14. The horizontal asymptote is 2y .

The vertical asymptote is 3x .

The x-intercept is 1

,02

.

The y-intercept is 1

0,3

.

15.

Increasing on ,1 Decreasing on 1,

16.

f x is concave up on 1.2,1.2

is concave down on , 1.2 1.2,

17. a. There is an absolute maximum of 12

at 5x .

There is an absolute minimum of 8 at

3x .

b. There is an absolute maximum of 7

at 6x .

There is an absolute minimum of 3 at

4x .

18.

a. The cost of the least expensive fence is $1200 .

b. A 2

-ft3

by 2

-ft3

square should be cut from each corner.

19.

a. 3

2 2

2 4

3 4

dy xy y

dx x y x

b. 2

3

8

dy

dx xy

c. 212 6 7

7 2

dy x x y

dx x y

d. 2

8 2

15 10 14

9 7

dy x x xy

dx y x

e.

2

6 3

2 3 2

dy x x

dx x

20.

a. 63

23

dy

dt

b. 22

dy

dt

21.

a. 34 in hdV

dt

b.

318 in mindV

dt

22.

a. 3 2 4 3 5 44 9 16

3 4 5

x x xC

b. 3

21612 9

3

yy y C

c. 4

25ln x C

x

d. 3 9

375

3 9

x xe ex C

e. 1231

324

xe C

f. 4

3

xe C

g. 34ln 9t t C

h.

45 22 3

8

x xC

i. 2 23 8 8x x C

f x

j. 2

3 3 3

2 4 4

3 9 27x x x

x xC

e e e

k. lnx x x C

l. ln 12x x x C

m. ln 1x

Cx x

23.

a. 21

b. 12

c. 29 11

2

e

d. 161

3

e. 19

15

f. 42 2e

24. 4 27

3 192 2

x xg x x

25. Left endpoints 276 units

Right endpoints 2172 units

Average 2124 units

Midpoints 2118 units

26. 4/3 sq. units

27. 32

3 sq. units

28. 37

12 sq. units

29. 8 sq. units

30.

a. 3, 24 6xxf x y x y

2, , 18xy yxf x y f x y xy

2, 18 4yyf x y x y

b. 2, 2 48x

xxf x y e xy

2, , 48xy yxf x y f x y x y

3, 16yyf x y x