m7 REVIEW.pdf

8/14/2019 m7 REVIEW.pdf

http://slidepdf.com/reader/full/m7-reviewpdf 1/6

Math 7 REVIEW

Part I: Problems

1. Using the precise definition of the limit, show that 1325 5

4lim 1 . x

x

[Find the that works for any arbitrarily chosen positive and show that it works]

2. Determine the that will most likely “work” for any arbitrarily chosen 0 in the proof to show that

2

5lim 7 18. x

x

3. Determine the following limits:

a)

4

2

16lim

4 8 x

x

x

b)

2 2

0

3( ) 3limh

a h a

h

c)

2

2

4 4 3lim

3 (2 1) x

x x

x x

d)

20

tanlim

2 8 x

x

x x

4. If

2

3

12 , 2( ) ,

4 8, 2

x x f x

x x x

determine22 2

lim ( ), lim ( ), and lim ( ). x x x

f x f x f x

5. Let

3, if 2

1( ) , if 2 .

1

1 , if 2

x

f x x x

x

a) Determine the set of all points where f will be continuous.

b) Determine the set of all points where f will be differentiable.

6. Find :dy

dx a)

2

2

7

2 1

x y

x

b)2tan(sin ) y x

7. Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to

determine intervals of increase/decrease, intervals of concavity), sketch the graph of

3

2 9

x y

x

.

8. If2

( ) ,4 1

x f x

x

determine lim ( ) and lim ( ).

x x f x f x

9. Using the same guidelines as in #7 above, sketch the graph of2

( 1)

( 1) 4 1

x x y

x x

.

10. If 4 y x c is a tangent line to the curve3 7 , y x x determine .c

11. The line 6 4 y x is tangent to the graph of g at 5, (5) .g Determine (5)g and '(5)g .

12. Find the local extremas of ( ) 2sin cos 2 f x x x in the interval 0,2 .



2

13. A particle moves on the -axis x in such a way that its position at time t is given by5 3( ) 3 25 60 .s t t t t

Determine when the particle will be moving to the left.

14. A ball is thrown vertically upward from the ground with an initial velocity of 128 ft/sec. If the positive

direction of the distance from the starting point is up, the equation of motion is216 128 .s t t

a) Find the average velocity of the ball during the time interval 3 194 4

.t

b) Find the instantaneous velocity of the ball at 34

seconds and at 194

seconds.

c) What is happening to the ball at 34 seconds and at 194 seconds? Why?

d) Find the speed of the ball at 34

seconds and at 194

seconds.

e) How long will it take the ball to reach its highest point?

f) Determine how high the ball will go.

g) How long will it take the ball to reach the ground?

h) What will be the instantaneous velocity of the ball when it reaches the ground?

15. Find 2' f if 2( ) sin cos cos3 f x x x x

16. Determine the equation of the tangent line that satisfies the following conditions.

a) Tangent to the graph of sin , where 0 2 , y x x x at its point of inflection.

b) Tangent to the curve3 3 9 at (4, 2). x y xy

17. Approximate3 123, to four decimal places, using

a) differentials b) Newton’s Method

18. A closed box with a rectangular base is needed to mail an item that requires 2883in of space. Because of the

unique shape of the item, the length of the base must be three times its width. Find the dimensions of the box

so that the least amount of material will be used.

19. An open right cylinder is inscribed in a sphere of radius 5 meters. Determine the dimensions of the cylinder

that will give maximum surface area.

20. A water trough 10 feet long has a cross section in the shape of an inverted equilateral triangle with an altitude

of 2 feet. Due to a crack at the bottom of the trough, water is leaking out at the rate of31 ft /hour.

a) Determine the rate at which the height of the water is changing when the depth of the water is 1 feet.

b) Determine the rate at which the height of the water is changing when the trough contains310 ft of water.

21. Evaluate:

a)3 3 2(3 4) 3 x x dx

b)

9

1

3

10

xdx

x

22. If

2

3

0

( ) 1 8 , x

F x t dt find '(1).F

23. Determine

0

2

sin

.1

x

d dt

dx t

24. If2"( ) , '(0) 7, and (0) 2, find ( ).G x x G G G x



3

25. Approximate

4

0

1 ,

16dx

x with four subintervals, using

a) Trapezoidal Rule b) Simpson’s Rule

26 Given the region bounded by3 23 3 y x x x and

2 1. y x

a) Find the area of the above region.

b) Find the volume of the solid generated when the region, where1 4 x , is rotated about the -axis. y

27. Find the average value of ( ) sin f x x on the closed interval2

0, .

28. Given the region in the first quadrant bounded by 1 y x and3 1. y x

a) Find the volume of the solid generated when this enclosed region is revolved about the x -axis.

b) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the solid

when the region is revolved about the line 2. x

29. Find the length of the arc of the curve2

3 y x from (1, 1) to (8, 4).

(For #30, 31, and 32) Set up, but do not integrate, an integral that would be used to find the

30. Volume of a solid S, where the base of S is a semi-circular disk described by 2 2( , ) : 4, 0 . x y x y x

Cross sections of S perpendicular to the -axis x are semicircles with diameters on the base.

31. Volume of the solid generated by revolving about the -axis x the region bounded by2 1 y x and the line

3. y x

32. Work done in pumping all the water over the top of a right circular conical tank of altitude 18 feet and radius

6 feet. The tank is full of water weighing362.5 lb/ft .

________________________________________________________________________________________________________

(Answers to Part I)

1. Want to find 0 such that 1325 5

1 x whenever 0 4 . x 13 82 2 25 5 5 5 5

1 4 . x x x

So we want 25

4 x whenever 0 4 x . This will be true if and only if 52

4 . x

Take 52

. If 52

and 0 4 , x then 13 8 52 2 2 2 25 5 5 5 5 5 5 2

1 4 . x x x

Therefore, 1325 5

1 x whenever 0 4 x , and so 1325 5

4lim 1 x

x

by precisendef of the limit.

2. minimum of 111, 3. a) 8 b) 6a c) 0 d) 18

4.22 2

lim ( ) 8; lim ( ) 8; lim ( ) does not exist x x x

f x f x f x

5. a) 2 2,1 1, b) , 2 2,1 1,2 2,

6. a)

31

2 22 2

15

7 2 1

x

x x

b) 2 2sin 2 sec sin x x



4

7. Domain: - ,-3 -3,3 3,

Intercept: (0,0)

Symmetry with respect to origin

VA at -3, 3 x x ; slant asymptote at y x ;

Crossover of slant asymptote at 0 x

Increasing on - ,-3 3 and 3 3,

Decreasing on -3 3, -3 , -3,3 , and 3,3 3

Local max at -3 3; x loc min at 3 3; x

HPI at 0 x

CD on , 3 (0,3); CU on -3,0 3,

PI at 0 x

8.2 2

1 1 1 1lim ; lim

2 24 1 4 1 x x

x x

x x

9. Domain: - ,-1 -1,

Intercept: (0,0) No symmetry

Hole at -1 x

No VA

HA at 1 12 2

- ; ; y y no crossovers

Increasing on - ,-1 and (-1, );

no local extremas

CU on - ,0 ; CD on 0, ; PI at (0,0)

10. 2, 2c 11. (5) 26; g'(5) 6g

12. loc max at 3 5 36 2 6 2

, & , ; loc min at 32 2

,1 & , 3

13. 1 2t 15. 3 16. a) 2 y x b) 54

3 y x

14. a) 40 ft/sec b) 3 194 4

( ) 104 ft/sec; 24 ft/secv v c) going up at 34

t secs; going down at 194

t secs

d) 3 194 4

( ) 104 ft/sec; 24 ft/secv v e) 4 sec f) (4) 256 fts g) 8 sec h) -128 ft/sec

17. 4.9733 18. 4” x 12” x 6” 19.5 2

radius= ;2

height=5 2

20. a) decreasing at3

20 ft/hr b) decreasing at4 3

20 ft/hr

21. a) 2

316

3 4 x C

b) 68 22. 6 23. sec x 24.41

127 2 x x

25. a) 0.22325 b) 0.22313 26. a) 25312

sq. units b) 4145

cu. units 27.2

28. a) 2942

cu. units b)

1

3

0

2 (2 ) ( 1) ( 1)V x x x dx 29. 7.6 30. 2

2

0

42

V x dx

31. 2

22 2

1

3 1V x x dx

32. 18

219

0

18 62.5W y y dy



5

Part II: Multiple Choice

1. If at a certain point '( ) 0, f x which statement concerning the point is always correct?

(A) It is either a relative maximum or a relative minimum.

(B) It is a point of inflection.

(C) It is a point at which the tangent line is parallel to the x -axis.

(D) It is a point at which the tangent line is parallel to the y -axis.

(E) None of these

2. If, for all values of , '( ) 0 and "( ) 0, x f x f x which of the following curves could be part of f ?

A) B) C) D) E)

3. For the function f defined in the adjacent figure,

At which of the points below is "( ) 0? f x

I) A II) B III) C IV) D V) E VI) F

(A) I and IV only (B) II and III only

(C) II, III, & V only (D) VI only

(E) none of these

y

D

E

C x

B A F

4. The radius of a sphere is increasing at the rate of 2 inches per second. When the radius is 10 inches, its volume

is increasing at the rate (in cubic inches per second) of

(A) 800 (B) 80 (C) 3200 (D) 40 (E) none of these

5. The set of all number(s) c in 1,3 satisfying the conclusion of the Mean Value Theorem (for derivatives) for the

function2( ) 5 3 1 f x x x on the interval 1,3 is

(A)3

2

(B)3

10

(C) 2 (D)3

,210

(E) none of these



6

6. y

'( ) y f x

x

-2 3

Let f be a function whose domain is the open interval ( 3, 4) and let the derivative of f have the graph shown

in the figure above. The function f is decreasing on which of the following intervals?

(A) (-3,-1) and (2,4) (B) (-2,0) and (3,4) (C) (-3,-2) and (0,3)

(D) (-1,2) only (E) (0,3) only

7. Referring to the graph of '( ) y f x in problem #6 above, f will have a point of inflection at

I. 2 x II. 1 x III. 0 x IV. 1 x V. 3 x

(A) I, III, & V (B) II & IV (C) II only (D) III only (E) none of these

8. The area of the region bounded by 2 cos y x , tan y x , and the -axis y can be found by

(A) 4

0

2 cos tan x x dx

(B) 2

0

2 cos tan x x dx

(C)

4

0

(tan 2 cos ) x x dx

(D)

2

0

(tan 2 cos ) x x dx

(E) none of these

9. The volume of the solid generated by revolving about the line 1 x the region bounded by 3 y x and the

lines 0 x and 8 y can be found by

(A)

2

3

0

2 ( 1)(8 )V x x dx (B)

2

3

0

2 ( 1)(8 )V x x x dx

(C)

8

3

0

2 ( 1)( 1)V x x dx (D)

8

3

0

( 1)( 2)V x x dx (E) none of these

10. The work done when a particle moves along the x -axis from 1 x to 3 x under the action of a force ( ) f x

pounds and x feet from the origin, where2

( ) (2 1) , f x x is

(A) 316 ft-lb (B)316

3ft-lb (C)

158

2ft-lb (D)

158

3ft-lb (E) none of these

-------------------------------------------------------------------------------------------------------------------------------------------

(Answers to Part II)

1. C 2. E 3. C 4. A 5. C 6. C 7. B 8. A 9. A 10. D

Documents

m7 REVIEW.pdf