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Midterm Exam Review Sheet 3

Multiple Choice – Non Calculator

1. The graph of y = 3x

2 – x

3 has a relative maximum at

(a) (0,0) only

(b) (1,2) only

(c) (2,4) only

(d) (4, −16) only

(e) (0,0) and (2,4)

2. 8 5 6 4 4 2

9 6 7 5 5 3

10 10 10

10 10 10lim

x

x x x

x x x

(a) 0 (b) 1 (c) −1 (d) 1/10 (e) −1/10

3. The figure to the right shows the graph of the velocity of

a moving object as a function of time. At which of the

marked points is the speed the greatest?

(a) A (d) D

(b) B (e) E

(c) C

4. What are all values of x for which the graph of y = x4

2 is concave downward?

(a) There are no values of x.

(b) x < 4

(c) x > – 4

(d) x < – 4

(e) x > 4

5. The equation of the tangent line to the curve x2 + y

2 = 169 at the point (5, −12) is

(a) 5y – 12x = −120

(b) 5x – 12y = 119

(c) 5x – 12y = 169

(d) 12x + 5y = 0

(e) 12x + 5y = 169

6. If the graph of 2( ) 2k

f x xx

has a point of inflection at 1x , then the value of k is

(a) −2

(b) −1

(c) 0

(d) 1

(e) 2

7. Piecewise functions f and g are shown to the right.

If h(x) = ( )f x ● ( )g x , then 3h

(a) 8

3

(b) 1

3

(c) 0

(d) 2

3

(e) 8

3

8. A particle moves along the x-axis in such a way that its position at time (t) is given by 1

( )1

tx t

t

.

(a) −4 (b) −2 (c) 3

5 (d) 2 (e) 4

9. If 2 2dyx y

dx , then

2

2

d y

dx =

(a) 22xy

(b) 3 34x y

(c) 2 32 2x x y

(d) 2 22 2x y xy

(e) 4 3 22 2x y xy

What is the acceleration of the particle at time t = 0?

10. The average rate of change of the function 1

( ) cos2

f x x

on the closed interval [−4, 0] is

(a) −1

2sin(2) (b) −

1

4 sin(2) (c)

1 cos(2)

4

(d)

1 cos(2)

4

(e)

1 sin(2)

4

11. If 6

2

0

2 2x x dx is approximated by three inscribed rectangles of equal width on the x-axis, then the

approximation is

(a) 24 (b) 26 (c) 28 (d) 48 (e) 76

Name: Date: Block:

Midterm Exam Review Sheet 4

Multiple Choice – Calculator Active

NOTE: The exact numerical value of the correct answer may not always appear among the choices given.

When this happens, select from among the choices the number that best approximates the exact

numerical value.

12. Let f be the function given by f(x) = tan x and let g be the function given by g(x) = x2. At what value

of x in the interval 0 ≤ x ≤ do the graphs of f and g have parallel tangent lines?

(a) 0 (b) 0.660 (c) 2.083 (d) 2.194 (e) 2.207

13. Let 1

( )f tt

for 0t . For what value of t is f t equal to the average rate of change of f on the

closed interval [a, b]?

(a) ab (d) ab

1

(b) ab (e)

ab

11

2

1

(c) ab

1

14. The figure above shows a road running in the shape of a parabola from the bottom of a hill at A to

point B. At B, it changes to a line and continues to on to C. The equation of the road is

2 , From A to B( )

, From BtoC

axR x

bx c

B is 1,000 feet from A and 100 feet higher. Since the road is smooth, R x is continuous. What is the

value of b?

(a) 0.2 (b) 0.02 (c) 0.002 (d) 0.0002 (e) 0.00002

15. The figure above shows the graph of the derivative of a function f. How many points of inflection

does f have in the interval shown?

(a) None (b) One (c) Two (d) Three (e) Four

16. The amount A t of a certain item produced in a factory is given by

A(t) = 4000 + 48(t – 3) – 4(t – 3)3

where t is the number of hours of production since the beginning of the workday at 8:00 a.m. At what

time is the rate of the production increasing most rapidly?

(a) 8:00 am

(b) 10:00 am

(c) 11:00 am

(d) 12:00 noon

(e) 1:00 pm

17. At how many points on the curve 5 4 24 3 15 6y x x x will the line tangent to the curve pass

through the origin?

(a) One

(b) Two

(c) Three

(d) Four

(e) Five

18. The graph of the derivative of a twice differentiable function is shown below.

If f (1) = −2, which of the following must be true?

(a) f (2) < f ′(2) < f ′′(2)

(b) f ′′(2) < f ′(2) < f (2)

(c) f ′(2) < f (2) < f ′′(2)

(d) f (2) < f ′′(2) < f ′(2)

(e) f ′ (2) < f ′′(2) < f (2)

19. The function tan 3xf x has a zero in the interval [0, 1.4]. The derivative at this point is

(a) 0.411 (b) 1.042 (c) 3.451 (d) 3.763 (e) undefined

20. Let f be a function that is everywhere differentiable. The value of f x is given for several values of

x in the table below.

x −10 −5 0 5 10

f x −2 −1 0 1 2

If f ′(x) is always increasing, which statement about f (x) must be true?

(a) f (x) has a relative min at x = 0.

(b) f (x) is concave down for all x.

(c) f (x) has a point of inflection at (0, f (0))

(d) f (x) passes through the origin

(e) f (x) is an odd function

21. The table below gives the values of a differentiable function f. what is the approximate value of 4 ?f

(a) 0.00234

(b) 0.289

(c) 0.427

(d) 2.340

(e) 4f cannot be approximated from the information given.

22. Which graph best represents the position of a particle, s(t), as a function of time, if the particle’s velocity

and acceleration are both positive?

23.

(a) I only (d) I and III only

(b) II only (e) II and III only

(c) I and II only

x f (x)

3.99800 1.15315

3.99900 1.15548

4.00000 1.15782

4.00100 1.16016

4.00200 1.16250