Practice Midterm Exam

Question 1

An object is dropped from the top of a tall building. At 2 seconds, it is 64 feet from the top of the building. At 4 seconds, it is 256 feet from the top of the building. What is the average rate the object was traveling in the interval between 2 and 4 seconds?

Answer 1

96 ft / s

Question 2

For which values of k will the line y = x + k meet the parabola of the equation y = −x ^2 + 4x − 8 in two distinct points?

Answer 2

k < −23/4

Question 3

What is the limit of the function in the graph at x = 4?

Answer 3

6

Question 4

Determine, if it exists, limx→1 x^2−2x+1/√x+3 −2

Answer 4

0

Question 5

What is the slope of the tangent line of the function f (x) = 4x ^2 − 2x + 1 at x = 3?

Answer 5

22

Question 6

Consider the function y = x^ 2 − 2x + 1. What is the slope of the tangent line at x = 2?

Answer 6

2

Question 7

The instantaneous rate of change of a ball (in ft/sec) is given by f′(x)=1/√x. When was the ball traveling at a rate of 1/4 ft/sec?

Answer 7

16 sec

Question 8

What is f ’ (x) if f (x) = x^64?

Answer 8

64x^63

Question 9

Compute the derivative of the function

f(x)=x−√x / (x^3−x+3).

Answer 9

(1−1/2x^−1/2)(x^3−x+3)−(x−√x)(3x^2−1)/(x^3−x+3)^2

Question 10

Find the derivative of:

P(t)=(3t^2/3−6t^1/3)^3⋅(3t^2−6t)^1/3

Answer 10

P′(t)=2(3t^2/3−6t^1/3)^3(3t^2−6t)^−2/3(t−1)+6(3t^2−6t)^

1/3(3t^2/3−6t^1/3)^2(t^−1/3−t^−2/3)

Question 11

What is the value of sin (π / 4)?

Answer 11

√2/2

Question 12

What is the derivative of the function

f(x) = e^x/2−tanx/x?

Answer 12

(e^x/2 /2−sec^2x)x−(e^x/2−tanx)/x^2

Question 13

Evaluate the following as true or false.

(ln(−x))′=1/x

Answer 13

true

Question 14

If dy / dx is undefined for a given value of x, then the line tangent to the curve y = f (x) at that value does not exist.

Answer 14

false

Question 15

Find an equation of the tangent line to the curve x^2/a^2−y^2/b^2=1, where a and b are constants, at the point (x0,y0).

Answer 15

x0x/a^2−y0y/b^2=1

Question 16

Below is the graph of a function f (x). Which graph could be the graph of its inverse f −1 (x)?

Answer 16

The graph of f −1 (x) looks like the reflection of the graph of f (x) across the line y = x. Another way to think of this is that whatever is true of the y-coordinates in the graph of f (x) must be true of the x-coordinates in the graph of f −1 (x). Because the graph of f (x) has no negative y-coordinates, the graph of f −1 (x) must not have any negative x-coordinates. Also in the original f (x) graph the y-coordinates decrease as the x-coordinates increase.

Question 17

What is the value of d/dx[f−1(x)] when x=0, given that f(x)=x1−x, and f−1(0)=0?

Answer 17

1

Question 18

Suppose I want to find an inverse to the function |cos x|. I intend to restrict the domain of the function to an interval whose left endpoint is x = 0. On which of the following intervals is |cos x| one-to one?

Answer 18

[0, π / 2]

Question 19

Find the derivative d/dx[arcsec2x].

Answer 19

2/|2x|√4x^2−1

Question 20

What does sech (0) equal?

Answer 20

1