Midterm Exam

Question 1

A projectile is fired straight up into the air. At 1 second, it has reached a height of 1,584 feet. At 4 seconds, it has reached a height of 6,144 feet. What is the average rate the projectile was traveling in the interval between 1 and 4 seconds?

Answer 1

1,520 ft / s

Question 2

The line of the equation y = x + 1 and the parabola of equation y = x ^2 − 1 intersect at two points, P and Q. What are the coordinates of the midpoint M of the segment PQ ?

Answer 2

M = (1/2, 3/2)

Question 3

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

Answer 3

The limit does not exist.

Question 4

Determine, if it exists, limx→2 √x+7−3/x^2−4x+4

Answer 4

The limit does not exist.

Question 5

What is the derivative of the function f (x) = 4x ^3 + 3x ^2 − 2 at x ?

Answer 5

12x^ 2 + 6x

Question 6

Consider the function y = x^ 2 − x + 7.

At what value of y is the slope of the tangent line equal to 3?

Answer 6

9

Question 7

Let f′(x)=3x^2+4x define the instanta-neous rate of change (in ft/min) of a car moving along the x-axis. What is the instantaneous rate of change at time 1 min?

Answer 7

7 ft / min

Question 8

What is the derivative of the function f(x)=12/x?

Answer 8

−12/x^2

Question 9

Compute the derivative of the function

f(x)=1/(x^2−1)^2

Answer 9

−4x/(x^2−1)^3

Question 10

Find the derivative of:

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

Answer 10

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

Question 11

How do you get the graph of sin (2x) from that of sin x ?

Answer 11

Shrink the graph of sin x horizontally by a factor of 2

Question 12

What is the derivative of the function

f(x) =sinx(1+e−x)/1+x?

Answer 12

(1+x)[(1+e^−x)cosx+(−e^−x)sinx]−(1+e^−x)sinx/(1+x)^2

Question 13

Evaluate the following as true or false.

(ln5)′=1/5

Answer 13

false

Question 14

If dy / dx = 0 for a given value of x, then the line tangent to the curve y = f (x) at that value is horizontal

Answer 14

true

Question 15

Find all the points on the curve x ^2 − xy + y^ 2 = 4 where the tangent line has a slope equal to −1.

Answer 15

(2, 2) and (−2, −2)

Question 16

Below are the graphs of four functions. Which function is invertible?

Answer 16

For a function to be invertible, it must pass the horizontal line test: any horizontal line must pass through one and only one point on the graph of the function. This is the only graph that satisfies this condition.

Question 17

What is the value of d/dx[f−1(x)]when x=2, given that f(x)=x^3+x and f−1(2)=1?

Answer 17

1/4

Question 18

Evaluate the following as true or false. arccotx=1/arctanx, as long as arccotx and arctanx are both defined.

Answer 18

false

Question 19

What is sinh (ln 2) equal to?

Answer 19

3/4

Question 20

Which of the following is not an equivalent statement of  “x approaches c ?”

Answer 20

x = c