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AP Calculus AB > Midterm Exam > Flashcards

Flashcards in Midterm Exam Deck (20)
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1
Q

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?

A

1,520 ft / s

2
Q

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 ?

A

M = (1/2, 3/2)

3
Q

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

A

The limit does not exist.

4
Q

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

A

The limit does not exist.

5
Q

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

A

12x^ 2 + 6x

6
Q

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

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

A

9

7
Q

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?

A

7 ft / min

8
Q

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

A

−12/x^2

9
Q

Compute the derivative of the function

f(x)=1/(x^2−1)^2

A

−4x/(x^2−1)^3

10
Q

Find the derivative of:

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

A

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

11
Q

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

A

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

12
Q

What is the derivative of the function

f(x) =sinx(1+e−x)/1+x?

A

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

13
Q

Evaluate the following as true or false.

(ln5)′=1/5

A

false

14
Q

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

A

true

15
Q

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

A

(2, 2) and (−2, −2)

16
Q

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

A

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.

17
Q

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?

A

1/4

18
Q

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

A

false

19
Q

What is sinh (ln 2) equal to?

A

3/4

20
Q

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

A

x = c

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