Free Printable AP calc Ch 4A Review

1.

$The\ \min.\ value\ of\ f\left(x\right)=x^2-6x+5\ is$

a)

4

b)

-6

c)

6

d)

-4

2.

The interval in which $y=x^2e^{–x}$ is increasing is

a)

$\left(-2,\ 0\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(2,\ \infty\right)$

d)

$\left(0,\ 2\right)$

3.

Which of the following functions is decreasing on $\left(0,\ \frac{\pi}{2}\right)$

a)

$\sin2x$

b)

$\tan x$

c)

$\cos x$

d)

$\cos3x$

4.

y = x (x – 3)² decreases for the values of x given by :

a)

$1<x<3$

b)

$x<0$

c)

$x>0$

d)

$0<x<\frac{3}{2}$

5.

The smallest value of the polynomial

x^3-18x^2+96x

in [0, 9] is

a)

126

b)

0

c)

135

d)

160

6.

The absolute maximum and the absolute minimum values of the function $f\left(x\right)=4x-\frac{x^2}{2}$ in [-2, 4.5] is

a)

8, -10

b)

4, - 2

c)

0, 3

d)

3, 6

7.

What is a vertical tangent?

a)

Has a positive slope.

b)

Has a negative slope.

c)

Has a slope of zero.

d)

Has an undefined slope.

8.

If a function has a derivative that is negative, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The function is constant

d)

Neither increasing nor decreasing

9.

If a function has a second derivative that is positive, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The function is concave up

d)

The function is concave down

10.

If f '(x) <0, what is true about f(x)?

a)

it is concave up there

b)

it's decreasing

c)

It's increasing

d)

it is concave down there

11.

a)

-1/4

b)

1/4

c)

π/4

d)

-π

12.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

13.

a)

a

b)

b

c)

c

d)

d

14.

a)

a

b)

b

c)

c

d)

d

15.

a)

1/2

b)

-1/2

c)

0

d)

3/2

16.

a)

A

b)

B

c)

C

d)

D

17.

Let f be the function given by f(x) = x³. What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [-1, 2]? (No calculator)

a)

0 only

b)

1 only

c)

√3 only

d)

-1 and 1

18.

If $x=0.5$ is a critical point of $f\left(x\right)$ and $f''\left(x\right)\ =\ \cos^2x\cdot\ln x$ then $x=0.5$ is a

a)

local maximum

b)

local minimum

c)

neither

19.

Which of the following is NOT a critical point of $f\left(x\right)\ =\ e^x\cdot x^2$ ?

a)

$x=0$

b)

$x=1$

c)

$x=-2$

20.

Where is the point of inflection for the function $f\left(x\right)\ =\ x^{3\ }+6x^2$ ?

a)

$x=0$

b)

$x=-4$

c)

$x=-2$

d)

$x=4$

21.

The following sign chart shows whether $f'\left(x\right)$ is positive, negative, or zero.
There is/are ...

a)

a local maximum at $x=-2$

b)

a local maximum at $x=4$

c)

local maxima at $x=-2$ and $x=4$

d)

no local extrema

22.

If $f\left(x\right)\ =\ \sqrt{1-x^2}$ ,

then $f'\left(1\right)$ is...

a)

$f\left(x\right)$ is not defined at $x=1$

b)

$1$

c)

$\frac{2}{\sqrt{3}}$

d)

$f'\left(1\right)\$ does not exist

23.

Critical numbers occur where the first derivative is what? (Check all that apply).

a)

endpoint

b)

zero

c)

maximum

d)

undefined

24.

Where does the global maximum of $f\left(x\right)\ =\ e^{-x^2}$ occur on $\left[-1,2\right]$ ?

a)

$x=-1$

b)

$x=0$

c)

$x=1$

d)

$x=2$

25.

Find the inflection point(s) of $f\left(x\right)\ =\ e^x+2x^2-3x$ .

a)

$x=\frac{3}{2}$

b)

$x=0,\ x=\frac{3}{2}$

c)

$x=0$

d)

$f\left(x\right)$ has no inflection points.

26.

Over what interval(s) is f(x) increasing?

a)

before -3 and after 1

b)

between -3 and 1

c)

(-5, 0) ∪ (2, ∞)

d)

(-5, ∞)

27.

For which value of x is f ' positive and increasing?

a)

a

b)

b

c)

c

d)

d

e)

e

28.

On the interval [−2, 2] the function f (x) = x⁴

a)

(a) has an absolute maximum but no local maxima.

b)

(b) has an absolute maximum at an interior point of the interval.

c)

(c) has a local minimum but no absolute minimum.

29.

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

a)

There is a local max at x=3

b)

There is a local min at x = 3

c)

There is an inflection point at x = 3

d)

There is an x-intercept at x = 3

30.

Identify the conclusion of the Mean Value Theorem.

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f'\left(c\right)=0$

31.

Identify ALL conditions of the Mean Value Theorem.

a)

f must be continuous on [a, b]

b)

$f\left(a\right)=f\left(b\right)$

c)

derivative must equal the slope of secant connecting the endpoints

d)

f must be differentiable on (a, b)

32.

$\frac{d}{dx}\left(\frac{\sin x}{1+\cos x}\right)=$

a)

$\frac{\cos x}{-\sin x}$

b)

$\frac{-\sin^2x-\cos x+\cos^2x}{\left(1+\cos x\right)^2}$

c)

$\frac{1}{1+\cos x}$

d)

$\frac{-\cos^2x+\cos x+\sin^2x}{\left(1+\cos x\right)^2}$

33.

f(x) = x²(x+1)²
Which of these is the linear approximation to f centered at x = 1?

a)

L(x) = 12x - 1

b)

L(x) = 6x + 5

c)

L(x) = 12x - 8

d)

L(x) = 12x - 16

34.

The function f is twice differentiable with f(2) = 1, f'(2)=4, and f''(2)=3. What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x = 2?

a)

0.4

b)

0.6

c)

0.7

d)

1.4

35.

Using the tangent line approximation for

f\left(x\right)=\sqrt{x}

at x = 9, approximate

\sqrt{8.2}

.

a)

3.133

b)

2.867

c)

2.863

d)

2.733

e)

2.712

36.

7.

a)

(A)

b)

(B)

c)

(C)

d)

(D)

37.

f' is given, which could be f?

a)

A

b)

B

c)

C

38.

Using this graph of f’(x), find the slope of the line tangent to f(x) at point c

a)

0

b)

1

c)

-1

d)

Not enough information

39.

f''(x) is pictured. Which x values are inflection points of f(x)?

a)

x=-5 and -1

b)

x=-3

c)

x=0

d)

no inflection points

40.

a)

2x

b)

3

c)

6

d)

9

41.

The function shown is the graph of f '(x), the derivative of f(x). The domain is [-3,5]. On which interval is the graph of f(x) concave upward?

a)

(-1,1) & (3,5]

b)

[-3,-2) & (4,5]

c)

[-3,-1) & (1,3)

d)

(-2,1) & (1,4)

42.

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec. How fast is the diameter changing when the diameter is 8 cm?

a)

-1/(16pi) cm/sec

b)

-1/16 cm/sec

c)

-1/(4pi) cm/sec

d)

1/(4pi) cm/sec

43.

(3 min) The area of a circular region is increasing at a rate of 96 π square meters per second. When the area of the region is 64 π square meters, how fast, in meters per second, is the radius of the region increasing?

a)

6

b)

8

c)

16

d)

4√3

44.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate are we looking for and when?

a)

$\frac{dA}{dt}\ when\ C=100π$

b)

$\frac{dC}{dt}\ when\ A\ =\ 100π$

c)

$\frac{dA}{dt}\ when\ A\ =\ 100π$

d)

$\frac{dC}{dt}\ when\ r\ =\ 100π$

45.

What is the derivative of circumference with respect to time?

a)

$\frac{dC}{dt}=2π\cdot\frac{dr}{dt}$

b)

$\frac{dC}{dt}=2π$

c)

$\frac{dC}{dt}=2πr$

d)

$\frac{dC}{dt}=π\cdot\frac{dr}{dt}$

46.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

At what rate is the radius changing when the circumference is 100π ft?

a)

$\frac{dr}{dt}=\frac{20}{π}$

b)

$\frac{dr}{dt}=\frac{50}{π}$

c)

$\frac{dr}{dt}=50$

d)

$\frac{dr}{dt}=20$

47.

A tank is in the form of an inverted cone having an altitude of 10 ft and a radius of 5 feet. Water is flowing into the tank at the rate of 1 ft³/min. How fast is the water level rising when the water is 3 ft deep?

a)

3.4 ft/min

b)

.14 ft/min

c)

1 ft/min

d)

1/2 ft/min

48.

You want to make a box to contain dirt and your pet earthworm. Using a 7 in by 10 in rectangle of cardboard, you cut congruent squares from the corners and fold up the sides.

Choose the equation would you use in order to do Calculus to find the maximum volume of dirt (including worm) the box can hold?

a)

$V=\left(7-2x\right)\left(10-2x\right)$

b)

$V=x\left(7-2x\right)\left(10-2x\right)$

c)

$V=x\left(7-x\right)\left(10-x\right)$

d)

$V=x\left(7+2x\right)\left(10+2x\right)$

49.

A rectangle is bounded by the x-axis and the parabola $y=12-x^2$ . What length and width should the rectangle have so that its area is a maximum?

Given the constraint equation above and the optimization equation $A=2xy$ , choose the DERIVATIVE of the merged (combined) equation.

a)

$A'=2xy$

b)

$A'=\ \ 24-6x^2$

c)

$A'=2x\left(12-x^2\right)$

d)

$A'=12-2x$

50.

If y=2x-8, what is the minimum value of the product xy?

a)

-16

b)

-8

c)

-4

d)

2

AP calc Ch 4A Review

Total questions: 50

Worksheet time: 2hrs 21mins

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