Free Printable Quarter 2 Exam Review BC Calculus

1.

Find the derivative of $f\left(x\right)\ =\ \tan\left(5x\right)^2$

a)

$f'\left(x\right)=10\tan\left(5x\right)$

b)

$f'\left(x\right)=10\sec^2\left(5x\right)\tan\left(5x\right)$

c)

$f'\left(x\right)=2\sec^2\left(x\right)\tan\left(x\right)$

d)

$f'\left(x\right)=10\sec^2\left(5x\right)$

2.

Given the point (0,2) occurs on the function f(x) = $e^x+\cos\left(x\right)$ , find the equation of the tangent line at that point.

a)

y + 2 = x

b)

y = x + 2

c)

y = 2x

d)

y = 1/2x

3.

Find the derivative with respect to x in the equation

$y^3+y^2-5y-x^2=-4$

a)

$\frac{dy}{dx}=\frac{3y^2+2y-5}{2x}$

b)

$\frac{dy}{dx}=\frac{2x-4}{3y^2+2y-5}$

c)

$\frac{dy}{dx}=\frac{2}{3y+2}$

d)

$\frac{dy}{dx}=\frac{2x}{3y^2+2y-5}$

4.

Suppose the radius of a circle is expanding at 2 meters per second. At 3 seconds, what rate is the Area increasing by?

a)

$24\pi\ \frac{m}{s}$

b)

$24\pi\ \frac{m^2}{s}$

c)

$24\ \frac{m^2}{s}$

d)

$12\pi\ \frac{m}{s}$

5.

$\lim_{x\rightarrow0}\frac{\sin\left(8x\right)}{2x}$

a)

4

b)

0

c)

8

d)

DNE

6.

Using the Mean Value Theorem, at what x-value does instantaneous rate of change equal the average rate of change for the function $f\left(x\right)=x^4-8x$ over the interval [0,2]?

a)

0

b)

$2^3$

c)

$2^{\frac{1}{3}}$

d)

DNE

7.

Use the first derivative to determine the interval at which $f\left(x\right)=\left(x+2\right)^{\frac{2}{3}}$ is decreasing.

a)

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

b)

$\left(-\infty,-2\right]$

c)

$\left[-2,\infty\right)$

d)

DNE

8.

Given the the function $F\left(t\right)=10t^3-120t^2+450t+1000$ determine the absolute minimum value on the closed interval [0,8]

a)

1000

b)

1540

c)

1500

d)

2040

9.

Solution to the differential equation

$\frac{dy}{dx}=1+2y$ with the initial condition $f\left(0\right)=1$ . What is the approxiamation for f(1) if Euler's method is used, starting at x = 0 with a step size of 0.5?

a)

2.5

b)

3.5

c)

4.0

d)

5.5

10.

Which Slope field is equivalent to equation,

$\frac{dy}{dx}=\frac{x}{y}$ ?

a)

b)

c)

d)

11.

Find the integral.

a)

(6e⁵ – 1) / 25

b)

(4e⁵ + 1) / 25

c)

(1 – e³) / 3

d)

e⁴

12.

Use a trapezoidal sum with four intervals from the table to estimate the integral of 0 to 9 of V(t).

a)

155/2

b)

128/2

c)

186/2

d)

151/2

13.

Find h'(3)

a)

-2

b)

0

c)

1

d)

3

14.

a)

A

b)

B

c)

C

d)

D

15.

The result of $\int_1^3\left(x-2\right)\ dx$ is:

a)

positive

b)

negative

c)

zero

16.

What is the slope of the line tangent to the curve $x^2+2xy+3y^2=2$ when y = 1?

a)

$-\frac{1}{8}$

b)

-1

c)

0

d)

$\frac{1}{8}$

17.

$\int_{-1}^1xe^{x^2}dx\ =$

a)

$-\frac{e}{2}$

b)

0

c)

$\frac{e}{2}$

d)

e

18.

$\int_{ }^{ }\ \frac{x}{x^2+5x+6}dx\ =$

a)

$\ln\left|\frac{\left(x+3\right)^2\ }{\left(x+2\right)^2}\right|+C$

b)

$\ln\left|\left(x+3\right)^3\left(x+2\right)^2\right|+C$

c)

$\ln\left|\frac{\left(x+2\right)^2\ }{\left(x+3\right)^3}\right|+C$

d)

$\ln\left|\left(x+3\right)^2\left(x+2\right)^3\right|+C$

19.

$\frac{d}{dx}\left(x^2\sin^2x\right)$

a)

2xsin2x

b)

2x $\cos^2x$

c)

$2x\sin^2x+x^2\cos^2x$

d)

$2x\sin^2x+2x^2\sin x\cos x$

20.

$y=\frac{x^2-1\ }{x^{2\ }+1}$

The line normal (perpendicular) to the curve above at x = 2 has a slope

a)

$-\frac{8}{25}$

b)

$-\frac{25}{8}$

c)

1

d)

$\frac{8}{25}$

21.

$\int_1^{e^2}x\ln xdx\ =$

a)

$\frac{e^2\ }{4}-\frac{1}{4}$

b)

$\frac{5e^4\ }{4}+\frac{1}{4}$

c)

$\frac{3e^2\ }{4}+\frac{1}{4}$

d)

$\frac{3e^4\ }{4}+\frac{1}{4}$

22.

$\int_4^{\infty}\ \ \frac{dx\ }{x^2+16}$

a)

$\frac{\pi}{16}$

b)

$\frac{\pi}{4}$

c)

$\frac{\pi}{2}$

d)

Divergent

23.

What is the equation of the line tangent to the graph of

$y=\sin^2x$ at $x=\frac{\pi}{4}$

a)

$y-\frac{1}{2}=-\left(x-\frac{\pi}{4}\right)$

b)

$y-\frac{1}{2}=\left(x-\frac{\pi}{4}\right)$

c)

$y-\frac{1}{\sqrt{2}}=\left(x-\frac{\pi}{4}\right)$

d)

$y-\frac{1}{2}=\frac{1}{2}\left(x-\frac{\pi}{4}\right)$

24.

$\int_2^{\infty}\ \frac{dt\ }{t^3}$

a)

0

b)

$\infty$

c)

$\frac{1}{8}$

d)

$-\frac{1}{2}$

25.

If

f\left(x\right)=\frac{x^2+5x-24\ }{x^2+10x+16}

, then the

\lim_{x\rightarrow-8}f\left(x\right)=

a)

0

b)

$-\frac{3}{2}$

c)

$\frac{11}{6}$

d)

Nonexistent

26.

A rock is thrown straight upward with an initial velocity of 50m/s from a point 100m above the ground. If the acceleration of the rock at any time is $a=-10\ \frac{m}{s^2\ }$ , what is the maximum height of the rock (in meters)?

a)

150

b)

175

c)

200

d)

225

27.

The average value of $\sec^2x$ on the interval $\left[\frac{\pi}{6},\frac{\pi}{4}\right]$

a)

$\frac{12\sqrt{3}-12}{\pi}$

b)

$\frac{12-4\sqrt{3}}{\pi}$

c)

$\frac{6\sqrt{2}-6}{\pi}$

d)

$\frac{6-6\sqrt{2}}{\pi}$

28.

Given the logistic differential equation $\frac{dz}{dt}=z\left(4-\frac{z}{100}\right)$ , where z(0) = 50, what is the $\lim_{z\rightarrow\infty}z\left(t\right)?$

a)

400

b)

200

c)

100

d)

4

29.

The graph of f is shown in the figure above. If $g\left(x\right)=\int_0^xf\left(t\right)dt$ , for what positive x-value of x does g(x) have a minimum?

a)

1

b)

2

c)

3

d)

4

30.

The side of a square is increasing at a constant rate of 0.5 cm/sec. In terms of the perimeter P, what is the rate of change of the area of the square in $\frac{cm^2}{\sec}?$

a)

0.05P

b)

0.2P

c)

0.4P

d)

6.4P

31.

What is the trapezoidal approximation of

$\int_0^3\ e^xdx$ using n = 4 subintervals?

a)

6.407

b)

19.972

c)

27.879

d)

34.944

32.

If $\int_{-2}^4\ f\left(x\right)dx=a$ and $\int_3^4\ f\left(x\right)dx=b$ , then $\int_3^{-2}\ f\left(x\right)dx\ =$

a)

a-2b

b)

a-b

c)

b-a

d)

2b-a

33.

Let R be the region in the first quadrant between the graphs of $y=e^{-x},\ y=\sin x$ , and the y-axis. The volume of the solid that results when R is revolved about the x-axis is

a)

-0.888

b)

-0.869

c)

0.869

d)

0.888

34.

Use Euler's method with h = 0.2 to estimate y(2.6), if $\frac{dy}{dx}=2y-4x$ and y(2)=6

a)

7.76

b)

8.944

c)

9.112

d)

8.653

35.

$\lim_{h\rightarrow0}\ \frac{\tan^{-1}\left(1+h\right)-\frac{\pi}{4}}{h}$ =

a)

2

b)

$\frac{4}{4+\pi^2}$

c)

$\frac{1}{2}$

d)

Nonexistent

36.

Given $x^2y+x^2=y^2+1,\ find\ \frac{d^2y\ }{dx^2}$ at (1,1)

a)

36

b)

12

c)

-12

d)

-36

37.

The value of c that satisfies the Mean Value Theorem for derivatives on the interval [0,5] for the function $f\left(x\right)=x^3-6x$

a)

$-\frac{5}{\sqrt{3}\ }$

b)

0

c)

1

d)

$\frac{5}{\sqrt{3}}$

38.

The slope field shown above corresponds to which of the following differential equations?

a)

\frac{dy}{dx}=\frac{2x}{y}

b)

$\frac{dy}{dx}=2xy$

c)

$\frac{dy}{dx}=\frac{y}{x}$

d)

$\frac{dy}{dx}=x+y$

39.

Equals

a)

_a∫^cf(x)dx

b)

_a∫^bf(x)dx

c)

_b∫^cf(x)dx

d)

_c∫^af(x)dx

40.

a)

Intermediate Value Theorem

b)

Rolle's Theorem

c)

Average Rate of Change

d)

Average Value of f

41.

Solution to the differential equation

$\frac{dy}{dx}=1+2y$ with the initial condition $f\left(0\right)=1$ . What is the approxiamation for f(1) if Euler's method is used, starting at x = 0 with a step size of 0.5?

a)

2.5

b)

3.5

c)

4.0

d)

5.5

42.

Which Slope field is equivalent to equation,

$\frac{dy}{dx}=\frac{x}{y}$ ?

a)

b)

c)

d)

43.

Find the region bounded by the curve

$y=\sqrt{x}$ , the x-axis and the line x = 4 is revolved about the x-axis to generate a solid. (Select two answers)

a)

$\pi\int_0^4xdx\$

b)

$2\pi\int_0^4xdx\$

c)

$\pi\int_0^2y\left(4-y^2\right)dy$

d)

$2\pi\int_0^2y\left(4-y^2\right)dy$

44.

Find the indefinite integral of $\int e^{2x}dx$

a)

$2e^{2x}+C$

b)

$\frac{1}{2}e^{2x}+C$

c)

$\frac{1}{4}e^{2x}+C$

d)

$2e^x+C$

45.

Find the indefinite integral of

a)

$\frac{1}{3}\operatorname{arcsec}x+C$

b)

$\frac{1}{3}\arcsin x+C$

c)

$\frac{1}{3}x^{-3}+C$

d)

$\frac{1}{3}\arctan x+C$

46.

Put your answer in fraction form.
$\lim_{x\rightarrow2}\left(\frac{x-2}{x^2-7x+10}\right)$

(a)

47.

If the length L of a rectangle is decreasing at a rate of 2 inches per minute while its width W is increasing at a rate of 2 inches per minute, which of the following must be true about the area A of the rectangle?

a)

A is always increasing

b)

A is always decreasing

c)

A is increasing only when L>W

d)

A is increasing only when L<W

e)

A remains constant

48.

The function $f\left(x\right)=\left(1-\sin x\right)^2$ is concave up at $x=\frac{\pi}{6}$ ? What is the estimate for $f\left(0.5\right)$ using the local linear approximation for $f$ at $x\ =\ \frac{\pi}{6}$ ? (round your answer to 3 decimal places)

(a)

49.

What method would you use here?

a)

antiderivatives

b)

u-substitution

c)

integration by parts

d)

slope fields

50.

What would you choose for your u here if you used integration by parts?

a)

t

b)

3t

c)

e^2t

d)

e^t

51.

a)

b)

c)

d)

52.

Let $f\left(x\right)=x^3-7x^2+25x-39$ and let $g$ be the inverse function of $f$ . What is the value of $g'\left(0\right)$ ?

a)

25

b)

10

c)

$\frac{1}{25}$

d)

$\frac{1}{10}$

53.

Oil flows into a concrete conical storage pit at the rate of 10 cubic feet per minute. The pit was built point down and has a depth of 15 feet and a ground level radius of 9 feet. How fast, in feet per minute, is the oil level rising when the oil is 10 feet deep?

a)

0.05

b)

0.09

c)

0.13

d)

0.44

54.

Let $f$ be the function defined above, where $a$ and $b$ are constants. If $f$ is differentiable at $x=0$ , what is the value of $a+b$ ?

a)

-3

b)

-2

c)

0

d)

2

55.

Let $f$ be a function whose derivative is given by $f'\left(x\right)=\frac{x}{15}+\sin\left(e^{0.2x}\right)$ . Which of the following is the approximate $x-$ value of a relative maximum point on the graph of $f$ ?

a)

2.830

b)

6.378

c)

8.673

d)

10.332

56.

Let $f$ be a function whose derivative is given by $f'\left(x\right)=\frac{x}{15}+\sin\left(e^{0.2x}\right)$ . Which of the following is the approximate $x-$ value of a relative maximum point on the graph of $f$ ?

a)

2.830

b)

6.378

c)

8.673

d)

10.332

57.

State the conditions under which the Squeeze Theorem can be applied to find the limit of a function.

a)

The Squeeze Theorem can be applied without knowing the behavior of other functions near the limit point

b)

The Squeeze Theorem can be applied to find the limit of a function when two other functions, one upper and one lower, are known to be close to the function of interest near the limit point. The upper and lower functions must converge to the same limit as the function of interest as they approach the limit point.

c)

The Squeeze Theorem can be applied when the upper and lower functions do not converge to the same limit as the function of interest

d)

The Squeeze Theorem can be applied when the functions are not close to the function of interest near the limit point

58.

Which of the following functions can be used in the Squeeze Theorem?

a)

a) Polynomial functions only.

b)

a) Polynomial functions only.

c)

c) Rational functions only.

d)

d) Any functions as long as they satisfy certain conditions.

59.

a)

IVT

b)

EVT

c)

MVT

60.

TRUE OR FALSE? If f is continuous on [-1,1], f(-1)=4 and f(1)= -2, then there is a zero between -1 and 1.

a)

TRUE

b)

FALSE

c)

CANNOT BE DETERMINED

Quarter 2 Exam Review BC Calculus

Total questions: 60

Worksheet time: 8hrs 40mins

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