Resources

Resources

Find derivatives of sine and cosine functions

Which of the following functions are continuous at

x = 1

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f(x) = e^x - 1

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g(x) = \ln(e^x - 1)

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1

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f

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g

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(Choice C) Both

f

g

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(Choice D) Neither

f

g

Find the unknown number in each equation:

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4n=2\cdot12

Find the magnitude and direction angle

\theta

\vec{v}+ \vec{w}

. Round your final answer to the nearest tenth. It's okay to round your intermediate calculations to the nearest hundredth.

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\begin{array}{l} || \vec{v}+ \vec{w}||~~,\ \theta~~ \end{array}

Logarithmic Functions:

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f(x)=a\log_{b}(x-h)+k

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f(x)=\log_{b}x

b > 1

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Vertical Asymptote:

Consider the surface integral

\int_{(S)} xyz \, dS

S

is the cone with parametric equations

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x=u\cos v

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y=u\sin v

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z=u

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0 \leq u \leq 2

0 \leq v \leq \frac{\pi}{2}

\begin{array}{l}f^{\prime}(x)=8 x^{3}-12 x \text { and } f(1)=1 \\ f(-2)=\end{array}

Find the volume of the solid generated when the region enclosed by

y=-\sqrt{x+1}, y=0

x=2

is revolved about the

\mathrm{x}

Find the particular solution,

y=f(x)

, to the differential equation

\frac{d y}{d x}=\frac{\cos x}{y}

f\left(\frac{3 \pi}{2}\right)=-1

Evaluate the expression, given functions

f

g

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\begin{array}{l} f(x)=3 x-2 \\ g(x)=5-x^{2} \end{array}

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3 f(1)-4 g(-2)=

\begin{array}{r}f(x)=8 e^{x \cos x} \\ f^{\prime}(x)=\end{array}

f(x)

g(x)

are differentiable. The function

h(x)

h(x)= \frac{f(x)}{g(x)}

f(-1)= 6

f'(-1)= -3

g(-1)= 2

g'(-1)= 5

h'(-1)

Find the following values of the function

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\begin{array}{l} f(x)=\left\{\begin{array}{ll} x+2 & x \leq 5 \\ 6-x & x>5 \end{array}\right. \\ f(2)=\square \\ f(5)=\square \\ f(11)=\square \end{array}

Determine whether or not

F

is a conservative vector field. If it is, find a function

f

F=f

. (If the vector field is not conservative, enter DNE.

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\begin{cases} F(x,y)=e^{x}\sin(y)i+e^{x}\cos(y)j \end{cases}

f(x,y)=

\cos x=\frac{3}{5}, x

I

, then find (without finding

x

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\begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}

\sin x=\frac{2}{3}, x

I

, then find the exact answers for the following (without finding

x

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\begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

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Multi-select Choices:

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y = 1 - x

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y = x

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x - y = 1

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x = 1

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y = -x - 1

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

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Multi-select Choices:

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y = 0.25x

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y = \frac{2}{5}

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x - y = 25

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y = 25 - 5x

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y = 5^x

x^{4} \leq f(x) \leq x^{2}

x

[-1,1]

x^{2} \leq f(x) \leq x^{4}

x < -1

x > 1

, at what points

c

do you automatically know

\lim_{x \rightarrow c}f(x)

? What can you say about the value of the limit at these points?

Find the equation of the normal to the curve.

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\frac{dy}{dx}=1.84e^{0.5 x} \text{ at } x=1

The derivative of a function

f

f^{\prime}(x)=\ln |x| \cdot x

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On which interval is the graph of

f

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Use a graphing calculator.

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1

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-1<x<0

x>1

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x>0

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x<0

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x<-0.368

x>0.368

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(E) All real numbers

The second derivative of a function

h

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h^{\prime \prime}(x)=e^{x^{2}}-\sin (x) \text {. }

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On which interval is the graph of

h

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Use a graphing calculator.

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1

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x<0.395

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x>0.395

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x<0

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x>0

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(E) All real numbers

The derivative of a function

h

h^{\prime}(x)=x^{2}-10 \sin (x)

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On which intervals is the graph of

h

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Use a graphing calculator.

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1

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\begin{array}{l}(-\infty,-3.837] \text { and } \\ {[-1.977,1.306]}\end{array}

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(-\infty, 0]

[2.479, \infty)

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[0,2.479]

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(E) All real numbers

Nyala is bungee jumping. The following function gives her elevation, in meters,

t

seconds after jumping:

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E(t)=\frac{80 \sin (0.6 t)}{t}+15

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What is the instantaneous rate of change of Nyala's elevation after

5

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1

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-9

96

meters per second

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-9

96

seconds per meter

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\mathbf{1 7 . 2 6}

meters per second

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\mathbf{1 7 . 2 6}

seconds per meter

The following function gives the temperature (in degrees Celsius) at the beach in Miami, Florida,

t

hours after midnight on a certain day:

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M(t)=-6 \cdot \sin \left(\frac{\pi}{12} t\right)+18

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What is the instantaneous rate of change of the temperature at

9

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1

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1

11

degrees Celsius per hour

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\mathbf{1 . 1 1}

degrees Celsius

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\mathbf{1 3 . 7 6}

degrees Celsius per hour

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\mathbf{1 3 . 7 6}

degrees Celsius

The side of the base of a square pyramid is increasing at a rate of

6

meters per minute and the height of the pyramid is decreasing at a rate of

1

meter per minute.

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At a certain instant, the base's side is

3

meters and the height is

9

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What is the rate of change of the volume of the pyramid at that instant (in cubic meters per minute)?

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1

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\mathbf{1 1 1}

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\mathbf{1 0 5}

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-105

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-111

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The volume of a square pyramid with base side

s

h

\frac{1}{3} s^{2} h

Find the value of

c

(x+1)

is a factor of the polynomial

p(x)

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\begin{array}{l} p(x)=5 x^{4}+7 x^{3}-2 x^{2}-3 x+c \\ c= \end{array}

The light from the moon, in lux, on the night of the

t^{\text {th }}

2016

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L(t)=0.25-\sin \left(\frac{2 \pi(t-2)}{28.5}\right) .

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What is the period of the light from the moon? Give an exact answer.

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U=\frac{1}{2} k x^{2}

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The elastic energy of an object,

U

, is determined by its absolute compression,

x

, and the spring constant of the object,

k

, as shown in the formula. Which of the following correctly shows the object's absolute compression in terms of its elastic energy and spring constant?

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1

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x=\sqrt{\frac{k U}{2}}

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x=\sqrt{\frac{2 U}{k}}

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x=\frac{\sqrt{2 U}}{k}

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x=\left(\frac{2 U}{k}\right)^{2}

h(t)

2

over every unit interval in

t

h(0) = 0

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Which could be a function rule for

h(t)

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h(t) = 2^t

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h(t) = -t + 2

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h(t) = 2t

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h(t) = \frac{-t}{2}

f^{\prime}(x)=2 f(x)

f(1)=5

f(3)=m e^{n}

for some integers

m

n

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m

n

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\begin{array}{l} m= \square \\ n= \square \end{array}

\begin{array}{l}f^{\prime}(x)=-\frac{320}{x^{6}} \text { and } f(1)=30 . \\ f(-2)=\end{array}

\begin{array}{l} f(3)=-1 \text { and } f^{\prime}(3)=0 \\ g(x)=x^{3} \end{array}

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\frac{d}{d x}[f(x) \cdot g(x)]

x=3

\begin{array}{l}F(x)=\int_{-2}^{\sin (x)} 3 t d t \\ F^{\prime}(x)=\end{array}

\begin{array}{l}H(x)=3 x^{2}-6 \\ h(x)=H^{\prime}(x) \\ \int_{-1}^{0} h(x) d x=\end{array}

\begin{array}{l}H(x)=\left|\frac{x}{2}\right| \\ h(x)=H^{\prime}(x) \\ \int_{-6}^{-2} h(x) d x=\end{array}

The height of a river above a fixed point on the riverbed was monitored over a

7

-day period. The height of the river,

H

t

days after monitoring began, was given by

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H=\frac{\sqrt{t}}{20}(20+6t-t^{2})+17\quad 0 \leq t \leq 7

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H

has a stationary value at

t=\alpha

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(a) use calculus to show that

\alpha

satisfies the equation

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5\alpha^{2}-18\alpha-20=0

f(x)=x

g(x)=-x

[f \circ g ](2)

[g \circ f] (2)

Write down the definite integral that represents the enclosed area AND find the area.

f(x)=-(x-3)(x+2)

, the vertical axis and the horizontal axis

f

f(x)=5 x^{6}-2 x^{3}-3

. Which of the following describes the end behavior of

f

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\lim _{x \rightarrow-\infty} f(x)=-\infty

\lim _{x \rightarrow \infty} f(x)=-\infty

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\lim _{x \rightarrow-\infty} f(x)=\infty

\lim _{x \rightarrow \infty} f(x)=\infty

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\lim _{x \rightarrow-\infty} f(x)=-\infty

\lim _{x \rightarrow \infty} f(x)=\infty

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\lim _{x \rightarrow-\infty} f(x)=\infty

\lim _{x \rightarrow \infty} f(x)=-\infty

Solve the exponential equation for

x

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\begin{array}{l} 32^{\frac{x}{5}}=\left(\frac{1}{16}\right)^{4 x-3} \\ x= \end{array}

Find the value of

4y - (y + 3)(2 + 10)

y = 5

Find the following trigonometric values.

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Express your answers exactly.

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\begin{array}{l} \cos \left(\frac{7 \pi}{4}\right)= \\ \sin \left(\frac{7 \pi}{4}\right)= \end{array}

What is the midline equation of the function

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h(x)=-3 \cos (\pi x+2)-6 ?

Use Lagrange multipliers to find the points on the given surface

y^{2}=16+x z

that are closest to the origin.

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\begin{array}{ll} \text { (smaller } y \text {-value) } & (x, y, z)=(\square) \\ \text { (larger } y \text {-value) } & (x, y, z)=(\square) \end{array}

What is the inverse of the function

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y=\log_{3}x

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1

y=x^{3}

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3

y=3^{x}

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2

y=\log_{x}3

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4

x=3^{1y}

What is the midline equation of the function

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\begin{array}{l} h(x)=5 \sin (4 x-2)-3 ? \\ y=\square \end{array}

Find the zeros of the function. Enter the solutions from least to greatest.

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\begin{array}{l} f(x)=(x-2)^{2}-9 \\ \text { lesser } x= \square \\ \text { greater } x=\square \end{array}

Solve for the exact value of

x

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2 \ln (5 x+5)+16=12

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Solve for the exact value of

x

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3 \ln (2 x-2)+14=5

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Solve for the exact value of

x

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5 \ln (8 x+1)-2=28

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...