Question 1

Sketch the graph of the function $f$ and evaluate $\lim _ { x \rightarrow - 3 ^ { + } } f ( x )$.
$$f ( x ) = \left\{ \begin{array} { l l } 
x + 5 , & \text { if } x \leq - 3 \
- 2 x - 1 , & \text { if } x > - 3
\end{array} \right.$$

Accepted Answer

The answer of Sketch the graph of the function $f$...

Question 2

$$\text { Use the graph of } f ( x ) = \frac { x ^ { 2 } + x - 2 } { x + 2 } \text { to guess at the limit } \lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + x - 2 } { x + 2 } \text {, if it exists. }$$

Accepted Answer

The answer of \[\text { Use the graph of }...

Question 3

Find the limit $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x }$, if it exists.
Select the correct answer.
A) Does not exist
B) $\frac { \sqrt { 6 } } { 12 }$
C) $\frac { \sqrt { 6 } } { 2 }$
D) $\frac { \sqrt { 6 } } { 6 }$

Accepted Answer

The limit can be evaluated using L'Hopital's rule:$$\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x } = \lim _ { x \rightarrow 0 } \frac { \frac { 1 } { 2 \sqrt { x + 6 } } } { 1 } = \frac { 1 } { 2 \sqrt { 6 } } = \frac { \sqrt { 6 } } { 12 }$$

Question 4

$$\text { What is the equation of this graph? }$$

Accepted Answer

The answer of \[\text { What is the equation of...

Question 5

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.
$$y = 4 + 2 x - x ^ { 2 }$$
Select the correct answer.
A)

B)

C)

D)

Accepted Answer

We can start with the graph of $f(x) = -x^2$ and apply the transformations:
\begin{align*}
g(x) &= f(x-1) + 4 \
&= -(x-1)^2 + 4
\end{align*}
which corresponds to shifting the parabola $f(x)=-x^2$ to the right by 1 unit and up by 4 units. Then, we can apply the transformation:
\begin{align*}
y &= 2 x + g(x) \
&= 2x - (x-1)^2 + 4 \
&= -x^2 + 2x + 3
\end{align*}
which corresponds to a vertical shift up by 3 units. Thus, the final graph is a parabola opening downwards with vertex at $(1,3)$.

Option C is the only graph that fits this description.

Question 6

If $\lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 7.9$, then if $\lim _ { x \rightarrow 2 } f ( x )$ exists, to what value does it converge?
Select the correct answer.
A) $6.9$
B) $8.9$
C) $9.9$
D) $7.9$
E) $5.9$

Accepted Answer

Since the right-hand limit exists and is equal to 7.9, the only way for the overall limit to not exist is if the left-hand limit approaches a different value. However, this cannot happen because the left-hand and right-hand limits must be equal for the overall limit to exist. Therefore, the limit as x approaches 2 must be 7.9.

Question 7

Evaluate
$$\frac { \cot \left( \frac { \pi } { 4 } + h \right) - 1 } { h } .$$

Accepted Answer

The answer of Evaluate
\[\frac { \cot \left( \frac { \pi...

Question 8

$$\text { If } 6 x - 1 \leq f ( x ) \leq x ^ { 2 } - 1 \text {,find } \lim _ { x \rightarrow 6 } f ( x )$$

Accepted Answer

The answer of \[\text { If } 6 x -...

Question 9

Classify the function as a Polynomial function, a Rational function, an algebraic function, or other.
$$f ( x ) = - 8 x ^ { - 7 } - x ^ { - 5 } - 7$$

Accepted Answer

The answer of Classify the function as a Polynomial function,...

Question 10

$$\text { Use the graph to determine where the function is discontinuous. }$$

Accepted Answer

The answer of \[\text { Use the graph to determine...

Question 11

Find the numbers, if any, where the function $f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \ 0 & \text { if } x > 1 \end{array} \right.$ is discontinuous. Select the correct answer.
A) $- 2$
B) 2
C) 1
D) 0

Accepted Answer

The function is continuous for $x>1$ since it is just constant $0$. Therefore, the potential discontinuity must occur at $x=1$. Checking from the left and right, we see that $\lim_{x	o 1^-}f(x)=3(1)-2=1$ and $\lim_{x	o 1^+}f(x)=0$. Since these limits are not equal, the function is discontinuous at $x=1$.

Question 12

Estimate the value of the following limit by graphing the function $f ( x ) = \frac { ( 5 \sin x ) } { ( \sin \pi x ) }$. $\lim _ { x \rightarrow 0 } \frac { 5 \sin x } { \sin \pi x }$
Round your answer correct to two decimal places.

Accepted Answer

The answer of Estimate the value of the following limit...

Question 13

A machinist is required to manufacture a circular metal disk with area $1000 \mathrm {~cm} ^ { 2 }$. If the machinist is allowed an error tolerance of $\pm 15 \mathrm {~cm} ^ { 2 }$ in the area of the disk, how close to the ideal radius must the machinist control the radius?
Round your answer to the nearest hundred thousandth. Select the correct answer.
A) $\delta \leq 0.13131 \mathrm {~cm}$
B) $\delta \leq 0.13281 \mathrm {~cm}$
C) $\delta \leq 0.13231 \mathrm {~cm}$
D) $\delta \leq 0.13331 \mathrm {~cm}$
E) $\delta \leq 0.13431 \mathrm {~cm}$

Accepted Answer

The answer of A machinist is required to manufacture a...

Question 14

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$.

Accepted Answer

The answer of Suppose that the graph of is given...

Question 15

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$, if it exists.
$$f ( x ) = \frac { x ^ { 2 } + x } { 2 \sqrt { x ^ { 3 } + x ^ { 2 } } }$$

Accepted Answer

The answer of Use the graph of the function to...

Question 16

Determine whether $f$ is even, odd, or neither.
$$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$$

Accepted Answer

The answer of Determine whether $f$ is even, odd, or...

Question 17

If a ball is thrown into the air with a velocity of $58 \mathrm { ft } / \mathrm { s }$, its height (in feet) after $t$ seconds is given by
$$H = 58 t - 9 t ^ { 2 } \text {. }$$
Find the velocity when $t = 9$.

Accepted Answer

The answer of If a ball is thrown into the...

Question 18

The point $P ( 16,4 )$ lies on the curve $y = \sqrt { x }$. If is the point $Q ( x , \sqrt { x } )$, use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the value $x = 3.89$.

Accepted Answer

The answer of The point $P ( 16,4 )$ lies...

Question 19

For what value of the constant $c$ is the function $f$ continuous on $( - \infty , \infty ) ?$
$$f ( x ) = \left\{ \begin{array} { l l l } 
c x + 5 & \text { for } & x \leq 2 \
c x ^ { 2 } - 5 & \text { for } x > 2
\end{array} \right.$$

Accepted Answer

The answer of For what value of the constant $c$...

Question 20

Differentiate.
$$y = \frac { \sin x } { 6 + \cos x }$$

Accepted Answer

The answer of Differentiate.
\[y = \frac { \sin x }...

Quiz 1: Functions and Models

Sketch the graph of the function $f$ and evaluate $\lim _ { x \rightarrow - 3 ^ { + } } f ( x )$ . $f ( x ) = \left\{ \begin{array} { l l } x + 5 , & \text { if } x \leq - 3 \\- 2 x - 1 , & \text { if } x > - 3\end{array} \right.$

Find the limit $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x }$ , if it exists. Select the correct answer.

$\text { What is the equation of this graph? }$ $\text { What is the equation of this graph? }$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$ Select the correct answer.

If $\lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 7.9$ , then if $\lim _ { x \rightarrow 2 } f ( x )$ exists, to what value does it converge? Select the correct answer.

Evaluate $\frac { \cot \left( \frac { \pi } { 4 } + h \right) - 1 } { h } .$

$\text { If } 6 x - 1 \leq f ( x ) \leq x ^ { 2 } - 1 \text {,find } \lim _ { x \rightarrow 6 } f ( x )$

Classify the function as a Polynomial function, a Rational function, an algebraic function, or other. $f ( x ) = - 8 x ^ { - 7 } - x ^ { - 5 } - 7$

$\text { Use the graph to determine where the function is discontinuous. }$ $\text { Use the graph to determine where the function is discontinuous. }$

Find the numbers, if any, where the function $f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \\ 0 & \text { if } x > 1 \end{array} \right.$ is discontinuous. Select the correct answer.

Estimate the value of the following limit by graphing the function $f ( x ) = \frac { ( 5 \sin x ) } { ( \sin \pi x ) }$ . $\lim _ { x \rightarrow 0 } \frac { 5 \sin x } { \sin \pi x }$ Round your answer correct to two decimal places.

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$ .

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$ , if it exists. $f ( x ) = \frac { x ^ { 2 } + x } { 2 \sqrt { x ^ { 3 } + x ^ { 2 } } }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

If a ball is thrown into the air with a velocity of $58 \mathrm { ft } / \mathrm { s }$ , its height (in feet) after $t$ seconds is given by $H = 58 t - 9 t ^ { 2 } \text {. }$ Find the velocity when $t = 9$ .

The point $P ( 16,4 )$ lies on the curve $y = \sqrt { x }$ . If is the point $Q ( x , \sqrt { x } )$ , use your calculator to find the slope of the secant line $P Q$ (correct to six decimal places) for the value $x = 3.89$ .

For what value of the constant $c$ is the function $f$ continuous on $( - \infty , \infty ) ?$ $f ( x ) = \left\{ \begin{array} { l l l } c x + 5 & \text { for } & x \leq 2 \\c x ^ { 2 } - 5 & \text { for } x > 2\end{array} \right.$

Differentiate. $y = \frac { \sin x } { 6 + \cos x }$

Quiz 1: Functions and Models

Find the limit lim⁡x→0x+6−6x\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x }limx→0​xx+6​−6​​ , if it exists. Select the correct answer.

What is the equation of this graph? \text { What is the equation of this graph? } What is the equation of this graph?

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. y=4+2x−x2y = 4 + 2 x - x ^ { 2 }y=4+2x−x2 Select the correct answer.

If lim⁡x→2+f(x) =7.9\lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 7.9limx→2+​f(x) =7.9 , then if lim⁡x→2f(x) \lim _ { x \rightarrow 2 } f ( x ) limx→2​f(x) exists, to what value does it converge? Select the correct answer.

Evaluate cot⁡(π4+h)−1h.\frac { \cot \left( \frac { \pi } { 4 } + h \right) - 1 } { h } .hcot(4π​+h)−1​.

If 6x−1≤f(x)≤x2−1,find lim⁡x→6f(x)\text { If } 6 x - 1 \leq f ( x ) \leq x ^ { 2 } - 1 \text {,find } \lim _ { x \rightarrow 6 } f ( x ) If 6x−1≤f(x)≤x2−1,find limx→6​f(x)

Classify the function as a Polynomial function, a Rational function, an algebraic function, or other. f(x)=−8x−7−x−5−7f ( x ) = - 8 x ^ { - 7 } - x ^ { - 5 } - 7f(x)=−8x−7−x−5−7

Use the graph to determine where the function is discontinuous. \text { Use the graph to determine where the function is discontinuous. } Use the graph to determine where the function is discontinuous.

Find the numbers, if any, where the function f(x) ={3x−2 if x≤10 if x>1f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \\ 0 & \text { if } x > 1 \end{array} \right.f(x) ={3x−20​ if x≤1 if x>1​ is discontinuous. Select the correct answer.

Suppose that the graph of is given fff is given. Describe how the graph of the function y=f(x−5)−5y = f ( x - 5 ) - 5y=f(x−5)−5 can be obtained from the graph of fff .

Use the graph of the function to state the value of lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x) , if it exists. f(x)=x2+x2x3+x2f ( x ) = \frac { x ^ { 2 } + x } { 2 \sqrt { x ^ { 3 } + x ^ { 2 } } }f(x)=2x3+x2​x2+x​

Determine whether fff is even, odd, or neither. f(x)=4x2x4+5f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }f(x)=x4+54x2​

If a ball is thrown into the air with a velocity of 58ft/s58 \mathrm { ft } / \mathrm { s }58ft/s , its height (in feet) after ttt seconds is given by H=58t−9t2. H = 58 t - 9 t ^ { 2 } \text {. }H=58t−9t2. Find the velocity when t=9t = 9t=9 .

The point P(16,4)P ( 16,4 )P(16,4) lies on the curve y=xy = \sqrt { x }y=x​ . If is the point Q(x,x)Q ( x , \sqrt { x } )Q(x,x​) , use your calculator to find the slope of the secant line PQP QPQ (correct to six decimal places) for the value x=3.89x = 3.89x=3.89 .

Differentiate. y=sin⁡x6+cos⁡xy = \frac { \sin x } { 6 + \cos x }y=6+cosxsinx​