Question 1

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$.
A) $\frac { 1 } { 18 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 2 } { 9 }$
E) limit does not exist

Accepted Answer

The limit can be simplified by factoring the denominator as $(x-9)(x+9)$ and then canceling out the $(x-9)$ term, leaving $\frac{1}{x+9}$. As $x$ approaches $9$ from the left, the expression simplifies to $\frac{1}{18}$.

Question 2

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$
A) 14
B) $\frac { 7 } { 2 }$
C) $\frac { \sqrt { 7 } } { 2 }$
D) $2 \sqrt { 7 }$
E) limit does not exist

Accepted Answer

The answer of Find \(\lim _ { x \rightarrow 0...

Question 3

A union contract guarantees a $16 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,500$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is givet by
$$f ( t ) = \left\{ \begin{array} { l l } 
31.50 , & 0 < t \leq 1 \
36.54 , & 1 < t \leq 2 \
42.39 , & 2 < t \leq 3
\end{array} \right.$$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.
A) 31.50
B) 36.54
C) 1.00
D) 42.39
E) The limit does not exist.

Accepted Answer

The limit does not exist because the function $f(t)$ has a jump discontinuity at $t = 1$. The left-hand limit as $t$ approaches 1 is 31.50, and the right-hand limit is 36.54, which are not equal.

Question 4

Find the slope of the graph of the following function at the point (-1,1) . $$3 x ^ { 2 } - 2$$
A) 4
B) -6
C) 6
D) 1
E) -8

Accepted Answer

The answer of Find the slope of the graph of...

Question 5

Find the following limit.
$$\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$$
A) 52
B) 36
C) 37
D) - 23
E) - 35

Accepted Answer

To find the limit as $x$ approaches $-4$, substitute $-4$ directly into the equation: $3(-4)^2 + 3(-4) + 1 = 3(16) - 12 + 1 = 48 - 12 + 1 = 37$.

Question 6

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .
A) 185
B) 110
C) 65
D) 75
E) 82

Accepted Answer

The answer of Use the limit process to find the...

Question 7

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).
A) $\frac { 9 } { 25 }$
B) $- 8$
C) 8
D) $\infty$
E) limit does not exist

Accepted Answer

As $x$ approaches $-\infty$, the term $\frac{x}{(5+x)^2}$ approaches 0 because the numerator grows linearly while the denominator grows quadratically. Therefore, the limit simplifies to $0 + 8 = 8$.

Question 8

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \
\hline f(x) & & & & & & & \
\hline
\end{array}$

A) 0
B) 4
C) $- 4$
D) $\infty$
E) limit does not exist

Accepted Answer

The limit of $f(x) = x - \sqrt{x^2 + 4}$ as $x$ approaches infinity is 0. This is because both $x$ and $\sqrt{x^2 + 4}$ grow indefinitely, but their difference approaches 0, as the $\sqrt{x^2 + 4}$ term becomes closer and closer to $x$ itself when $x$ becomes very large.

Question 9

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$, where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$. Round your answer to the nearest cent.
A) $\$ 51,650.00$
B) $\$ 244.75$
C) $\$ 13.50$
D) $\$ 258.25$
E) $\$ 244.82$

Accepted Answer

The average cost per unit can be found by dividing the total cost by the number of units produced. When $x=200$, the total cost is:
$$C = 13.50(200) + 48,950 = 51,950$$
Therefore, the average cost per unit is:
$$\frac{51,950}{200}= 259.75 \approx \boxed{\text{(D) }\$ 258.25}$$

Note that the answer is rounded to the nearest cent.

Question 10

Use the figure below to approximate the slope of the curve at the point $( x , y )$.

A) $- \frac { 1 } { 2 }$
B) The slope is undefined.
C) 1
D) $\frac { 1 } { 2 }$
E) 0

Accepted Answer

The answer of Use the figure below to approximate the...

Question 11

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$.
$$g ( x ) = - 7 - 4 x$$
A) $- 11$
B) $- 27$
C) $- 3$
D) $- 4$
E) 5

Accepted Answer

We use the limit definition of slope:
\begin{align*}
m &= \lim_{h 	o 0} \frac{f(5+h)-f(5)}{h} \
&= \lim_{h 	o 0} \frac{-7-4(5+h)-(-7-4(5))}{h} \
&= \lim_{h 	o 0} \frac{-4h}{h} \
&= -4
\end{align*}
Therefore, the slope of the graph of $g(x)$ at $(5,-27)$ is $-4$.

Question 12

Find the following limit, if it exists. $$\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }$$
A) -3
B) 9
C) 7
D) -11
E) The limit does not exist.

Accepted Answer

The answer of Find the following limit, if it exists....

Question 13

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$.
A) $8 x + 2$
B) $4 x + 2$
C) $8 x + 5$
D) $4 x + 5$
E) $8 x$

Accepted Answer

The derivative of a polynomial function is found by applying the power rule, which states that $d/dx [x^n] = nx^{n-1}$. Applying this rule to each term in the function $f(x) = 4x^2 + 2x + 3$, we get the derivative as $f'(x) = 8x + 2$, since the derivative of a constant term (like 3) is 0.

Question 14

Find the following limit. Round your answer to two decimals.
$$\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$$
A) $0.22$
B) $1.32$
C) $0.44$
D) $0.18$
E) $1.41$

Accepted Answer

To find the limit as $x$ approaches 3, substitute $x = 3$ directly into the expression: $\frac{\sqrt{3^2 + 7}}{2 \cdot 3^2} = \frac{\sqrt{9 + 7}}{2 \cdot 9} = \frac{\sqrt{16}}{18} = \frac{4}{18} = \frac{2}{9} \approx 0.22$.

Question 15

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$.
A) $\frac { 1 } { 3 }$
B) 3
C) $\infty$
D) $\frac { 1 } { 6 }$
E) the slope is undefined at this point

Accepted Answer

The slope of the graph of a function at a given point can be found using the derivative of the function at that point. The derivative of $\sqrt{x + 4}$ with respect to $x$ can be found using the limit definition of the derivative or by applying the chain rule directly. The derivative of $\sqrt{x + 4}$ is $\frac{1}{2\sqrt{x + 4}}$. Evaluating this derivative at $x = 5$ gives $\frac{1}{2\sqrt{5 + 4}} = \frac{1}{2\sqrt{9}} = \frac{1}{2 \cdot 3} = \frac{1}{6}$. Therefore, the slope of the graph at the point $(5,3)$ is $\frac{1}{6}$.

Question 16

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$-axis on the interval $[ 1,4 ]$.
A) $\frac { 143 } { 2 }$
B) $\frac { 81 } { 4 }$
C) $\frac { 47 } { 2 }$
D) 78
E) $\frac { 51 } { 4 }$

Accepted Answer

The answer of Use the limit process to find the...

Question 17

Find
$$\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$$
by direct substitution.
A) $\frac { 8 } { \sqrt { 10 } }$
B) $10 \sqrt { 2 }$
C) $40 \sqrt { 10 }$
D) 1
E) $\frac { 40 } { \sqrt { 10 } }$

Accepted Answer

Direct substitution of $x = 8$ into the given limit expression yields $\frac{5 \cdot 8}{\sqrt{8 + 2}} = \frac{40}{\sqrt{10}}$, which simplifies to the answer in choice E.

Question 18

Find the following limit, if it exists. $$\lim _ { x \rightarrow - 3 } x ^ { 5 }$$
A) -3
B) -15
C) 3
D) -243
E) The limit does not exist.

Accepted Answer

As $x$ approaches $-3$, $x^5$ will approach $(-3)^5=-243$ from the negative side. Thus, the limit is $-243$. Therefore, the answer is $\boxed{	ext{(D)}}$.

Question 19

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$.

A) 24
B) $\infty$
C) 12
D) 0
E) limit does not exist

Accepted Answer

The answer of Use the graph to find \(\lim _...

Question 20

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$$
A) $( - 1,2 ) , ( 0,0 ) , ( 1,2 )$
B) $( - 1 , - 2 ) , ( 1 , - 8 )$
C) $( - 1,2 ) , ( 1,2 )$
D) $( 0,0 ) , ( 1,0 )$
E) $( 0,0 ) , ( 1,2 )$

Accepted Answer

The tangent line is horizontal where the derivative $f'(x) = 0$. Solving $-8x^3 + 8x = 0$ gives $x = 0, \pm 1$. Substituting these values into $f(x)$ gives the points $(-1, 2), (0, 0), (1, 2)$.

Quiz 11: Limits and an Introduction to Calculus

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$ .

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Find the slope of the graph of the following function at the point (-1,1) . $3 x ^ { 2 } - 2$

Find the following limit. $\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists) .

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$ . $\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\\hline f(x) & & & & & & & \\\hline\end{array}$

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

Use the figure below to approximate the slope of the curve at the point $( x , y )$ . $Use the figure below to approximate the slope of the curve at the point ( x , y ) . A) - \frac { 1 } { 2 } B) The slope is undefined. C) 1 D) \frac { 1 } { 2 } E) 0$

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$ . $g ( x ) = - 7 - 4 x$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }$

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$ -axis on the interval $[ 1,4 ]$ .

Find $\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$ by direct substitution.

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } . A) 24 B) \infty C) 12 D) 0 E) limit does not exist$

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$

Quiz 11: Limits and an Introduction to Calculus

Find lim⁡x→9−x−9x2−81\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }limx→9−​x2−81x−9​ .

Find lim⁡x→0−xx+7−7\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }limx→0−​x+7​−7​x​

Find the slope of the graph of the following function at the point (-1,1) . 3x2−23 x ^ { 2 } - 23x2−2

Find the following limit. lim⁡x→−43x2+3x+1\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1limx→−4​3x2+3x+1

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .

Find lim⁡x→−∞[x(5+x) 2+8]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]limx→−∞​[(5+x) 2x​+8] (if it exists) .

The cost function for a certain model of a digital camera given by C=13.50x+48,950C = 13.50 x + 48,950C=13.50x+48,950 , where CCC is the cost (in dollars) and xxx is the number of cameras produced. Find the average cost per unit when x=200x = 200x=200 . Round your answer to the nearest cent.

Use the figure below to approximate the slope of the curve at the point (x,y) ( x , y ) (x,y) .

Use the limit process to find the slope of the graph of the following function at the point (5,−27) ( 5 , - 27 ) (5,−27) . g(x) =−7−4xg ( x ) = - 7 - 4 xg(x) =−7−4x

Find the following limit, if it exists. lim⁡x→−39−3x−2x23+x\lim _ { x \rightarrow - 3 } \frac { 9 - 3 x - 2 x ^ { 2 } } { 3 + x }limx→−3​3+x9−3x−2x2​

Find the derivative of f(x) =4x2+2x+3f ( x ) = 4 x ^ { 2 } + 2 x + 3f(x) =4x2+2x+3 .

Find the following limit. Round your answer to two decimals. lim⁡x→3x2+72x2\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }limx→3​2x2x2+7​​

Use the limit process to find the slope of the graph of x+4\sqrt { x + 4 }x+4​ at (5,3) ( 5,3 ) (5,3) .

Use the limit process to find the area of the region between f(x) =14(x2+4x) f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right) f(x) =41​(x2+4x) and the xxx -axis on the interval [1,4][ 1,4 ][1,4] .

Find lim⁡x→85xx+2\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }limx→8​x+2​5x​ by direct substitution.

Find the following limit, if it exists. lim⁡x→−3x5\lim _ { x \rightarrow - 3 } x ^ { 5 }limx→−3​x5

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​ .

Use the function below and its derivative to determine any points on the graph of fff at which the tangent line is horizontal. f(x) =−2x4+4x2,f′(x) =−8x3+8xf ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 xf(x) =−2x4+4x2,f′(x) =−8x3+8x