Exam 1: Functions and Models

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of $45 \mathrm {~cm} / \mathrm { s }$ . Express the radius $r$ of this circle as a function of the time $t$ (in seconds) and find $A \circ r$ , if $A$ is the area of this circle as a function of the radius.

(Short Answer)

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Question 61

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

(Short Answer)

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Question 62

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$ , if it exists. $f ( x ) = \frac { 1 } { 1 + 4 ^ { 1 / x } }$

(Short Answer)

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Question 63

$\text { Find the function } f \cdot g \text { and its domain if } f ( x ) = \sqrt { x + 7 } \text { and } g ( x ) = \sqrt { x - 7 } \text {. }$

(Short Answer)

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Question 64

Find the domain of the function $f ( x ) = \frac { x } { - 2 \sin x - 3 }$ .

(Multiple Choice)

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Question 65

The monthly cost of driving a car depends on the number of miles driven. Julia found that in October it cost her $\$ 200$ to drive $300 \mathrm { mi }$ and in July it cost her $\$ 350$ to drive $600 \mathrm { mi }$ . Express the monthly $\operatorname { cost } C$ as a function of the distance driven $d$ assuming that a linear relationship gives a suitable model.

(Short Answer)

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Question 66

A spherical balloon with radius $r$ inches has volume $4 \frac { \pi r ^ { 3 } } { 3 }$ . Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r + 3$ inches.

(Short Answer)

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Question 67

$\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$ $\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$

(Essay)

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Question 68

Find the interval(s) where $f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }$ is continuous.

(Multiple Choice)

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Question 69

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 $x = 3.89$ .

(Multiple Choice)

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Question 70

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 $f$ .

(Short Answer)

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Question 71

$\text { How close to } 2 \text { do we have to take } x \text { so that } 5 x + 3 \text { is within a distance of } 0.01 \text { from } 13 \text { ? }$

(Short Answer)

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Question 72

Which of the following graphs is neither even nor odd?

(Multiple Choice)

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Question 73

Find the limit. $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } \left( \frac { 2 } { x } \right)$

(Short Answer)

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Question 74

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

(Multiple Choice)

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Question 75

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) ,\lim _ { x \rightarrow 5 ^ { - } } F ( x )$

(Short Answer)

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Question 76

$\text { Let } f ( x ) = x ^ { 2 } - 18 x + 75 \text { and } g ( x ) = \sqrt { x + 7 } \text {. Find } ( f \circ g ) ( 74 ) ( g \circ g ) ( 74 ) \text {. }$

(Short Answer)

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Question 77

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 $f$ .

(Multiple Choice)

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Question 78

Consider the following function. $f ( x ) = \left\{ \begin{array} { c c } 3 - x & x < - 1 \\x & - 1 \leq x < 3 \\( x - 3 ) ^ { 2 } & x \geq 3\end{array} \right.$ Determine the values of $a$ for which $\lim _ { x \rightarrow a } f ( x )$ exists.

(Short Answer)

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Question 79

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

(Multiple Choice)

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Question 80

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of $45 \mathrm {~cm} / \mathrm { s }$ . Express the radius $r$ of this circle as a function of the time $t$ (in seconds) and find $A \circ r$ , if $A$ is the area of this circle as a function of the radius.

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

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$ , if it exists. $f ( x ) = \frac { 1 } { 1 + 4 ^ { 1 / x } }$

$\text { Find the function } f \cdot g \text { and its domain if } f ( x ) = \sqrt { x + 7 } \text { and } g ( x ) = \sqrt { x - 7 } \text {. }$

Find the domain of the function $f ( x ) = \frac { x } { - 2 \sin x - 3 }$ .

A spherical balloon with radius $r$ inches has volume $4 \frac { \pi r ^ { 3 } } { 3 }$ . Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r + 3$ inches.

$\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$ $\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$

Find the interval(s) where $f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }$ is continuous.

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 $x = 3.89$ .

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 $f$ .

$\text { How close to } 2 \text { do we have to take } x \text { so that } 5 x + 3 \text { is within a distance of } 0.01 \text { from } 13 \text { ? }$

Which of the following graphs is neither even nor odd?

Find the limit. $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } \left( \frac { 2 } { x } \right)$

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) ,\lim _ { x \rightarrow 5 ^ { - } } F ( x )$

$\text { Let } f ( x ) = x ^ { 2 } - 18 x + 75 \text { and } g ( x ) = \sqrt { x + 7 } \text {. Find } ( f \circ g ) ( 74 ) ( g \circ g ) ( 74 ) \text {. }$

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 $f$ .

Consider the following function. $f ( x ) = \left\{ \begin{array} { c c } 3 - x & x < - 1 \\x & - 1 \leq x < 3 \\( x - 3 ) ^ { 2 } & x \geq 3\end{array} \right.$ Determine the values of $a$ for which $\lim _ { x \rightarrow a } f ( x )$ exists.

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

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

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Exam 1: Functions and Models

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

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)=11+41/xf ( x ) = \frac { 1 } { 1 + 4 ^ { 1 / x } }f(x)=1+41/x1​

Find the function f⋅g and its domain if f(x)=x+7 and g(x)=x−7. \text { Find the function } f \cdot g \text { and its domain if } f ( x ) = \sqrt { x + 7 } \text { and } g ( x ) = \sqrt { x - 7 } \text {. } Find the function f⋅g and its domain if f(x)=x+7​ and g(x)=x−7​.

Find the domain of the function f(x)=x−2sin⁡x−3f ( x ) = \frac { x } { - 2 \sin x - 3 }f(x)=−2sinx−3x​ .

A spherical balloon with radius rrr inches has volume 4πr334 \frac { \pi r ^ { 3 } } { 3 }43πr3​ . Find a function that represents the amount of air required to inflate the balloon from a radius of rrr inches to a radius of r+3r + 3r+3 inches.

The graph of the function f follows. Choose the graph of y=12f(x−1). \text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. } The graph of the function f follows. Choose the graph of y=21​f(x−1).

Find the interval(s) where f(x)=x2−2x+3f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }f(x)=x2−2x+3​ is continuous.

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 .

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 .

Which of the following graphs is neither even nor odd?

Find the limit. lim⁡x→0+tan⁡−1(2x)\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } \left( \frac { 2 } { x } \right)limx→0+​tan−1(x2​)

Evaluate the limit. lim⁡x→1(x+5)3(x2−6)\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)limx→1​(x+5)3(x2−6)

Let F(x)=x−5∣x−5∣F ( x ) = \frac { x - 5 } { | x - 5 | }F(x)=∣x−5∣x−5​ . Find the following limits. lim⁡x→5+F(x),lim⁡x→5−F(x)\lim _ { x \rightarrow 5 ^ { + } } F ( x ) ,\lim _ { x \rightarrow 5 ^ { - } } F ( x )limx→5+​F(x),limx→5−​F(x)

Let f(x)=x2−18x+75 and g(x)=x+7. Find (f∘g)(74)(g∘g)(74). \text { Let } f ( x ) = x ^ { 2 } - 18 x + 75 \text { and } g ( x ) = \sqrt { x + 7 } \text {. Find } ( f \circ g ) ( 74 ) ( g \circ g ) ( 74 ) \text {. } Let f(x)=x2−18x+75 and g(x)=x+7​. Find (f∘g)(74)(g∘g)(74).

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 .

If 6x−1≤f(x)≤x2−16 x - 1 \leq f ( x ) \leq x ^ { 2 } - 16x−1≤f(x)≤x2−1 ,find lim⁡x→6f(x)\lim _ { x \rightarrow 6 } f ( x )limx→6​f(x)

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

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

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$ , if it exists. $f ( x ) = \frac { 1 } { 1 + 4 ^ { 1 / x } }$

$\text { Find the function } f \cdot g \text { and its domain if } f ( x ) = \sqrt { x + 7 } \text { and } g ( x ) = \sqrt { x - 7 } \text {. }$

Find the domain of the function $f ( x ) = \frac { x } { - 2 \sin x - 3 }$ .

A spherical balloon with radius $r$ inches has volume $4 \frac { \pi r ^ { 3 } } { 3 }$ . Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r + 3$ inches.

$\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$ $\text { The graph of the function } f \text { follows. Choose the graph of } y = \frac { 1 } { 2 } f ( x - 1 ) \text {. }$

Find the interval(s) where $f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }$ is continuous.

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 $x = 3.89$ .

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 $f$ .

Find the limit. $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } \left( \frac { 2 } { x } \right)$

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) ,\lim _ { x \rightarrow 5 ^ { - } } F ( x )$

$\text { Let } f ( x ) = x ^ { 2 } - 18 x + 75 \text { and } g ( x ) = \sqrt { x + 7 } \text {. Find } ( f \circ g ) ( 74 ) ( g \circ g ) ( 74 ) \text {. }$

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 $f$ .

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