Exam 1: Functions and Models

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. $F = \frac { 9 } { 5 } C + 32$ What is the $F$ -intercept and what does it represent?

Which of the given functions is discontinuous?

Is there a number $a$ such that $\lim _ { x \rightarrow - 3 } \frac { 3 x ^ { 2 } + a x + a + 3 } { x ^ { 2 } + x - 6 }$ exists? If so, find the value of $a$ and the value of the limit. Select the correct answer.

The graph shown gives the weight of a certain person as a function of age. Find the age at which the person started an exercise program.

How would you define $f ( 7 )$ in order to make $f$ continuous at 7 ? $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x - 7 }$

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.

Sketch the graph of the function $f$ and evaluate $\lim _ { x \rightarrow 3 } f ( x )$ . $f ( x ) = \left\{ \begin{array} { l l } x - 4 , & \text { if } x \leq 3 \\- 2 x + 5 , & \text { if } x > 3\end{array} \right.$

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

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

An open rectangular box with volume $2 \mathrm {~m} ^ { 3 }$ has a square base. Express the surface area of the box as a function $S ( x )$ of the length $x$ of a side of the base.

Evaluate the limit. $\lim _ { x \rightarrow \infty } \frac { 10 x ^ { 2 } - 3 x + 1 } { 7 x ^ { 2 } + 3 x - 3 }$

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.

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

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.

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

Find the limit if $g ( x ) = x ^ { 4 }$ . $\lim _ { x \rightarrow 2 } \frac { g ( x ) - g ( 2 ) } { x - 2 }$

Find the limit. $\lim _ { t \rightarrow \infty } \frac { t ^ { 2 } + 3 } { t ^ { 3 } + t ^ { 2 } - 7 }$

Find the range of the function. $y = 2 + \cos x$ 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.