Exam 1: Functions and Models

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$

The position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 s (meters) 0 21.9 25.8 69.2 92.2 Find the average velocity for the time period beginning when $t = 2$ and lasting 2 seconds.

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

Write an equation that expresses the fact that a function $f \text { is continuous at the number } 4 .$

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

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

Find the domain. $g ( u ) = \sqrt { u } - \sqrt { 3 - u }$

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.

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

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

Define the function $f ( x ) = \frac { 6 x ^ { 3 } + x } { 4 x }$ at 0 so as to make it continuous at 0 .

The graphs of $f ( x )$ and $g ( x )$ are given. a) For what values of $x$ is $f ( x ) = g ( x )$ ? b) Find the values of $f ( - 2 )$ and $g ( 4 )$ .

$\text { Use the graph of } f ( x ) = \frac { \sin 4 x } { \tan 5 x } \text { to guess at the limit } \lim _ { x \rightarrow 0 } \frac { \sin 4 x } { \tan 5 x } \text {, if it exists. }$

The position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 5 s (feet) 0 16 35 71 112 179 Estimate the instantaneous velocity when $t = 2$ by averaging the velocities for the periods $[ 1,2 ]$ and $[ 2,3 ]$ .

$\text { Determine whether the function whose graph is given is even, odd, or neither. }$

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