Exam 1: Functions and Models

$\text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. }$

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

Evaluate the function $f ( x ) = 7 \left( \frac { \sqrt { x } - \sqrt { 2 } } { x - 2 } \right)$ at the given numbers (correct to six decimal places). Use the results to guess the value of the limit $\lim _ { x \rightarrow 2 } f ( x )$ . x f(x) 1.6 1.8 1.9 1.99 1.999 2.4 2.2 2.1 2.01 2.001 Limit

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.

Let $f ( x ) = \left\{ \begin{array} { c c } x - 4 & \text { if } x \leq 5 \\k x ^ { 2 } - 24 x + 46 & \text { if } x > 5\end{array} \right.$ Find the value of $k$ that will make $f$ continuous on $( - \infty , \infty )$ .

You are given $\lim _ { x \rightarrow a } f ( x ) = L$ and a tolerance $\varepsilon$ . Find a number $\delta$ such that $| f ( x ) - L | < \varepsilon$ whenever $0 < | x - a | < \delta$ . $\lim _ { x \rightarrow 3 } 4 x = 12 ; \quad \varepsilon = 0.01$

Find the limit. $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 12 } { x - 2 }$

If $f$ and $g$ are continuous functions with $f ( 9 ) = 6$ and $\lim _ { x \rightarrow 9 } [ 2 f ( x ) - g ( x ) ] = 9$ , find $g ( 9 )$ .

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.

Which of the given functions is discontinuous?

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

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

Differentiate. $y = \frac { \sin x } { 3 + \cos x }$

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 $x = 3.89$ .

The volume of a right circular cone of radius $r$ and height $h$ is $V = \frac { \pi } { 3 } r ^ { 2 } h$ . Suppose that the radius and height of the cone are changing with respect to time $t$ . a. Find a relationship between $\frac { d V } { d t } , \frac { d r } { d t }$ , and $\frac { d h } { d t }$ . b. At a certain instant of time, the radius and height of the cone are 12 in. and 13 in. and are increasing at the rate of $0.2 \mathrm { in } . / \mathrm { sec }$ and $0.5 \mathrm { in } . / \mathrm { sec }$ , respectively. How fast is the volume of the cone increasing?

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

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. Show that the curves of the given equations are orthogonal. $y - \frac { 7 } { 4 } x = \frac { \pi } { 2 } , \quad x = \frac { 7 } { 4 } \cos y$

Find the point at which the given function is discontinuous. $f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { x - 7 } , & x \neq 7 \\7 , & x = 7\end{array} \right.$

Find the limit. $\lim _ { x \rightarrow \frac { 10 } { x } } \tan ^ { - 1 } \left( \frac { 5 } { x } \right)$