Exam 1: Functions and Models

If a ball is thrown into the air with a velocity of $58 \mathrm { ft } / \mathrm { s }$ , its height (in feet) after $t$ seconds is given by $H = 58 t - 9 t ^ { 2 } .$ Find the velocity when $t = 9$ .

Find an expression for the function $y = f ( x )$ whose graph is the bottom half of the parabola $x + ( 6 - y ) ^ { 2 } = 0$ .

Find the derivative of the function. $f ( x ) = x \sin ^ { 8 } x$

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species $S$ of bats living in caves in central Mexico has been related to the surface area $A$ measured in $m ^ { 2 }$ of the caves by the equation $S = 0.7 A ^ { 03 }$ (a) The cave called mission impossible near puebla, mexico, has suface area of $A = 90 \mathrm {~m} ^ { 2 }$ . How many species of bats would expect to find in that cave? (b) If you discover that 5 species of bats live in cave estimate the area of the cave.

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

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.

The level of nitrogen dioxide present on a certain June day in downtown Megapolis is approximated by $A ( t ) = 0.03 t ^ { 3 } ( t - 7 ) ^ { 4 } + 64.8 \quad 0 \leq t \leq 7$ where $A ( t )$ is measured in pollutant standard index and $t$ is measured in hours with $t = 0$ corresponding to 7 a.m. What is the average level of nitrogen dioxide in the atmosphere from 1 a.m. to 2 p.m. on that day? Round to three decimal places.

Sketch the graph of $y = - 1 - \cos x$ over one period.

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 $x = 3.89$ .

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

Suppose the distance $s$ (in feet) covered by a car moving along a straight road after $t$ sec is given by the function $s = f ( t ) = 3 t ^ { 2 } + 13 t$ . Calculate the (instantaneous) velocity of the car when $t = 35$ .

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

Suppose that $f$ and $g$ are functions that are differentiable at $x = 1$ and that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - 3 , g$ $( 1 ) = 2$ , and $g ^ { \prime } ( 1 ) = 5$ . Find $h ^ { \prime } ( 1 )$ . $h ( x ) = \frac { x f ( x ) } { x + g ( x ) }$

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

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

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