Exam 1: Functions and Models

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.

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

Scientists have discovered that a linear relationship exists between the amount of flobberworm mucus secretions and the air temperature. When the temperature is $65 ^ { \circ } \mathrm { F }$ , the flobberworms each secrete 16 grams of mucus a day; when the temperature is $95 ^ { \circ } \mathrm { F }$ , they each secrete 22 grams of mucus a day. Find a function $M ( t )$ that gives the amount of mucus secreted on a given day, where $t$ is the temperature of that day in degrees Fahrenheit.

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

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.

A spherical balloon with radius $r$ inches has volume $\frac { 4 } { 3 } \pi r ^ { 3 } \text {. }$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r + 1$ inches.

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

Find the domain of the function. $f ( x ) = \frac { 5 } { 4 x - 1 }$

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?

If a rock is thrown upward on the planet Mars with a velocity of $12 \mathrm {~m} / \mathrm { s }$ , its height in meters $t$ seconds later is given by $y = 12 t - 1.92 t ^ { 2 }$ Find the average velocity over the time interval $[ 2,3 ]$ .

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

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

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.

By graphing the function $f ( x ) = \frac { ( \cos x - \cos 5 x ) } { x ^ { 2 } }$ and zooming in toward the point where the graph crosses the $y$ -axis, estimate the value of $\lim _ { x \rightarrow 0 } f ( x )$ .

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after $t$ minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with $t = 38$ and $t = 42$ . t (mins) 36 38 40 42 44 Heartbeats 2570 2720 2840 3020 3070

Which of the given functions is discontinuous?

The graph of the function $f$ is given. State the value of $f ( - 0.4 )$ .

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

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

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