Exam 1: Functions and Models

Suppose the total cost in maunufacturing $x$ units of a certain product is $C ( x )$ dollars. a. What does $C ^ { \prime } ( x )$ measure? Give units. b. What can you say about the sign of $C ^ { \prime }$ ? c. Given that $C ^ { \prime } ( 3000 ) = 11$ , estimate the additional cost in producing the 3001 st unit of the product.

Identify the "inside function" $u = f ( x )$ and the "outside function" $y = g ( u )$ . Then find $d y / d x$ using the Chain Rule. $y = \sqrt { x ^ { 2 } - 2 }$

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

Evaluate the limit. $\lim _ { x \rightarrow 0 } \frac { ( 6 + x ) ^ { - 1 } - 6 ^ { - 1 } } { x }$

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.

Find the limit. $\lim _ { x \rightarrow 2 } \sqrt { \frac { 4 x ^ { 2 } + 1 } { 3 x - 2 } }$

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.

Determine whether f is even, odd, or neither. $f ( x ) = \frac { 8 x ^ { 2 } } { x ^ { 4 } + 1 }$

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

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 $H = 58 t - 9 t ^ { 2 } \text {. }$ Find the velocity when $t = 9$ .

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

Find the function $f \cdot g$ and its domain if $f ( x ) = \sqrt { x + 7 }$ and $g ( x ) = \sqrt { x - 7 }$ .

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$ .

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

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

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

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is $20 \mathrm { ft }$ and is increasing at the rate of $\frac { 1 } { 6 } \mathrm { ft } / \mathrm { sec }$ . Round to the nearest tenth if necessary.