Exam 1: Functions and Models

If $1 \leq f ( x ) \leq x ^ { 2 } + 6 x + 6$ , for all $x$ , find $\lim _ { x \rightarrow - 1 } f ( x )$

Find the limit. $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } - 25 } } { 2 x - 6 }$

Differentiate. $y = \frac { \sin x } { 6 + \cos x }$

Find the domain of the function. $f ( x ) = \frac { 7 x + 1 } { x ^ { 2 } }$

Let $f ( x ) = x \left| x ^ { 3 } \right|$ . a. Sketch the graph of $f$ . b. For what values of $x$ is $f$ differentiable? c. Find a formula for $f ^ { \prime } ( x )$ .

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.

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

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 $[ 2,3 ]$ .

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of $45 \mathrm {~cm} / \mathrm { s }$ . Express the radius $r$ of this circle as a function of the time $t$ (in seconds) and find $A \circ r$ , if $A$ is the area of this circle as a function of the radius.

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

If $\lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 7.9$ , then if $\lim _ { x \rightarrow 2 } f ( x )$ exists, to what value does it converge? Select the correct answer.

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

Find $a$ , such that the function $f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }$ has the domain $( - 4,4 )$ . Select the correct answer.

Find the value of $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ . $f ( x ) = \frac { 1 } { 1 + 6 ^ { 1 / x } }$

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

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

The sides of a square baseball diamond are $90 \mathrm { ft }$ long. When a player who is between the second and third base is $30 \mathrm { ft }$ from second base and heading toward third base at a speed of $24 \mathrm { ft } / \mathrm { sec }$ , how fast is the distance between the player and home plate changing? Round to two decimal places.

The following figure shows a portion of the graph of a function $f$ defined on the interval $[ - 1,1 ]$ . Sketch the complete graph of $f$ if it is known $f$ is odd.