Exam 2: A Preview of Calculus

$\text { Consider the function } f ( x ) = \sqrt { x } \text { and the point } P ( 4,2 ) \text { on the graph of } f \text {. }$ Graph $f$ and the secant line passing through $P ( 4,2 )$ and $Q ( x , f ( x ) )$ for $x = 3$ .

A ring has a inner circumference of 10 centimeters. What is the radius of the ring? Round your answer to four decimal places.

Determine the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { \sin ^ { 4 } x } { x ^ { 3 } }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow - 10 } \frac { \sqrt { - 6 x - 54 } - \sqrt { 6 } } { x + 10 }$ x -10.1 -10.01 -10.001 -9.999 -9.99 -9.9 f(x)

Find the constant a such that the function $f ( x ) = \left\{ \begin{array} { l l } 6 , & x \leq - 5 \\a x + b , & - 5 < x < 1 \\- 6 , & x \geq 1\end{array} \right.$ is continuous on the entire real line.

Determine the following limit. (Hint: Use the graph to calculate the limit.) $\lim _ { x \rightarrow 1 } \left( x ^ { 2 } + 4 \right)$

$\text { Find the } x \text {-values (if any) at which the function } f ( x ) = \frac { x + 2 } { x ^ { 2 } + 6 x + 8 } \text { is not continuous. }$ Which of the discontinuities are removable?

Find the limit (if it exists). $\lim _ { \Delta x \rightarrow 0 } \frac { ( x + \Delta x ) ^ { 2 } - 9 ( x + \Delta x ) + 2 - \left( x ^ { 2 } - 9 x + 2 \right) } { \Delta x }$

$\text { Find all values of } c \text { such that } f \text { is continuous on } ( - \infty , \infty ) \text {. }$ $f ( x ) = \left\{ \begin{array} { l l } 4 - x ^ { 2 } , & x \leq c \\x , & x > c\end{array} \right.$

$\text { Use a graphing utility to graph the function } f ( x ) = \csc \frac { \pi x } { 2 } \text { and determine the }$ following one-sided limit. $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = - 13 \text { and } \lim _ { x \rightarrow c } g ( x ) = - 10 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

A 30 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of $r = \frac { 2 x } { \sqrt { 900 - x ^ { 2 } } } \mathrm { ft } / \mathrm { sec }$ , where $x$ is the distance between the base of the ladder and the house. Find the rate $r$ when $x$ is 18 feet.

$\text { Find } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x ) - f ( x ) } { \Delta x } \text { where } f ( x ) = 4 x - 3 \text {. }$

$\text { Discuss the continuity of the function } f ( x ) = \frac { x ^ { 2 } - 4 } { x - 2 } \text {. }$

Find the value of c guaranteed by the Intermediate Value Theorem. $f ( x ) = \frac { x ^ { 2 } - 5 x } { x - 3 } , \left[ \frac { 9 } { 2 } , 18 \right] , f ( c ) = 6$

$\text { A long distance phone service charges } \$ 0.35 \text { for the first } 10 \text { minutes and } \$ 0.1 \text { for }$ each additional minute or fraction thereof. Use the greatest integer function to write the cost $C$ of a call in terms of time $t$ (in minutes).

the rectangles in the following graph to approximate the area of the region bounded by $y = \sin x , y = 0 , x = 0$ , and $x = \pi$ .

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = 7 \text { and } \lim _ { x \rightarrow c } g ( x ) = 3 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } \frac { f ( x ) } { g ( x ) }$

Find the constant $a$ such that the function $f ( x ) = \left\{ \begin{array} { r r } - 4 \cdot \frac { \sin x } { x } , & x < 0 \\a + 7 x , & x \geq 0\end{array} \right.$ is continuous on the entire real line.

$\text { Consider the function } f ( x ) = \sqrt { x } \text { and the point } P ( 81,9 ) \text { on the graph of } f \text {. }$ Find the slope of the secant line passing through $P ( 81,9 )$ and $Q ( x , f ( x ) )$ for $x = 1$ . Round your answer to four decimal places.