Exam 13: A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right)$

Solve the problem. -An explosion causes debris to rise vertically with an initial velocity of 96 feet per second. The function $s ( t ) = - 16 t ^ { 2 } + 96 t$ describes the height of the debris above the ground, s(t), in feet, t seconds after the explosion. What is the instantaneous speed of the debris 4.8 second(s) after the explosion?

Choose the one alternative that best completes the statement or answers the question. Find the slope of the tangent line to the graph at the given point. - $f ( x ) = x ^ { 2 } + 5 x \text { at } ( 4,20 )$

Use the graph of y = g(x) to answer the question. -What is the range of g? A) all real numbers B) $\{ y \mid 2 \leq y \leq 4 \}$ C) $\{ y \mid - 2 \leq y \leq 4 \}$ D) $\{ y \mid - 2 \leq y \leq 5 \}$

Find the one-sided limit. - $\lim _ { x \rightarrow 4 ^ { + } } \frac { 16 - x ^ { 2 } } { 4 - x }$

Find the limit algebraically. - $\lim _ { x \rightarrow 1 } \frac { 3 x ^ { 2 } + 7 x - 2 } { 3 x ^ { 2 } - 4 x + 2 }$

Use the graph of y = g(x) to answer the question. -Find f(-4).

Determine whether f is continuous at c. - $f ( x ) = \frac { x ^ { 2 } - 36 } { x - 6 } ; \quad c = - 6$

Approximate the area under the curve and above the x-axis using n rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f ( x ) = 2 x ^ { 2 } + x + 3 \text { from } x = 0 \text { to } x = 6 ; n = 6$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _ { x \rightarrow 8 } f ( x )$

Find the limit algebraically. - $\lim _ { x \rightarrow - 0 } ( \sqrt { x } - 2 )$

Choose the one alternative that best completes the statement or answers the question. Find the slope of the tangent line to the graph at the given point. - $f ( x ) = x ^ { 2 } + 11 x - 15 \text { at } ( 1 , - 3 )$

Determine whether f is continuous at c. - $f ( x ) = \frac { x - 2 } { ( x + 9 ) ( x + 8 ) } ; c = - 9$

Find the limit algebraically. - $\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 2 \right) ^ { 3 }$

Determine whether f is continuous at c. - $f ( x ) = \left\{ \begin{array} { r l } - 2 x + 9 , & x < 1 \\- 7 x + 14 , & x > 1\end{array} ; \quad c = 1 \right.$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrowπ / 2 } f ( x ) , f ( x ) = \sin x - 2$

Find the numbers at which f is continuous. At which numbers is f discontinuous? - $f ( x ) = 4 \tan x$

Determine whether f is continuous at c. - $f ( x ) = 3 x ^ { 4 } - 7 x ^ { 3 } + x - 2 ; \quad c = 2$

Solve the problem. -Given the function f defined over the interval [a, b], graph the function indicating the area A under f from a to b. Then express the area A as an integral. $f ( x ) = x ^ { 2 } + 3 , \quad [ - 4,2 ]$

Choose the one alternative that best completes the statement or answers the question. Find the slope of the tangent line to the graph at the given point. - $f ( x ) = 5 x ^ { 2 } + x \text { at } ( - 4,76 )$