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

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x |$

Find the derivative of the function at the given value of x. - $f ( x ) = - 7 x + 11 ; x = 6$

Determine whether f is continuous at c. - $f ( x ) = \frac { x + 6 } { ( x - 2 ) ( x - 3 ) } ; c = - 6$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -The volume of a rectangular box with square base and a height of 5 feet is $V ( x ) = 5 x ^ { 2 }$ , where $x$ is the length of a side of the base. Find the instantaneous rate of change of volume with respect to $x$ when $x = 3$ feet.

Find the numbers at which f is continuous. At which numbers is f discontinuous? - $f ( x ) = 5 x ^ { 2 } - 3 x$

Find the one-sided limit. - $\lim _ { x \rightarrow 8 ^ { - } } \frac { x ^ { 2 } - 9 } { x - 3 }$

Choose the one alternative that best completes the statement or answers the question. 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. - $f ( x ) = 3 x ^ { 2 } - 2 \text { from } x = 1 \text { to } x = 5 ; n = 4$

Choose the one alternative that best completes the statement or answers the question. 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. - $f ( x ) = 2 x ^ { 2 } + x + 3 \text { from } x = - 2 \text { to } x = 1 ; n = 3$

Use a graphing utility to find the indicated limit rounded to two decimal places. - $\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }$

Find the limit algebraically. - $\lim _ { x \rightarrow 10 } - 8 x ^ { 3 }$

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

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

Determine whether f is continuous at c. - $f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + x - 7 ; \quad c = 0$

Use the graph of y = g(x) to answer the question. -Find $\mathrm { f } ( - 4 )$ .

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

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

Find the numbers at which f is continuous. At which numbers is f discontinuous? - $f ( x ) = \left\{ \begin{aligned}\frac { x ^ { 2 } - 16 } { x - 4 } & \text { if } x \neq 4 \\8 & \text { if } x = 4\end{aligned} \right.$

Find the limit algebraically. - $\lim _ { x \rightarrow 6 } x$

Determine whether f is continuous at c. - $f ( x ) = \frac { 5 } { x + 2 } ; c = - 2$

Find the limit algebraically. - $\lim _ { x \rightarrow 0 } \frac { 3 \tan x } { 7 x }$