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

Solve the problem. - $f ( x ) = \tan x$ is defined over the interval $\left[ 0 , \frac { 3 } { 2 } \right]$ (a) Approximate the area A by partitioning $\left[ 0 , \frac { 3 } { 2 } \right]$ into 4 subintervals of equal length and choosing u as the left endpoint of each subinterval. (b) Approximate the area A by partitioning $\left[ 0 , \frac { 3 } { 2 } \right]$ into 8 subintervals of equal length and choosing $\mathrm { u }$ as the right endpoint of each subinterval. (c) Express the area A as an integral. (d) Use a graphing utility to approximate this integral to three decimal places.

Find the one-sided limit. - $\lim _ { x \rightarrow \theta ^ { + } } ( 2 \cos x )$

Find the limit algebraically. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Solve the problem. -The volume $\mathrm { V }$ of a right circular cylinder of height 2 and radius $\mathrm { r }$ is $\mathrm { V } ( \mathrm { r } ) = 2 \pi \mathrm { r } ^ { 2 }$ . Find the instantaneous rate of change of volume with respect to radius $r$ when $r = 7$ .

Determine whether f is continuous at c. - $f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x - 2 } , & x > 2 \\x ^ { 2 } + 5 x , & x \leq 2\end{array} ; \quad c = 2 \right.$

Find the numbers at which f is continuous. At which numbers is f discontinuous? - $f ( x ) = \left\{ \begin{aligned}x - 5 & \text { if } x \leq 5 \\2 x - 10 & \text { if } x > 5\end{aligned} \right.$

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. - $f ( x ) = 2 x + 3 \text { from } x = 0 \text { to } x = 2 ; n = 4$

Find the derivative of the function at the given value of x. - $f ( x ) = x ^ { 3 } - 20 ; x = 3$

Find the one-sided limit. - $\lim _ { x \rightarrow 0 ^ { - } } ( 4 - 5 x )$

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. - $f ( x ) = 2 x ^ { 2 } + x - 3 \text { at } ( 4,33 )$

Find the limit algebraically. - $\lim _ { x \rightarrow - 1 } - 6$

Find the limit algebraically. - $\lim _ { x \rightarrow0 } ( x - \sqrt { 5 } ) ( x + \sqrt { 5 } )$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right)$

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. - $f ( x ) = 2 x ^ { 2 } + x + 3 \text { from } x = - 2 \text { to } x = 1 ; n = 3$

Find the equation of the tangent line to the graph of f at the given point. - $f ( x ) = 5 x ^ { 2 } + x \text { at } ( - 4,76 )$

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

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

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

Write the word or phrase that best completes each statement or answers the question. Determine where the rational function is undefined. Determine whether an asymptote of a hole appears at such numbers. - $R ( x ) = \frac { x ^ { 3 } - x ^ { 2 } + x - 1 } { x ^ { 4 } - x ^ { 3 } + 3 x - 3 }$

Find the derivative of the function at the given value of x using a graphing utility. If necessary, round to four decimal places. - $f ( x ) = x \sin ( 10 x ) ; x = 8$