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

Choose the one alternative that best completes the statement or answers the question. - $f ( x ) = x \sin ( 10 x ) ; x = 8$

Write the word or phrase that best completes each statement or answers the question. - $f ( x ) = \sqrt [ 3 ] { x }$ is defined on the interval $[ 0,6 ]$ (a) Approximate the area A under the graph of f by partitioning $[ 0,6 ]$ into three subintervals of equal length an choose $u$ as the left endpoint of each subinterval. (b) Approximate the area A under the graph of f by partitioning $[ 0,6 ]$ into three subintervals of equal length an choose $u$ 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 limit algebraically. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2$

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

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. - $f ( x ) = 2 x ^ { 2 } + x + 3 \text { from } x = - 2 \text { to } x = 1 ; n = 6$

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. - $f ( x ) = 2 x ^ { 3 } - 1$ from $x = 1$ to $x = 6 ; n = 5$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -The function $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 }$ describes the volume of a cube, $\mathrm { f } ( \mathrm { x } )$ , in cubic inches, whose length, width, and height each measure $x$ inches. Find the instantaneous rate of change of the volume with respect to $x$ when $x = 4$ inches.

Find the one-sided limit. - $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 }$

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

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$

Determine whether f is continuous at c. - $f ( x ) = \frac { 8 } { x ^ { 2 } + 3 x } ; \quad c = - 8$

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. - $\lim _ { x \rightarrow 4 } \left( x ^ { 2 } + 4 x + 1 \right)$

Find the equation of the tangent line to the graph of f at the given point. - $f ( x ) = - 4 x ^ { 2 } + 7 x \text { at } ( 5,65 )$

Determine whether f is continuous at c. - $f ( x ) = \frac { 5 } { x + 7 } ; c = 0$

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

Find the numbers at which f is continuous. At which numbers is f discontinuous? - $f ( x ) = \frac { 3 x - 1 } { x ^ { 2 } - 25 }$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow-1} f(x)$

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. - $f ( x ) = x ^ { 2 } + 11 x - 15 \text { at } ( 1 , - 3 )$