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

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _ { x \rightarrow 2 } f ( x )$

Find the limit algebraically. - $\lim _ { x \rightarrow - 4 } \left( 4 x ^ { 3 } - 2 x + 123 \right) ^ { 2 / 3 }$

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

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

Determine whether f is continuous at c. - $f ( x ) = \frac { x - 6 } { ( x - 1 ) ( x - 9 ) } ; \quad c = 0$

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

Find the derivative of the function at the given value of x using a graphing utility. If necessary, round to four decimal places. - $f ( x ) = \sin ( 8 x ) ; x = 7$

Find the limit algebraically. - $\lim _ { x \rightarrow 1 } \sqrt { 3 x - 2 }$

Determine whether f is continuous at c. - $f ( x ) = \frac { x - 7 } { x + 9 } ; c = 7$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 0 } f ( x ) , \quad f ( x ) = \left\{ \begin{array} { r l } x + 1 & x < 0 \\4 x + 1 & x \geq 0\end{array} \right.$

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

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

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

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. - $f ( x ) = - 4 x ^ { 2 } + 7 x \text { at } ( 5,65 )$

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

Use the graph of y = g(x) to answer the question. - $\text { Find } \lim _ { x \rightarrow 1 } g ( x ) \text {. }$

Find the limit as x approaches c of the average rate of change of the function from c to x. -c = 3; $f ( x ) = 3 x ^ { 2 } + 39 x$

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. - $f ( x ) = 5 x - 1 \text { from } x = 1 \text { to } x = 4 ; n = 2$

Find the one-sided limit. - $\lim _ { x \rightarrow ( 3 \pi / 4 ) ^ { - } } \sin x$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }$