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

Find the limit as x approaches c of the average rate of change of the function from c to x. - $c = - 4 ; \quad f ( x ) = x ^ { 3 }$

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

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. - $R ( x ) = \frac { x ^ { 3 } - x ^ { 2 } + x - 1 } { x ^ { 4 } - x ^ { 3 } + 3 x - 3 }$

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

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

Find the limit algebraically. - $\lim _ { x \rightarrow 0 } \frac { 4 + x } { x ^ { 3 } }$

Find the limit algebraically. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 }$

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. - $f ( x ) = 5 x ^ { 2 } + x \text { at } ( - 4,76 )$

Use the graph of y = g(x) to answer the question. -Does $\lim _ { x \rightarrow 4 } g ( x )$ exist? If it does, what is it?

Find the equation of the tangent line to the graph of f at the given point. - $f ( x ) = x ^ { 2 } + 11 x - 15$ at $( 1 , - 3 )$

Find the limit algebraically. - $\lim _ { x \rightarrow - 1 } \left( 3 x ^ { 2 } - 19 \right) ^ { 2 }$

Write the word or phrase that best completes each statement or answers the question. - $\mathrm { f } ( \mathrm { x } ) = 2 \mathrm { x } - 2$ is defined on the interval $[ 1,5 ]$ . In (a) and (b), approximate the area under $\mathrm { f }$ as follows: (a) Partition $[ 1,5 ]$ into four subintervals of equal length and choose $u$ as the left endpoint of each subinterval. (b) Partition $[ 1,5 ]$ into four subintervals of equal length and choose $u$ as the right endpoint of each subinterval. (c) What is the actual area A?

Find the derivative of the function at the given value of x. - $f ( x ) = x ^ { 2 } + 9 x + 9 ; x = 8$

Use the graph of y = g(x) to answer the question. -Find $f ( 1 )$ .

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 112 feet per second. The function $s ( t ) = - 16 t ^ { 2 } + 112 t + 4$ describes the ball's height above the ground, $s ( t )$ , in feet, $t$ seconds after it was hit. What is the instantaneous speed of the ball $0.5$ seconds after it was hit?

Find the equation of the tangent line to the graph of f at the given point. - $f ( x ) = 2 x ^ { 2 } + x - 3 \text { at } x = ( 4,33 )$

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

Write the word or phrase that best completes each statement or answers the question. -s $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ is defined on the interval $[ 0,4 ] .$ In (a) and (b), approximate the area under $f$ (to three decimal place follows: (a) Partition $[ 0,4 ]$ into four subintervals of equal length and choose $u$ as the left endpoint of each subinterval. (b) Partition $[ 0,4 ]$ into four subintervals of equal length and choose $u$ as the right endpoint of each subinterval. (c) Use a graphing utility to approximate the actual area A.

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