Exam 8: Limits and Derivatives

Use the given information to find $f ^ { \prime } ( 2 )$ of the function $f ( x ) = g ( x ) h ( x )$ . $g ( 2 ) = 3 \text { and } g ^ { \prime } ( 2 ) = - 2 , h ( 2 ) = - 1 \text { and } h ^ { \prime } ( 2 ) = 4$

Find $\frac { d y } { d x }$ of $y = \sqrt { u }$ , $u = 9 - x ^ { 2 }$ .

Find the x-values (if any) at which the function $f ( x ) = \frac { x + 6 } { x ^ { 2 } + 10 x + 24 }$ is not continuous. Which of the discontinuities are removable?

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow - 5 } \frac { x + 5 } { x ^ { 2 } + 2 x - 15 }$ x -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 f(x)

Find the derivative of the following function using the limiting process. $f ( x ) = \sqrt { 3 x - 4 }$

Find the x-values (if any) at which the function $f ( x ) = - 13 x ^ { 2 } + 6 x + 5$ is not continuous. Which of the discontinuities are removable?

Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing the function with a graphing utility, if one is available. $y = \frac { 8 x - 7 } { x ^ { 2 } + 36 }$

Find the derivative of the function. $f ( x ) = - 8 x ^ { 3 } - 2 x ^ { 2 } - 1$

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 2 } { x + 7 }$

A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by $P = 275 \left( 1 + \frac { 7 t } { 47 + t ^ { 2 } } \right)$ where t is the time (in hours). Find the rate of change of the population when $t = 5.00$ .

Find the marginal revenue for producing x units. (The revenue is measured in dollars.) $R = 50 x - 0.5 x ^ { 2 }$

Find the derivative of the function $h ( x ) = x ^ { 5 / 3 }$ .

Find an equation of the line that is tangent to the graph of f and parallel to the given line. $f ( x ) = 5 x ^ { 2 } , \quad 20 x - y + 3 = 0$

Use the demand function $x = 250 \left( 1 - \frac { 5 p } { 7 p + 2 } \right)$ to find the rate of change in the demand x for the given price $p = \$ 2.00$ . Round your answer to two decimal places.

Find the derivative of the function. $h ( x ) = 18 x ^ { 23 } + 19 x ^ { 13 } - 8 x ^ { 10 } + 20 x - 2$

Find an equation of the tangent line to the graph of f at the given point. $f ( s ) = ( s - 5 ) \left( s ^ { 2 } - 3 \right),$ at $( 1,8 )$

Differentiate the given function. $y = \frac { 5 } { 4 x ^ { 4 } }$