Exam 3: Differentiation

Find the slope of the curve at the indicated point. -A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant $80.0 \mathrm {~mm}$ . Find the rate of change of surface area with respect to radius when the radius is $0.090 \mathrm {~mm}$ . (Answer can be left in terms of $\pi$ ).

Find an equation for the tangent to the curve at the given point. - $y = x ^ { 2 } + 2 , \quad ( 3,11 )$

The graph of a function is given. Choose the answer that represents the graph of its derivative -

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. - u(2)=10,(2)=2,v(2)=-1,(2)=-4 (3v-u) at x=2

Find an equation for the tangent to the curve at the given point. - $y = x ^ { 2 } - x , ( - 3,12 )$

Estimate the slope of the curve at the indicated point. -

Provide an appropriate response. -Under standard conditions, molecules of a gas collide billions of times per second. If each molecule has diameter average distance between collisions is given by $\mathrm { L } = \frac { 1 } { \sqrt { 2 } \pi \mathrm { t } ^ { 2 } \mathrm { n } } \text {, }$ where $\mathrm { n }$ , the volume density of the gas, is a constant. Find $\frac { \mathrm { dL } } { \mathrm { dt } }$ .

Provide an appropriate response. -Is there any difference between finding the derivative of f(x) at x = a and finding the slope of the line tangent to f(x) at x = a? Explain.

Given the graph of f, find any values of x at which f ' is not defined. -The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f.

Given the graph of f, find any values of x at which f ' is not defined. -

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. - u(2)=6,(2)=3,v(2)=-1,(2)=-4. at x=2

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. - $u ( 1 ) = 4 , u ^ { \prime } ( 1 ) = - 7 , v ( 1 ) = 7 , v ^ { \prime } ( 1 ) = - 3 \text {. }$ $\frac { d } { d x } \left( \frac { u } { v } \right)$ at $x = 1$

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$ - $g ( x ) = 5 x + \sqrt { x }$

Calculate the derivative of the function. Then find the value of the derivative as specified. - $f ( x ) = \frac { 8 } { x + 2 } ; f ^ { \prime } ( 0 )$

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. - $u ( 1 ) = 5 , u ^ { \prime } ( 1 ) = - 7 , v ( 1 ) = 6 , v ^ { \prime } ( 1 ) = - 2 \text {. }$ $\frac { d } { d x } ( 3 v - u )$ at $x = 1$

Find the derivative. - $y = \sqrt [ 8 ] { x ^ { 7 } } + x ^ { 7 e }$

Find the derivative. - $s = \frac { 5 e ^ { t } } { 2 e ^ { t } + 1 }$

Find the indicated derivative. - $\frac { d v } { d t }$ if $v = t + \frac { 6 } { t }$

Find the derivative. - $r = \frac { 2 } { s ^ { 3 } } - \frac { 8 } { s }$

Provide an appropriate response. -Over what intervals of x-values, if any, does the function y $y = 2 x ^ { 2 }$ increase as x increases? For what values of x, if any, is y' positive? How are your answers related?