Exam 4: Extrema on an Interval

A company introduces a new product for which the number of units sold S is $S ( t ) = 300 \left( 7 - \frac { 10 } { 5 + t } \right)$ where $t$ is the time in months since the product was introduced. Find the average value of $S ( t )$ during the first year.

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = ( x + 3 ) ( x + 2 ) ^ { 2 }$ on the closed interval $( - 3 , - 2 )$ . If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $( - 3 , - 2 )$ such that $f ^ { t} ( c ) = 0$ .

Find two positive numbers whose product is 181 and whose sum is a minimum.

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 7 x + 6 } { \sqrt { 64 x ^ { 2 } + 2 x } }$

$\text { Find the value of the derivative (if it exists) of } f ( x ) = ( x - 2 ) ^ { 4 / 5 } \text { at the indicated }$ extremum.

$\text { Use differentials to approximate the value of } \sqrt { 25.1 } \text {. Round your answer to four }$ decimal places.

The resistance R of a certain type of resistor is $R = \sqrt { 0.001 T ^ { 4 } - 3 T + 400 } ,$ where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra System to find $\frac { d R } { d T } .$

$\text { Determine the open intervals on which the graph of } f ( x ) = 3 x ^ { 2 } + 7 x - 3 \text { is concave }$ downward or concave upward.

Locate the absolute extrema of the function $g ( x ) = \frac { 4 x + 5 } { 5 } \text { on the closed interval }$ $( 0,5 )$

Determine whether the Mean Value Theorem can be applied to the function $f ( x ) = 2 \sin x + \sin 2 x$ on the closed interval $[ 4 \pi , 5 \pi ]$ . If the Mean Value Theorem can be applied, find all numbers $c$ in the open interval $( 4 \pi , 5 \pi )$ such that $f ^ {t } ( c ) = \frac { f ( 5 \pi ) - f ( 4 \pi ) } { 5 \pi - 4 \pi }$ .

A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at altitude h (in kilometers). The data are shown below. Find the rate of change of the pressure with Respect to altitude $\text { when } h = 10 \text { using the relation } \ln P = - 0.1499 h + 10.3016 \text {. Round your answer to }$ one decimal place. h 0 5 10 15 20 p 28,085 15,169 6,458 3,371 1405

Analyze and sketch a graph of the function $y = x \sqrt { 4 - x ^ { 2 } }$

Analyze the graph of the function $f ( x ) = - x + 4 \arctan ( x ) . \text { Determine any intercepts, }$ relative extrema, points of inflection and asymptotes. Also determine where the graph is increasing or decreasing and concave up or concave down. Then identify the graph from the choices below.

A company introduces a new product for which the number of units sold S is $S ( t ) = 300 \left( 5 - \frac { 10 } { 3 + t } \right)$ where $t$ is the time in months since the product was introduced. During what month does $S ^ { t } ( t )$ equal the average value of $S ( t )$ during the first year?

$\text { Find the relative maxima of } f ( x ) = \cos ^ { 2 } ( 4 x ) \text { on the interval } ( 0,3.142 ) \text { by applying }$ the First Derivative Test. Round numerical values in your answer to three decimal places.

$\text { Find all critical numbers of the function } f ( x ) = \sin ^ { 2 } 6 x + \cos 6 x , 0 < x < \frac { \pi } { 3 } \text {. }$

Find the relative minima of $f ( x ) = \cos ^ { 2 } ( 9 x ) \text { on the interval } ( 0,1.396 ) \text { by applying }$ the First Derivative Test. Round numerical values in your answer to three decimal places.

$\text { Compare } d y \text { and } \Delta y \text { for } y = - 4 x ^ { 2 } - 3 \text { at } x = - 1 \text { with } \Delta x = d x = - 0.06 \text {. Give your }$ answers to four decimal places.

The graph of a function f is is shown below. Sketch the graph of the derivative $f ^ { \prime } \text {. }$

The vsum of the perimeters of an equilateral triangle and a square is 19. Find the dimensions of the triangle and the square that produce a minimum total area.