Exam 4: Extrema on an Interval

A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

$\text { Locate the absolute extrema of the function } f ( x ) = \sin \pi x \text { on the closed interval }$ $\left[ 0 , \frac { 1 } { 3 } \right]$

Sketch a graph of the function $f ( x ) = x \tan x \text { over the interval } - \frac { 3 \pi } { 2 } < x < \frac { 3 \pi } { 2 }$

Sketch a graph of the function $f ( x ) = 3 x - 6 \sin x \text { over the interval } 0 \leq x \leq 2 \pi$

$\text { Locate the absolute extrema of the function } f ( x ) = x ^ { 3 } - 12 x \text { on the closed interval }$ $[ 0,4 ] .$

A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Suppose the deflection D of a beam of length L is $D = 6 x ^ { 4 } - 9 L x ^ { 3 } + 2 L ^ { 2 } x ^ { 2 }$ where $x$ is the distance from one end of the beam. Find the value of x that yields the maximum deflection.

$\text { The graph of } f \text { is shown. Graph } f , f ^ { \prime } \text { and } f ^ { \prime \prime } \text { on the same set of coordinate axes. }$

The graph of f is shown below. On what interval is $f ^ {t } \text { an increasing function? }$

$\text { For the function } f ( x ) = ( x - 1 ) ^ { \frac { 2 } { 3 } } \text { : }$ (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Use a graphing utility to confirm your results.

Find the open interval(s) on which the function $f ( x ) = \cos ^ { 2 } ( 3 x )$ is increasing in the interval Round numerical values in your answer to three decimal places.

$\text { Determine the open intervals on which the graph of } y = - 6 x ^ { 3 } + 8 x ^ { 2 } + 6 x - 5 \text { is }$ concave downward or concave upward.

Match the function $f ( x ) = \frac { 4 x + 3 } { 5 x + 6 }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 1 f(x)

Find all relative extrema of the function $f ( x ) = x ^ { 2 / 3 } - 6$ . Use the Second Derivative Test where applicable.

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is $\theta = \frac { \pi } { 9 }$ radians. The distance (in meters) the ball bearing rolls in $t$ seconds is $s ( t ) = 4.9 ( \sin \theta ) t ^ { 2 }$ . Determine the value of $s ^ { t } ( t )$ after one second. Round numerical values in your answer to one decimal place.

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = \cos \pi x \text { on }$ the closed interval $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ . If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ such that $f ^ { t } ( c ) = 0$ .

$\text { The graph of } f \text { is shown. Graph } f , f ^ { \prime } \text { and } f ^ { \prime \prime } \text { on the same set of coordinate axes. }$

Find the limit. $\lim _ { x \rightarrow \infty } \left( \frac { 3 } { 4 } x + \frac { 2 } { x ^ { 4 } } \right)$

Analyze and sketch a graph of the function $f ( x ) = \frac { x } { 1 + x ^ { 4 } }$

$\text { Identify the open intervals where the function } f ( x ) = 6 x ^ { 2 } - 6 x + 4 \text { is increasing or }$ decreasing.