Exam 4: Extrema on an Interval

Find the points of inflection and discuss the concavity of the function. $f ( x ) = 4 x ^ { 3 } - 5 x ^ { 2 } + 5 x - 7$

Match the function $f ( x ) = \frac { 2 \sin x } { x ^ { 2 } + 1 }$ with one of the following graphs.

Identify the open intervals where the function $f ( x ) = x \sqrt { 30 - x ^ { 2 } }$ is increasing or decreasing.

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

Find all relative extrema of the function $f ( x ) = - 4 x ^ { 2 } - 32 x - 62$ . Use the Second Derivative Test where applicable.

Determine the open intervals on which the graph of $f ( x ) = 5 x + 7 \cos x$ is concave downward or concave upward.

Find measurements of the base and altitude of a triangle are found to be 54 and 33 centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate The propagated error in computing the area of the triangle. Round your answer to four decimal places.

The graph of a function f is is shown below. Sketch the graph of the derivative .

The height of an object t seconds after it is dropped from a height of 250 meters is $s ( t ) = - 4.9 t ^ { 2 } + 250$ . Find the time during the first 8 seconds of fall at which the instantaneous velocity equals the average velocity.

Find all relative extrema of the function $f ( x ) = 2 x ^ { 4 } - 32 x ^ { 3 } + 4$ . 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 $\theta$ radians. The distance (in meters) the ball bearing rolls in $t$ seconds is $s ( t ) = 4.1 ( \sin \theta ) t ^ { 2 }$ Determine the speed of the ball bearing after $t$ seconds.

$\text { Match the function } f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } + 2 } \text { with one of the following graphs. }$

Sketch the graph of the function $f ( x ) = \frac { 1 + x } { 1 - x }$ using any extrema, intercepts, symmetry, and asymptotes.

$\text { Find all critical numbers of the function } g ( x ) = x ^ { 4 } - 4 x ^ { 2 } \text {. }$

Find the point of inflection of the graph of the function $f ( x ) = 2 \sin \frac { x } { 7 }$ on the interval $[ 0,14 \pi ]$

$\text { Find the relative extremum of } f ( x ) = - 9 x ^ { 2 } + 54 x + 2 \text { by applying the First }$ Derivative Test.

Determine whether the Mean Value Theorem can be applied to the function $f ( x ) = x ^ { 2 }$ on the closed interval $[ 3,9 ]$ . If the Mean Value Theorem can be applied, find all numbers $c$ in the open interval $( 3,9 )$ such that $f ^ { t} ( c ) = \frac { f ( 9 ) - f ( 3 ) } { 9 - ( 3 ) }$ .

Sketch a graph of f is shown below. For which values of x is $f ^ { t } ( x ) \text { zero? }$

A container holds 3 liters of a 25% brine solution. A model for the concentration C of the mixture after adding x liters of a 0.67 % brine solution to the container and then draining x liters Of the well-mixed solution is given as $C = \frac { 3 + 2 x } { 12 + 3 x } . \text { Find } \lim _ { x \rightarrow \infty } C . \text { Round your answer to two decimal }$ places.

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