Exam 4: Extrema on an Interval

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

Find measurement of the side of a square floor tile is 14 inches, with a possible error of $\frac { 1 } { 78 }$ inch. Use differentials to approximate the possible propagated error in computing the area of the square.

Use a graphing utility to graph the function $f ( x ) = \sqrt { x } + \cos \frac { x } { 2 } \text { and locate the }$ absolute extrema of the function on the interval $[ 0,2 \pi ]$

Analyze and sketch a graph of the function $f ( x ) = \frac { x ^ { 2 } - 2 x + 9 } { x }$

$\text { Use a graphing utility to graph the function } f ( x ) = \frac { 8 } { 8 - x } \text { and locate the absolute }$ extrema of the function on the interval $( 0,8 )$ .

Find all points of inflection on the graph of the function $f ( x ) = - 5 e ^ { - 7 x ^ { 2 } }$

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded For the first two minutes is given by $T = \frac { 1351 + 78 t } { 54 + t } . \text { Find } \lim _ { t \rightarrow \infty } T$

Sketch the graph of the relation $x y ^ { 2 } = 2$ using any extrema, intercepts, symmetry, and asymptotes.

Determine the open intervals on which the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 625 }$ is concave upward or concave downward.

$\text { The resistance } R \text { of a certain type of resistor is } R = \sqrt { 0.009 T ^ { A } - 4 T + 100 } \text {, where } R \text { is }$ measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the Nearest whole number.

Sketch the graph of the relation $x ^ { 2 } y = 4$ using any extrema, intercepts, symmetry, and asymptotes. A) B) D) E)

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 36%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)

$\text { Find the value of the derivative (if it exists) of the function } f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 49 } \text { at the }$ extremum point . (0,0)

Determine whether the Mean Value Theorem can be applied to the function $f ( x ) = x ^ { 3 }$ on the closed interval $[ 0,16 ]$ . If the Mean Value Theorem can be applied, find all numbers $c$ in the open interval $( 0,16 )$ such that $f ^ { t } ( c ) = \frac { f ( 16 ) - f ( 0 ) } { 16 - 0 }$ .

Determine the open intervals on which the graph of the function $y = 3 x - \tan x , \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \text { is concave upward or concave downward. }$

Find the limit. $\lim _ { x \rightarrow \infty } \frac { - 6 x } { \sqrt { 64 x ^ { 2 } - 5 } }$

Find all critical numbers of of the function $f ( \theta ) = 2 \sec \theta + \tan \theta , 14 \pi < \theta < 16 \pi .$

Find a function f that has derivative $f ^ {t } ( x ) = 12 x - 6 \text { and with graph passing through }$ the point (5,6).

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 529 square meters.

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