Exam 4: Applications of Derivatives

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \theta ) = \csc \theta \cot \theta - 3 , \mathrm { P } \left( \frac { 3 \pi } { 4 } , 0 \right)$

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. -

Use the following function and a graphing calculator to answer the questions. $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 3 x + 2 , [ - 0.5,1.8 ]$ a). Plot the function over the interval to see its general behavior there. Sketch the graph below. b). Find the interior points where f' = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f' as well. List the points as ordered pairs (x, y). c). Find the interior points where f' does not exist. List the points as ordered pairs (x, y). d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y). e). Find the function's absolute extreme values on the interval and identify where they occur.

Solve the problem. -A marathoner ran the 26.2 mile New York City Marathon in 2.7 hrs. Did the runner ever exceed a speed of 9 miles per hour?

Identify the function's local and absolute extreme values, if any, saying where they occur. - $g ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 7 } { 3 } x ^ { 3 } + 7 x ^ { 2 } - 8 x - 4$

Find the location of the indicated absolute extremum for the function. -Maximum

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \mathrm { e } ^ { 5 \mathrm { x } } , \mathrm { P } \left( 0 , \frac { 11 } { 5 } \right)$

Find the largest open interval where the function is changing as requested. -Decreasing $f ( x ) = x ^ { 3 } - 4 x$

Find the absolute extreme values of the function on the interval. - $f ( x ) = | x - 7 | , \quad 5 \leq x \leq 10$

Find the location of the indicated absolute extremum for the function. -Maximum

Find the location of the indicated absolute extremum for the function. -Minimum

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. - $s ( t ) = \sqrt { t ( 4 - t ) } , \quad [ - 1,5 ]$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 5 } - 15 x ^ { 4 } - 3 x ^ { 3 } - 172 x ^ { 2 } + 135 x - 0.037$

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval. - $f ( x ) = x + \frac { 18 } { x } , [ 2,9 ]$

Find the largest open interval where the function is changing as requested. -Decreasing $f ( x ) = | x - 8 |$

Solve the problem. -Find the graph that matches the given table. () -1 0 1 does not exist 3 0

Find the largest open interval where the function is changing as requested. -Decreasing $f ( x ) = \sqrt { 4 - x }$

Find the extreme values of the function and where they occur. - $\mathrm { y } = \frac { 1 } { \mathrm { x } ^ { 2 } - 1 }$

Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur. -