Exam 4: Applications of Derivatives

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. - $\text { The function } \mathrm { P } ( \mathrm { x } ) = 2 \mathrm { x } + \frac { 200 } { \mathrm { x } } , 0 < \mathrm { x } < \infty$ models the perimeter of a rectangle of dimensions x by $\frac{100}{x}$ (a) Find the extreme values for P. (b) Give an interpretation in terms of perimeter of the rectangle for any values found in part (a).

Find the absolute extreme values of the function on the interval. - $f ( x ) = \csc x , - \frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$

Find the extreme values of the function and where they occur. - $y = ( x - 5 ) ^ { 2 / 3 }$

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = ( x + 5 ) ^ { 2 } e ^ { - x }$

Solve the problem. -On our moon, the acceleration of gravity is $1.6 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later?

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \theta ) = 8 - \csc ^ { 2 } \theta , \mathrm { P } \left( \frac { \pi } { 4 } , 0 \right)$

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } 4 - x , & x < 0 \\4 + 3 x - x ^ { 2 } , & x \geq 0\end{array} \right.$

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. - $f ( x ) = x ^ { 2 } + 5 x + 2 , [ 1,2 ]$

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \theta ) = 4 + \sec ^ { 2 } \theta , \mathrm { P } ( \pi , 0 )$

Find the absolute extreme values of the function on the interval. - $f ( x ) = \ln ( - x ) , - 7 \leq x \leq - 1$

Provide an appropriate response. -Let f have a derivative on an interval I. f' has successive distinct zeros at x = 1 and x = 5. Prove that there can be at most one zero of f on the interval (1, 5).

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

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 12 x + 2$

Find the absolute extreme values of the function on the interval. - $F ( x ) = - \frac { 3 } { x ^ { 2 } } , 0.5 \leq x \leq 3$

Find the largest open interval where the function is changing as requested. -Increasing y = 7x - 5

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = ( 8 - x ) ( 9 - x )$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.01 x ^ { 5 } - x ^ { 4 } + x ^ { 3 } + 8 x ^ { 2 } - 7 x + 94$

Find all possible functions with the given derivative. - $y ^ { \prime } = \frac { 3 } { 2 \sqrt { t } }$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $h ( x ) = \frac { x - 2 } { x ^ { 2 } + 3 x + 6 }$

Find the function with the given derivative whose graph passes through the point P. - $r ^ { \prime } ( t ) = \sec ^ { 2 } t - 4 , P ( 0,0 )$