Exam 4: Applications of Derivatives

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 53 x ^ { 2 } - 86 x + 5$

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = x - 9 , P ( 2 , - 10 )$

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

Find the absolute extreme values of the function on the interval. -g(x) = -x² + 8x - 16, 4 $\leq$ x $\leq$ 4

Solve the problem. -A trucker handed in a ticket at a toll booth showing that in 3 hours he had covered 222 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

Find the largest open interval where the function is changing as requested. -Increasing $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 4 } \mathrm { x } ^ { 2 } - \frac { 1 } { 2 } \mathrm { x }$

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. -Imagine there is a function for which f'(x) = 0 for all x. Does such a function exist? Is it reasonable to say that all values of x are critical points for such a function? Is it reasonable to say that all values of x are extreme values for such a function. Give reasons for your answer.

Find the absolute extreme values of the function on the interval. - $f ( x ) = 6 x ^ { 2 / 3 } , - 27 \leq x \leq 8$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positios $\mathrm { t }$ . $v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$

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

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. -Let $f ( x ) = \left| x ^ { 3 } - 9 x \right|$ (a) Does $f ^ { \prime } ( 0 )$ exist? (b) Does $f ^ { \prime } ( 3 )$ exist? (c) Does $f ^ { \prime } ( - 3 )$ exist? (d) Determine all extrema of $f$ .

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. - $f ( x ) = x ^ { 1 / 3 }$ , $[-5,2]$

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

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = \sqrt { x ^ { 2 } + 12 x + 72 }$

Solve the problem. -A rocket lifts off the surface of Earth with a constant acceleration of 30 m/sec². How fast will the rocket be going 2.5 minutes later?

Determine all critical points for the function. - $f ( x ) = x ^ { 2 } + 6 x + 9$

Solve the problem. -At about what velocity do you enter the water if you jump from a $15 \mathrm {~meter}$ cliff? (Use $\mathrm { g } = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ .)

Find the function with the given derivative whose graph passes through the point P. - $g ^ { \prime } ( x ) = \frac { 4 } { x ^ { 2 } } + 6 x , P ( - 2,5 )$

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. - $f ( x ) = \ln ( x - 1 ) , [ 2,6 ] \quad$ Round to the nearest thousandth.

Solve the problem. -Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time $t$ , fint body's position at time $t$ . $a = 3.2 , v ( 0 ) = - 14 , s ( 0 ) = - 5$