Exam 5: Applications of Derivatives

Solve the problem. -Find the number of units that must be produced and sold in order to yield the maximum profit, given the follow equations for revenue and cost: R(x)=2x G(x)=0.01+0.9x+40

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time $t$ , find the body's position $\mathrm { t }$ . $v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$

Find the extreme values of the function and where they occur. - $y = \frac { 6 } { \sqrt { 1 - 3 x ^ { 2 } } }$

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

Find the absolute extreme values of the function on the interval. - $g ( x ) = 9 - 8 x ^ { 2 } , - 2 \leq x \leq 3$

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

Using the derivative of f(x) given below, determine the critical points of f(x). - $f ^ { \prime } ( x ) = ( x - 6 ) ^ { 2 } ( x + 2 )$

Find an antiderivative of the given function. - $\mathrm { e } ^ { - 9 x / 7 }$

Solve the problem. -At noon, ship A was 15 nautical miles due north of ship B. Ship A was sailing south at 15 knots (nautical miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was sailing east at 6 knots and Continued to do so all day. The visibility was 5 nautical miles. Did the ships ever sight each other?

Find all possible functions with the given derivative. - $y ^ { \prime } = x ^ { 3 } - 8 x$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow - 0 } \frac { x } { \sin x }$

Solve the problem. -Use Newton's method to estimate the one real solution of $- 3 x ^ { 3 } - 2 x - 5 = 0$ . Start with $x _ { 1 } = - 0.5$ and then find $x _ { 2 }$

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

Find the function with the given derivative whose graph passes through the point P. - $\mathrm { r } ^ { \prime } ( \theta ) = \csc \theta \cot \theta - 3 , \mathrm { P } \left( \frac { 2 \pi } { 3 } , 0 \right)$

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

Solve the problem. -The 8 ft wall shown here stands 27 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 6 )$

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 3 } \mathrm { y } } { \mathrm { dx } ^ { 3 } } = 4 ; \mathrm { y } ^ { \prime \prime } ( 0 ) = 3 , \mathrm { y } ^ { \prime } ( 0 ) = - 3 , \mathrm { y } ( 0 ) = 5$

Answer the question. -It took 29 seconds for the temperature to rise from $1^{\circ} \mathrm{F}$ to $164^{\circ} \mathrm{F}$ when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some time, the temperature was increasing at a rate of $\frac{163}{29} \circ \mathrm{F} / \mathrm{sec}$ . Explain.

Provide an appropriate response. - $\text { The function } f ( x ) = \left\{ \begin{array} { l l } 2 x & 0 \leq x < 1 \\0 & x = 1\end{array} \text { is zero at } x = 0 \text { and } x = 1 \text { and differentiable on } ( 0,1 ) \text {, but its derivative on ( } 0,1 \right)$ is never zero. Does th is example contradict Rolle's Theorem?