Exam 5: Applications of Derivatives

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - $g ( t ) = \frac { t ^ { 3 } } { 3 } - \frac { 11 } { 2 } t ^ { 2 } + 24 t , 0 \leq t < \infty$

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

Solve the problem. -Use Newton's method to estimate the solutions of the equation $- 4 \mathrm { x } ^ { 2 } - 2 \mathrm { x } + 1 = 0$ . Start with $\mathrm { x } _ { 1 } = 0.5$ for the right-hand solution and with $\mathrm { x } _ { 0 } = - 2$ for the solution on the left. Then, in each case find $\mathrm { x } _ { 2 }$ .

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - $f ( x ) = ( x + 7 ) ^ { 2 } , \Leftrightarrow < x \leq 0$

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

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Solve the problem. -From a thin piece of cardboard 50 in. by $50 \mathrm { in }$ ., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary.

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

Find the most general antiderivative. - $\int \frac { \sec \theta } { \sec \theta - \cos \theta } d \theta$

Graph the rational function. - $y=\frac{x^{3}}{x^{2}-25}$

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

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = e ^ { 3 x } , P \left( 0 , \frac { 7 } { 3 } \right)$

Solve the problem. -If the price charged for a candy bar is $\mathrm { p } ( \mathrm { x } )$ cents, then $\mathrm { x }$ thousand candy bars will be sold in a certain city, where $\mathrm { p } ( \mathrm { x } ) = 24 - \frac { \mathrm { x } } { 22 }$ . How many candy bars must be sold to maximize revenue?

Graph the rational function. - $y = \frac { 3 } { x ^ { 2 } }$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 16 } { x - 4 }$

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. - $f ( x ) = \tan ^ { - 1 } x , \left[ - \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right] \quad$ Round to the nearest thousandth.

Use a computer algebra system (CAS) to solve the given initial value problem. - $y ^ { \prime } = 10 x ^ { 2 } \sin x , y ( 0 ) = 1$

Answer the problem. -Consider the quartic function $f ( x ) = a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e , a \neq 0$ . Must this function have at least one critical point? Give reasons for your answer. (Hint: Must $f ^ { \prime } ( x ) = 0$ for some $x$ ?) How many local extreme values can $f$ have?

Solve the problem. -How close does the curve $y = \sqrt { x }$ come to the point $( 3,0 )$ ? (Hint: If you minimize the square of the distance, you can avoid square roots.)

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x). - $y ^ { \prime } = 4 | x | = \left\{ \begin{array} { c c } - 4 x , & x \leq 0 \\4 x , & x > 0\end{array} \right.$