Exam 5: Applications of Derivatives

For the given expression y, find y'' and sketch the general shape of the graph of y = f(x). - =

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Solve the problem. -You are planning to close off a corner of the first quadrant with a line segment 17 units long running from ( $x , 0$ ) to $( 0 , y )$ . Show that the area of the triangle enclosed by the segment is largest when $x = y$ .

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)=30x-0.5 C(x)=2x+8

Solve the problem. -The acceleration of gravity near the surface of Mars is $3.72 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . If a rock is blasted straight up from the surface with an initial velocity of $85 \mathrm {~m} / \mathrm { sec }$ (about $190 \mathrm { mph }$ ), how high does it go? (Hint: When is velocity zero?)

Solve the problem. -Use Newton's method to estimate the solutions of the equation $3 x ^ { 2 } + 2 x - 5 = 0$ . Start with $x _ { 1 } = 0.5$ for the right-hand solution and with $x _ { 0 } = - 2$ for the solution on the left. Then, in each case find $x _ { 2 }$ .

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 59 x + 7$

Determine all critical points for the function. - $f ( x ) = x ^ { 2 } + 12 x + 36$

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

Solve the problem. -Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius $3 .$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $g ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 4 } { 3 } x ^ { 3 } - \frac { 9 } { 2 } x ^ { 2 } + 36 x - 4$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = - x ^ { 3 } + 1.5 x ^ { 2 } + 36 x + 2$

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow \theta } \frac { \sin 9 x } { 5 x }$

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. - $h ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 8 x + 2 , \Leftrightarrow < x \leq 0$

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

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x + \frac { 15 } { 4 } , & x \leq 1 \\x ^ { 3 } - 6 x ^ { 2 } + 8 x , & x > 1\end{array} \right.$

Solve the problem. - $\text { The function } y = \cot x - \frac { 2 \sqrt { 3 } } { 3 } \csc x \text { has an absolute maximum value on the interval } 0 < x < \pi \text {. Find it. }$

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 absolute extreme values of the function on the interval. - $f ( x ) = \ln ( - x ) , - 9 \leq x \leq - 1$

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. -Which one is correct, and which one is wrong? Give reasons for your answers. (a) $\lim _ { x \rightarrow 2 } \frac { x + 2 } { x ^ { 2 } - 4 } = \lim _ { x \rightarrow 2 } \frac { 1 } { 2 x } = - \frac { 1 } { 4 }$ (b) $\lim _ { x \rightarrow 2 } \frac { x + 2 } { x ^ { 2 } - 4 } = \frac { 0 } { - 8 } = 0$