Exam 5: Applications of Derivatives

Solve the problem. -Explain why the following four statements ask for the same information. (i) Find the roots of $f ( x ) = 3 x ^ { 3 } - 4 x - 1$ (ii) Find the $x$ -coordinates of the intersections of the curve $y = x ^ { 3 }$ with the line $y = 4 x + 1$ . (iii) Find the $x$ -coordinates of the points where the curve $y = x ^ { 3 } - 4 x$ crosses the horizontal line $y = 1$ . (iv) Find the values of $x$ where the derivative of $g ( x ) = \frac { 3 } { 4 } x ^ { 4 } - 2 x ^ { 2 } - x + 5$ equals zero.

Provide an appropriate response. -Find the absolute maximum and minimum values of $f ( x ) = \ln ( \sin x )$ on $\left[ \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right]$ .

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. - $f ( x ) = \ln ( x - 1 ) , [ 2,4 ]$ Round to the nearest thousandth.

Find the absolute extreme values of the function on the interval. - $f ( x ) = x ^ { 8 / 3 } , - 1 \leq x \leq 8$

Solve the problem. -Use Newton's method to estimate the one real solution of the equation $5 x ^ { 5 } - 2 x - 3 = 0$ . Start with $x _ { 1 } = 1$ . Then, in each case find $x _ { 2 }$ .

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as the estimated solution. - $3 x ^ { 2 } + 2 x - 1 = 0 ; x _ { 0 } = 1 ; \text { Find the right-hand solution. }$

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = x - 6 , P ( 3 , - 8 )$

Solve the problem. -Select an appropriate graph of a twice-differentiable function $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ that passes through the points $( - \sqrt { 2 } , 1 )$ , $\left( - \frac { \sqrt { 6 } } { 3 } , \frac { 5 } { 9 } \right) , ( 0,0 ) , \left( \frac { \sqrt { 6 } } { 3 } , \frac { 5 } { 9 } \right)$ and $( \sqrt { 2 } , 1 )$ , and whose first two derivatives have the following sign patterns.

Solve the problem. -Can anything be said about the graph of a function y = f(x) that has a second derivative that is always equal to zero? Give reasons for your answer.

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

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as the estimated solution. - $x ^ { 4 } - 3 = 0 ; x _ { 0 } = 1 ; \text { Find the negative solution. }$

Find the extrema of the function on the given interval, and say where they occur. - $\sin 4 x , 0 \leq x \leq \frac { \pi } { 2 }$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $h ( x ) = \frac { x - 1 } { x ^ { 2 } + 5 x + 10 }$

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. -y $=2 x^{3}-3 x^{2}-12 x$

Solve the problem. -Use Newton's method to estimate the solutions of the equation $3 \mathrm { x } ^ { 4 } + 2 \mathrm { x } - 4 = 0$ . Start with $\mathrm { x } _ { 1 } = 1$ for the right-hand solution and with $x _ { 0 } = - 1.5$ for the solution on the left. Then, in each case find $x _ { 2 }$ .

Solve the problem. -A manufacturer uses raw materials to produce p products each day. Suppose that each delivery of a particular material is $d, whereas the storage of that material is x dollars per unit stored per day. (One unit is the amount required to produce one product). How much should be delivered every x days to minimize the average daily cost in the production cycle between deliveries?

Solve the problem. -Marcus Tool and Die Company produces a specialized milling tool designed specifically for machining ceramic components. Each milling tool sells for $\$ 4$ , so the company's revenue in dollars for $x$ units sold is $R ( x ) = 4 x$ . The company's cost in dollars to produce $x$ tools can be modeled as $C ( x ) = 304 + 30 x ^ { 5 / 8 }$ . Use Newton's method to find the break-even point for the company (that is, find $x$ such that $C ( x ) = R ( x )$ ). Use $x = 370$ as your initial guess and show all your work.

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. - $y=x^{1 / 3}\left(x^{2}-63\right)$

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. -y $= \frac { 16 x } { x ^ { 2 } + 16 }$

Answer each question appropriately. -Suppose the velocity of a body moving along the s-axis is $\frac { \mathrm { ds } } { \mathrm { dt } } = 9.8 \mathrm { t } - 1$ . Find the body's displacement over the time interval from $t = 2$ to $t = 8$ given that $s = s _ { 0 }$ when $t = 0$ .