Exam 3: The Derivative and the Tangent Line Problem

Identify the graph which has the following characteristics. f(0)=-2 (x)=2,-\infty

Find the derivative of the function. $f ( \theta ) = \frac { 7 } { 5 } \sin ^ { 2 } 2 \theta$

Find the derivative of the function. $f ( x ) = x ^ { 8 } \sqrt { 5 - 3 x }$

$\text { Find an equation of the line that is tangent to the graph of the function } f ( x ) = \frac { 7 } { \sqrt { x } }$ and parallel to the line $7 x + 2 y - 18 = 0$ .

The ordering and transportation cost C for the components used in manufacturing a product is $C = 160 \left( \frac { 280 } { x ^ { 2 } } + \frac { x } { x + 40 } \right) , x \geq 1$ where $C$ is measured in thousands of dollars and $x$ is the order size in hundreds. Find the rate of change of $C$ with respect to $x$ for $x = 24$ . Round your answer to two decimal places.

$\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } }$

ladder feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of feet per second. Find the rate at which the angle Between the ladder and the wall of the house is changing when the base of the ladder is feet from The wall. Round your answer to three decimal places.

$\text { Find the derivative of the function } f ( x ) = x ^ { 6 } e ^ { x } \text {. }$

A man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure). When he is feet from the base of the light, at what rate is the Tip of his shadow moving?

$\text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right)$

$\text { Find the derivative of the trigonometric function } f ( s ) = s ^ { 3 } \tan s \text {. }$

Suppose the position function for a free-falling object on a certain planet is given by $s ( t ) = - 14 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ . A silver coin is dropped from the top of a building that is 1370 feet tall. Find velocity of the coin at impact. Round your answer to the three decimal places.

man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet above the ground (see figure). When he is feet from the base of the light, at what rate is the Length of his shadow changing?

Determine all values of , (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 4 } - 4 x + 4$

$\text { Evaluate the derivative of the function } f ( t ) = \frac { 6 t + 3 } { 5 t - 1 } \text { at the point } \left( 5 , \frac { 11 } { 8 } \right) \text {. }$

$\text { Use the quotient rule to differentiate the following function } f ( s ) = \frac { 2 s } { s ^ { 5 } + 7 } \text { and }$ evaluate $f ^ { t } ( - 2 )$

$\text { Find the derivative of the function } f ( x ) = 13 \arcsin ( x - 1 ) \text {. }$

Complete two iterations of Newton's Method for the function using the initial guess . Round all numerical values in your answer to four decimal places.

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 3 } + 8 x + x ^ { 13 } y - y ^ { 6 } = 4$

$\text { Find an equation to the tangent line to the graph of the function } f ( x ) = \tan ^ { 8 } x$ at the point $\left( \frac { 4 \pi } { 5 } , 0.078 \right)$ . The coefficients below are given to two decimal places.