Exam 2: Limits and Derivatives

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is $20 \mathrm { ft }$ and is increasing at the rate of $\frac { 1 } { 6 } \mathrm { ft } / \mathrm { sec }$ . Round to the nearest tenth if necessary.

The cost (in dollars) of producing x units of a certain commodity is $C ( x ) = 4,280 + 13 x + 0.03 x ^ { 2 }$ Find the average rate of change with respect to $x$ when the production level is changed from $x = 102$ to $x = 122$ .

Differentiate. $K ( x ) = \left( 3 x ^ { 5 } + 1 \right) \left( x ^ { 6 } - 4 x \right)$

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s ( t ) = t ^ { 10 } e ^ { - t } ; \quad t = 1$

Find the derivative of the function. $f ( x ) = x \sin ^ { 8 } x$

$\text { Sketch the graph of the derivative } f ^ { \prime } \text { of the function } f \text { whose graph is given. }$

The position function of a particle is given by $s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , t \leq 0$ When does the particle reach a velocity of $22 \mathrm {~m} / \mathrm { s }$ ?

The slope of the tangent line to the graph of the exponential function $y = 6 ^ { x }$ at the point $( 0,1 )$ is $\lim _ { x \rightarrow 0 } \frac { 6 ^ { x } - 1 } { x }$ . Estimate the slope to three decimal places.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 3 x = x \cos 3 y , \left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)$

If $g ( x ) = \sqrt { 2 - 3 x }$ , use the definition of derivative to find $g ^ { \prime } ( x )$ .

Suppose that $F ( x ) = f ( g ( x ) )$ and $g ( 14 ) = 2 , g ^ { \prime } ( 14 ) = 4 , f ^ { \prime } ( 14 ) = 15$ , and $f ^ { \prime } ( 2 ) = 13$ . Find $F ^ { \prime } ( 14 )$ .

Find equations of the tangent lines to the curve $y = \frac { x - 10 } { x + 10 }$ that are parallel to the line $x - y = 10$

Differentiate. $y = \frac { \sin x } { 6 + \cos x }$

Find an equation of the tangent line to the curve $120 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2312 \left( x ^ { 2 } - y ^ { 2 } \right)$ at the point $( 4,1 )$ .

Find the differential of the function at the indicated number. $f ( x ) = 13 \sin x + 4 \cos x ; \quad x = \frac { \pi } { 4 }$

Gravel is being dumped from a conveyor belt at a rate of $34 \mathrm { ft } / \mathrm { min }$ and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is $13 \mathrm { ft }$ high? Round the result to the nearest hundredth.