Exam 2: Limits and Derivatives

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.

Find the points on the curve $y = 2 x ^ { 3 } + 3 x ^ { 2 } - 36 x + 19$ where the tangent is horizontal.

Find the derivative of the function. $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)$

In an adiabatic process (one in which no heat transfer takes place), the pressure P and volume V of an ideal gas such as oxygen satisfy the equation $P ^ { 5 } V ^ { 7 } = C$ where $C$ is a constant. Suppose that at a certain instant of time, the volume of the gas is $2 \mathrm {~L}$ , the pressure is $100 \mathrm { kPa }$ , and the pressure is decreasing at the rate of $5 \mathrm { kPa } / \mathrm { sec }$ . Find the rate at which the volume is changing.

Suppose the total cost in maunufacturing $x$ units of a certain product is $C ( x )$ dollars. a. What does $C ^ { \prime } ( x )$ measure? Give units. b. What can you say about the sign of $C ^ { \prime }$ ? c. Given that $C ^ { \prime } ( 3000 ) = 11$ , estimate the additional cost in producing the 3001 st unit of the product.

Calculate $y ^ { \prime }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

Find the derivative of the function. $y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)$

If $f ( x ) = 6 \cos x + \sin ^ { 2 } x$ , find $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ .

Find the instantaneous rate of change of the function $f ( x ) = \sqrt { 3 x }$ when $x = 3$ .

Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$ . $\tan x \approx x$

Let $f ( x ) = x \left| x ^ { 3 } \right|$ . a. Sketch the graph of $f$ . b. For what values of $x$ is $f$ differentiable? c. Find a formula for $f ^ { \prime } ( x )$ .

Find an equation of the tangent line to the given curve at the indicated point. $\frac { 1 } { 7 } y ^ { 2 } - x ^ { 3 } - x ^ { 2 } = 0 ; \quad ( 1 , \sqrt { 14 } )$

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S = 4 x ^ { 2 }$ Find the linear density when $x$ is $3 \mathrm {~m}$ .

A point moves along the curve $3 y + y ^ { 2 } - 8 x = 2$ . When the point is at $\left( - \frac { 1 } { 2 } , - 1 \right)$ , its $x$ -coordinate is increasing at the rate of 3 units per second. How fast is its $y$ -coordinate changing at that instant of time?

Water flows from a tank of constant cross-sectional area $50 \mathrm { At } ^ { 2 }$ through an orifice of constant cross-sectional area $\frac { 1 } { 4 } \mathrm { ft } ^ { 2 }$ located at the bottom of the tank. Initially, the height of the water in the tank was $20 \mathrm { ft }$ , and $t$ sec later it was given by the equation $2 \sqrt { h } + \frac { 1 } { 25 } t - 2 \sqrt { 20 } = 0 \quad 0 \leq t \leq 50 \sqrt { 20 }$ How fast was the height of the water decreasing when its height was $2 \mathrm { ft }$ ?

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

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.

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

Find the derivative of the function. $f ( x ) = ( 4 x + 9 ) ^ { 9 }$