Exam 2: Limits and Derivatives

If a cylindrical tank holds 10000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after $t$ minutes as $V ( t ) = 10000 \left( 1 - \frac { 1 } { 60 } t \right) ^ { 2 } , 0 \leq t \leq 60$ Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$ ) as a function of $t$ .

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

Find the derivative of the function. $f ( x ) = 2 \cos x - 2 x - 8$

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

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

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

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.

The top of a ladder slides down a vertical wall at a rate of $0.15 \mathrm {~m} / \mathrm { s }$ . At the moment when the bottom of the ladder is $1.5 \mathrm {~m}$ from the wall, it slides away from the wall at a rate of $0.3 \mathrm {~m} / \mathrm { s }$ . How long is the ladder?

If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$ . Select the correct answer.

Find the derivative of the function. Select the correct answer. $f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }$

Find the linearization L (x) of the function at a. $f ( x ) = x ^ { 2 / 3 } ; \quad a = 64$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = x ^ { 5 } g ( x )$ Select the correct answer.

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = x ^ { 7 } g ( x )$

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.

Differentiate. $y = \frac { \sin x } { 3 + \cos x }$

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.

The equation of motion is given for a particle, where $s$ is in meters and $t$ is in seconds. Find the acceleration after 5 seconds. $s = t ^ { 3 } - 3 t$

Find the differential of the function at the indicated number. $f ( x ) = e ^ { 7 x } + \ln ( x + 8 ) ; x = 0$