Exam 2: Differentiation: Basic Concepts

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

The equation for the tangent line to the curve $x ^ { 2 } + 2 x y = y ^ { 3 }$ at the point (1, -1) is y = -1.

If $f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }$ , what is $f ^ { \prime } ( x )$ ?

The gross national product (GNP) of a certain country is $N ( t ) = t ^ { 2 } + 3 t + 121$ billion dollars where t is the number of years after 1990. At what percentage rate will the GNP be changing with respect to time in 1995? Round your answer to one hundredth of a percent, if necessary.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 9 } { \sqrt { x } }$

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the normal line to $f ( x ) = 4 x ^ { 3 } - 8 x + 49$ at the point with x-coordinate -2.

You measure the side of a cube to be 12 centimeters long and conclude that the volume of the cube is $12 ^ { 3 } = 1,728$ cubic centimeters. If your measurement of the side is accurate to within 6%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary.

Differentiate: $f ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }$

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 3$ that passes through (1, 4) is y = 2x + 3.

It is estimated that t years from now, the population of a certain community will be $p ( t ) = 14 - \frac { 6 } { 3 + t }$ thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be $C ( p ) = 0.5 \sqrt { p ^ { 2 } + 2 p + 30 }$ units when the population is p thousand. The rate at which the level of carbon monoxide will be changing 3 years from now is

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 3$ at the point (3, 12).

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 2$ that passes through (25, 7) is y = 2x + 2.

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }$ at the point (1, 7).

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 3 } y ^ { 2 } - 3 x y = 4 x + y - 19$ ; (0, 19)

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters. If your measurement of the side is accurate to within 5%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary.