Exam 2: Differentiation: Basic Concepts

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 5 is

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 6 }$

Differentiating $f ( x ) = x ^ { 6 } - 5 x + 2$ gives $6 x ^ { 5 }$ .

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 3 )$

Find an equation for the tangent line to the curve $y = \sqrt { 4 + \frac { x } { 4 } }$ at the point where x = -1. Round numbers to two decimal places.

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 6 x ^ { 2 } + 3 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -3) is y = 3x - 3.

Suppose the output at a certain factory is $Q = 2 x ^ { 3 } + 2 x ^ { 3 } y ^ { 2 } + 6 y ^ { 4 }$ units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 20 hours of skilled labor and 30 hours of unskilled labor. Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level. Round you answer to two decimal places, if necessary.

What is the rate of change of $f ( t ) = \frac { 2 t - 3 } { t + 5 }$ with respect to t when t = 5?

Find an equation for the tangent line to the curve $x ^ { 3 } + x y + y ^ { 3 } = x$ at the point (1, 0).

Find the equation of the line that is tangent to the curve $f ( x ) = \frac { 5 x ^ { 2 } - 7 x + 1 } { 5 - 4 x ^ { 3 } }$ at the point (1, -1).

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1. f (x) = x³ + 4x - 3

In a certain factory, output Q is related to inputs x and y by the equation $Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }$ . If the current levels of input are x = 255 and y = 155, use calculus to estimate the change in input y that should be made to offset a decrease of 0.6 unit in input x so that output will be maintained at its current level. Round your answer to two decimal places, if necessary.

Find $f ^ { \prime \prime \prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 1 } { x ^ { 2 } } + \sqrt { 6 }$ .

The gross annual earnings of a certain company were $E ( t ) = 0.2 t ^ { 2 } + 9 t + 30$ thousand dollars where t is the number of years since its formation in 1990. The gross annual earnings with respect to t in 1995 are growing at 13.75%.

The temperature in degrees Fahrenheit inside an oven t minutes after turning it on can be modeled with the function $F ( t ) = \frac { 400 t + 70 } { t + 1 }$ Find $F ^ { \prime } ( 5 )$ and interpret what it tells us about the temperature. Round your answer to 2 decimal places.

Differentiate: $f ( x ) = \left( x ^ { 2 } + 4 \right) ( x + 4 )$

An environmental study of a certain suburban community suggests that t years from now the average level of carbon monoxide in the air will be $Q ( t ) = 0.07 t ^ { 2 } + 0.6 t + 2.8$ parts per million (ppm). The rate that the carbon monoxide level will change with respect to time 3 years from now will be 0.42 ppm/yr.

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4$

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 2 x ^ { 4 } + 10 x ^ { 3 } - 2 x ^ { 2 } + 11 x - 2$ .