Exam 2: Differentiation: Basic Concepts

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 7 x ^ { 4 } + 4 x ^ { 3 } - 6 x ^ { 2 } + 11 x - 7$ .

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (2, 5).

If the displacement of a moving object is $s ( t ) = t ^ { 3 }$ , the acceleration is 6t.

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f ( x ) = - x ^ { 3 } + 10 x ^ { 2 } - 3 x$ transistor radios x hours later. Approximately how many radios will the worker assemble between 10:00 and 10:15 A.M.?

If $g ( y ) = \sqrt { 5 y + y ^ { 2 } }$ represents the height in inches of a sapling y weeks after germination, find $g ^ { \prime } ( 6 )$ and interpret what it tells us about the height of the tree. Round your answer to 1 decimal place.

When toasters are sold for p dollars apiece, local consumers will buy $D ( p ) = \frac { 57,600 } { p }$ toasters a month. It is estimated that t months from now, the price of the toasters will be $p ( t ) = 0.03 t ^ { 3 / 2 } + 22.08$ dollars. Compute the rate at which the monthly demand for the toasters will be changing with respect to time 16 months from now.

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 30 - \frac { 7 } { 2 t + 1 }$ thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be $C ( p ) = 0.3 \sqrt { p ^ { 2 } + p + 30 }$ parts per million (ppm) when the population is p thousand. The rate at which the level of pollution is changing with respect to time 3 years from now is about 0.084 ppm per year.

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

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 50 - \frac { 7 } { 2 t + 1 }$ thousand people. At what rate, in people/year will the population be growing 3 years from now?

An equation for the tangent line to the curve $f ( x ) = x ^ { 3 } ( 1 - 3 x ) ^ { 2 }$ at the point where x = -1 is y = 72x + 56.

An appliance store manager estimates that for x television ads run per day, $R ( x ) = - 0.01 x ^ { 3 } + x ^ { 2 } - 3 x + 200$ refrigerators will be sold per month. Find $R ^ { \prime } ( 4 )$ and interpret what it tells us about sales.

When a certain commodity is sold for p dollars per unit, consumers will buy $D ( p ) = \frac { 30,000 } { p }$ units per month. It is estimated that t months from now, the price of the commodity will be $p ( t ) = 0.3 t ^ { 5 / 2 } + 5.4$ dollars per unit. The monthly demand will be decreasing 40 months from now.

If $f ( x ) = \sqrt [ 3 ] { x } - \frac { 1 } { \sqrt { x } }$ , differentiate f (x).

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at $\left( 2 , \frac { 1 } { 2 } \right)$ .

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

Find the equation of the tangent line to the given curve at the specified point: $x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8$ ; (0, 8)

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1.f (x) = x³ + 5x + 2

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x + 9$ at the point (1, 12).

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$ .