Exam 3: Additional Applications of the Derivative

Find the dimensions of the rectangle of largest area that can be inscribed in a semi-circle of radius R, assuming that one side of the rectangle lies on a diameter of the semi-circle.

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Find all critical numbers of the function $f ( x ) = x ^ { 5 } + 5 x ^ { 6 }$ .

The function $f ( x ) = x ^ { 2 } + \frac { 2 } { x }$ has a relative minimum at x = 1.

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula $C = 0.1 x ^ { 2 } + 20 x + 500$ . Find the number of units that will minimize the average cost.

Determine the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5$ on the interval -2 $\le$ x $\le$ 1.

Find all the critical numbers of the function $f ( x ) = 2 x ^ { 2 } - 8 x + 7$ .

Graph $f ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - x$ .

Locate all inflection points of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ .

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (7, 34) and crosses the y-axis at (0, 6).

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Find A and B so that the graph of $f ( x ) = \frac { 29 - A x } { B x + 15 }$ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Graph $f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }$ .