Exam 3: Differentiation Rules

Estimate the absolute maximum value of the function $y = x \sqrt { 3 x - x ^ { 2 } }$ to two decimal places on the interval $[ 0,3 ]$ .

You are given the graph of the derivative $f ^ { \prime }$ of a function $f$ . (a) Determine the intervals on which $f$ is increasing or decreasing. (b) Find the $x$ -coordinate of the relative maxima and relative minima of $f$ .

A piece of wire $14 \mathrm {~m}$ long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

Consider the function $f ( x ) = \frac { 8 x } { \ln x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

For what values of $c$ does the curve have maximum and minimum points? $F ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 x$

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 2 , \quad [ 2,4 ]$

Find $f$ . $f ^ { \prime \prime } ( x ) = 18 x + 24 x ^ { 2 }$

Find a cubic function $f ( x ) = a x ^ { 3 } - b x ^ { 2 } + c x - d$ that has a local maximum value of 112 at 1 and a local minimum value of -1,184 at 7.

The quantity demanded per month of an item is related to the unit price by the demand equation $p = \frac { 45 } { 0.04 x ^ { 2 } + 1 } , \quad 0 \leq x \leq 20$ where $p$ is measured in dollars and $x$ is measured in units of a thousand. How many items must be sold by the manufacturer to maximize its revenue? Hint: Recall that the revenue is given by $R = p x$ .

Use the graph of $f$ to estimate the values of $c$ that satisfy the conclusion of the Mean Value Theorem for the interval $[ 0,7 ]$ . Select all that apply.

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = 2 x ^ { 3 } + c x ^ { 2 } + 8 x ?$

The size of the monthly repayment $k$ that amortizes a loan of $A$ dollars in $N$ years at an interest rate of $r$ per year, compounded monthly, on the unpaid balance is given by $k = \frac { A r } { 12 \left[ 1 - \left( 1 + \frac { r } { 12 } \right) ^ { - 12 N } \right] }$ The value of $r$ can be found by performing the iteration $r _ { n + 1 } = r _ { n } - \frac { A r _ { n } + 12 k \left[ \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N } - 1 \right] } { A - 12 N k \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N - 1 } }$ A family secured a loan of $\$ 360,000$ from a bank to finance the purchase of a house. They have agreed to repay the loan in equal monthly installments of \$2476 over 25 years. Find the interest rate on this loan. Round the rate to one decimal place.

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find all the critical numbers of the function. $g ( x ) = 7 x + \sin ( 7 x )$

Find the relative extrema, if any, of $f ( t ) = e ^ { t } - 7 t - 3$ . Use the Second Derivative Test, if possible.

Find the critical number(s), if any, of the function $g ( t ) = t ^ { 2 } \ln 9 t$ .

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $\mathrm { f }$ is shown. On what intervals is $f$ decreasing?

Find the most general antiderivative of the function. $f ( x ) = 15 x ^ { 2 } - 16 x + 9$