Exam 3: Differentiation Rules

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.

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 $R = p x$ .

Find the critical number(s), if any of the function $f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$ .

The function $f ( t ) = \sin ^ { 2 } 2 t$ satisfies the hypotheses of Rolle's Theorem on the interval $\left[ 0 , \frac { \pi } { 2 } \right]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

Use Newton's method to approximate the indicated root of $x ^ { 4 } + x - 4 = 0$ in the interval $[ 1,2 ]$ , correct to six decimal places. Use $x _ { 1 } = 1.5$ as the initial approximation.

Find the dimensions of the rectangle enclosed in the semicircle $y = \sqrt { 144 - x ^ { 2 } }$ with the largest possible area.

Consider the function $f ( x ) = x ^ { 2 } e ^ { - 9 x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

Given $f ( x ) = x + \frac { 36 } { x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

Find the dimensions of a rectangle of area $400 \mathrm { ft } ^ { 2 }$ that has the smallest possible perimeter.

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

Estimate the extreme values of the function.Round the answers to the nearest hundredth. $y ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 40 x + 15$

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length $L = 8$ and width $W = 3$ .

Find the minimum points of the function. $F ( x ) = x ^ { 4 } - 108 x$

A woman at a point $A$ on the shore of a circular lake with radius $4 \mathrm { mi }$ wants to arrive at the point $C$ diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of $6 \mathrm { mi } / \mathrm { h }$ and row a boat at $2 \mathrm { mi } / \mathrm { h }$ . How should she proceed? (Find $\theta$ ). Round the result, if necessary, to the nearest hundredth.