Exam 3: Differentiation Rules

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

A production editor decided that a promotional flyer should have a 1 -in. margin at the top and the bottom, and a $\frac { 1 } { 2 }$ -in. margin on each side. The editor further stipulated that the flyer should have an area of $72 \mathrm { in. } ^ { 2 }$ . Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

Sketch the curve. $y = \sqrt { \frac { x } { x - 2 } }$

An aircraft manufacturer wants to determine the best selling price for a new airplane. The company estimates that the initial cost of designing the airplane and setting up the factories in which to build it will be 700 million dollars. The additional cost of manufacturing each plane can be modeled by the function $m ( x ) = 1,600 x + 40 x ^ { 4 / 5 } + 0.15 x ^ { 2 }$ where $x$ is the number of aircraft produced and $m$ is the manufacturing cost, in millions of dollars. The company estimates that if it charges a price $p$ (in millions of dollars) for each plane, it will be able to sell $x ( p ) = 390 - 5.8 p$ Find the cost function.

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem. $f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$

For what values of $a$ and $b$ is $( 2,2.5 )$ is an inflection point of the curve $x ^ { 2 } y + a x + b y = 0$ ? What additional inflection points does the curve have?

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.

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side $L = 9 \mathrm {~cm}$ if one side of the rectangle lies on the base of the triangle. Round your answer to the nearest tenth.

$\text { Find the absolute maximum value of } y = 7 \sin \left( \frac { \pi } { 10 } \right)$

Find the maximum and minimum points of the function. $F ( x ) = \left( 1 + x ^ { 2 } \right) ^ { 3 } + 6 x ^ { 4 }$

Estimate the value of $\sqrt { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 2 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places. Select the correct answer.

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.

A production editor decided that a promotional flyer should have a 1 -in. margin at the top and the bottom, and a $\frac { 1 } { 2 }$ -in. margin on each side. The editor further stipulated that the flyer should have an area of 72 in. $^ { 2 }$ . Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

Use the Second Derivative Test to find the relative extrema, if any, of the function $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 36 x - 5$

Find $f$ . $f ^ { \prime \prime } ( x ) = x ^ { 3 } + \sinh x , f ( 0 ) = 6 , f ( 2 ) = \frac { 13 } { 5 }$

Find the critical number(s) of the function. $y = x ^ { 2 } + 10 x$