Exam 8: Appendix: Algebra Review

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

It is projected that t years from now the population of a city will be changing at the rate of $t ^ { 2 } e ^ { 0.01 t }$ thousand people per year. If the current population is 1 million, what will the population be 4 years from now?

Money is transferred continuously into an account at the constant rate of $1,600 per year. Assume the account earns interest at the annual rate of 5% compounded continuously. Compute the future value of the income stream over a 16 year period.

Write an equation for the line with the given properties.Through (3, -1) with slope 2

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 4$ that passes through (1, 5) is y = 2x + 4.

Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + $\infty$ or - $\infty$ . $\lim _ { x \rightarrow 5^- } f ( x )$ where $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & \text { if } x \leq 5 \\x + 8 & \text { if } x > 5\end{array} \right.$

Evaluate $\int x \ln \sqrt { x } d x$

After t weeks, a charity is raising money at the rate of 5,000 t ln(t + 1) dollars per week. How much money is raised during the first 10 weeks? Round to the nearest ten dollars.

An appliance manufacturer can sell refrigerators for $600 apiece. The manufacturer's total cost consists of a fixed overhead of $12,000 plus production cost of $400 per refrigerator. How many refrigerators must be sold for the manufacturer to break even?

Find an equation for the tangent line to the curve $y = \sqrt { 3 + \frac { x } { 4 } }$ at the point where x = -1. Round numbers to two decimal places.

An investment will generate income continuously at the constant rate of $2,300 per year for 5 years. If the prevailing annual interest rate remains fixed at 11% compounded continuously, what is the present value of the investment?

Compute $f _ { x }$ for $f ( x , y ) = 9 x y ^ { 3 }$ .

Compute f_x for f (x, y) = 3x⁵y - 8x +e^xy.

Simplify: $\frac { 4 ^ { 6 } } { 4 ^ { 5 } }$

Find an equation for the tangent line to the curve $x ^ { 2 } + y ^ { 3 } = x y + 1$ at the point (1, -1).

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 7 }$ the ratio in the atmosphere. Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round to the nearest whole year.

A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. (For example, if only a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market, the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad. If the manufacturer can produce the machines at a cost of $10,000 apiece, how many should be supplied to each market to generate the largest possible profit?

Compute f (8, 5) if $f ( x , y ) = 4 x y ^ { 3 }$

If $f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }$ , then $f ^ { \prime } ( x ) = x + 2 e ^ { - x }$