MATH 122 Spring 2016 Review Exam

Let \( y = f(x) = x^3 + 10 \). What values of x give a value of 135?

It costs a toy company $500 per month to operate. This cost includes electricity, paying employees, etc. The toy sells for $10; however, it cost the company $2 to make each toy. Write a function that computes the company’s profit per month.

The daily revenue of ChickfilA in Russell House can be estimated by the function \( R = 1.2t + 25\), where R is thousands of dollars and t is days since January 1st, 2005. What is the slope of this function? What are the units of the slope? Interpret the slope in terms of ChickfilA’s daily revenue.

Find the relative change of a population that changes from 2,000 to 8,000.

A new car is sold for $22,000 and depreciates linearly to zero in 25 years.
(a) Find a formula for its value V as a function of time, t, in years.
(b) How much is the car worth 6 years after it is sold?

An online tshirt company sells tshirts for $15 dollars each. It costs $4 to make the tshirt; however, the company has a start up price of $850. Answer the following questions using units.
(a) What is the cost function?
(b) What is the revenue function?
(c) What is the breakeven point?

A new tourist site has opened up in the Mediterranean. The initial population, P, of the site is 3000 people at year 0. There is an inflow of workers to take up new job opportunities. Write formula(s) for the population P in the year t (beginning with t = 0), if… (a) The population increases by 600 people per year (b) The population increases by 7% each year

The following are the populations of three different towns, where t is time in years: \( P=700(1.45)^t, P=1200(1.28)^t, P=300(.9)^t \)
(a) Which town has the largest initial population?
(b) Which town has the largest growth rate?
(c) Are any of the towns decreasing in size?

Convert the function \( P = 643e^{0.18t} \) to the form \( P = Pa^t \).

What is the initial quantity of the following function? What is the growth rate? Is this quantity growing or decaying? \( P = 85(0.91)^t \).

A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour.
(a) Write a formula for the amount, A (in milligrams), of caffeine in the body t hours after drinking a cup of coffee. (b) Find the halflife of caffeine.

A bank offers an interest rate of 5% per year compounded continuously. If you deposit $10,000, how much money will you have in the account after 6 years? How much would be in the account after 6 years if it was compounded annually at the same rate?

You get a check for $18,000 for graduation and deposit it into your account paying 8% interest per year, compounded continuously, how long will it take for your balance to be $33,000 for a new Jeep?

A Cowboys fan bets you $1,000 that the Panthers wouldn’t go 110. To pay his bet he asks you if you would rather receive five equal payments of $200 starting now and reoccurring on this day for the next 4 years, or $1,000 right now. Assume a 7% interest rate per year, compounded continuously, and ignoring taxes, which option would you choose? (Which option (account) would yield more money at the end of the 4 years?)

For the following polynomials, indicate degree and leading coefficient.
(a) \( y = 6x^8 + x^4 \)
(b) \( y = 5x^{12}  2x^6  198x +1 \)
(c) \( \frac{1}{2}x^3  \frac{5}{6} x^9 + \frac{1}{7} x^4 \)
(d) \( y = \pi \)

Given the following functions, indicate whether they are power functions. In case they are, find a suitable constant of proportionality \( k \), and power \( p \).
(a) \( 7\sqrt{x} \)
(b) \( 22^x \)
(c) \( \big( 6x^7 \big)^2 \)
(d) \( \dfrac{9}{4\sqrt{x}} \)

Write the equation of a function whose graph is obtained by vertically compressing the graph of \( y = x^2 \) by a factor of 4, followed by a vertical downward shift of 6 units.

Let \( f(x) = 8^x \). Use a small interval (say \( x=4 \) to \( x=4.01 \) ) to estimate \( f’(4) \)

The time t (in hours) that a drug stays in a person’s system is a function of the quantity administered, k, (im mg).
(a) Interpret the statement \( f(7) = 5 \). Give units for the numbers 7 and 5.
(b) Write the derivative of the function \( t = f(k) \) in Leibnitz notation.
(c) If \( f’(7) = 0.2 \), what are the units of \( 0.2 \).
(d) Interpret \( f’(7) = 0.2 \) in terms of dose and duration.

If \( q = f(p) \) gives the number of thousands of tons of zinc produced when the price is p in dollars per ton, then what are the units and meaning of \( \dfrac{dq}{dp} = 0.2 \) when \( p = 900 \)

Find the derivative of the following functions:
(a) \( f(t) = \frac{6}{t} + \frac{3}{t^2} \).
(b) \( f(x) = 3^x + \frac{3}{x^2} \).
(c) \( f(x) = \dfrac{e^{2x}}{x^3} \).
(d) \( f(x) = (3x^46)(x^3+2x^5+3) \).

Given \( f(x) = x^4 + 2x^3 + 5x  14 \), compute \( f’(0), f’(1), f’(2) \).

Find an equation of the tangent line to the graph of \( y = \ln x \) at the point where \( x=1 \).

Find constants \( a, b \) so that the minimum of the parabola \( y = x^2 + ax + b \) is at the point \( (2,6) \).

The function \( f(x) = x^4  4x^3 +8x \) has a critical point at \( x=1 \). Use the second derivative test to identify it as a local maximum or local minimum.

Find the inflection points of \( f(x) = 6x^4 + 2x^3  4x \).

Find the global maximum and global minimum of the function \( f(x) = x^3  12x^2 + 3x  17 \) for \( 5 \leq x \leq 15\).

When the price of a laptop is $1,500, 100 people purchase it. With every $50 decrease in price, 20 more people purchase the laptop. What price gives the largest revenue?

Total revenue is given by the function \( R(q) = 12q + 0.6 q^2 \). Total cost is given by the function \( C(q) = 150 + 14q \). Find the quantity that maximizes profit when \( 0 \leq q \leq 1,000 \).

Compute the following antiderivatives:
(a) \( \int \big( 13 e^x +2^x +x^3 \big) \, dx \)
(b) \( \int \big( \sqrt{x^7} + x^7 + 8x^6 \big) \ \, dx \)
(c) \( \int 9x \sqrt{34x^29}\, dx \)
(d) \( \int x e^{4x} \, dx \)
(e) \( \int x^2 e^x \, dx \)
(f) \( \int x^6 \ln x \, dx \)

Find the consumer surplus for the demand curve \( p = 100  q^2/2 \) when \( q = 4 \)

For the function given below, compute left and right Riemann sums, and average them.
t  0  2  4  8  12  15  16 
f(t)  1  4  1  2  4  5  6 