Problem Set #6 Due Friday, April 12Th
Name: ______Date: 03/22/13
AP Calculus ABMs. Wilson
Problem Set #6 – Due Friday, April 12th
Answer all questions on looseleaf or graph paper. Make sure all answers are complete and show all work where necessary.
1.) Let .
a.) Show that is a solution to the differential equation in the interval.
b.) Show that is a solution to the differential equation in the interval .
c.) Explain why is a solution to the differential equation in the domain .
d.) Show that the function is a solution to the differential equation for any values of C1 and C2.
2.) Find the specific solution to each of the following second-order initial value problems by first finding dy/dx and then finding y.
a.) .
b.) .
c.) .
3.) Suppose .
a.) Use the substitution to show that
.
b.) Use another trig identity to evaluate the definite integral without a calculator.
4.) To see the effect of a relatively small error in the estimate of the amount of carbon-14 in a sample being dated, answer the following questions about this hypothetical situation.
a.) A fossilized bone found in central Illinois in the year A.D. 2000 contains 17% of its original carbon-14 content. Estimate the year the animal died.
b.) Repeat part (a) assuming 18% instead of 17%.
c.) Repeat part (a) assuming 16% instead of 17%.
5.) John Napier (1550-1617), the Scottish laird who invented logarithms, was the first person to answer the question “What happens if you invest an amount of money at 100% yearly interest, compounded continuously?”
a.) What does happen? Explain.
b.) How long does it take to triple your money?
c.) How much can you earn in a year?
6.) The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation , where k is a constant. As a diver you know from experience that diving to 18 feet in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below a tenth of the surface value. About how deep can you expect to work without artificial light?
The next 2 problems are from past AP Exams.
7.) Let f be a differentiable function, defined for all real numbers x, with the following properties:
(i)
(ii)
(iii)
Find f(x). Show your work.
8.) Let f be a function with f(1) = 4 such that for all points (x, y) on the graph of f the slope is given by .
a.) Find the slope of the graph of f at the point where x = 1.
b.) Write and equation of the line tangent to the graph of f at x = 1 and use it to approximate f(1.2).
c.) Find f(x) by solving the separable differential equation with the initial condition f(1) = 4.
d.) Use your solution from part (c) to find f (1.2).