9 questions about Applied Calculus II, due within 18 hours

Math 112 Applied Calculus II Final Assignment

INSTRUCTIONS:

• Complete the problems listed below. Write your answers neatly on blank or lined paper (messy and/or un-readable work will not be graded).

• Scan your work and save as a single PDF (I recommend to download the free app called Adobe Scan).

• Submit the PDF on C4 by Wednesday, April 8th at 12:00 PM (note: email submissions and non-PDF submissions will not be graded).

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Question 1 AA/4

Determine which, if any, of the following areas are equal. Prove your claims by showing the appropriate calculations.

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Question 2 AA/6

Suppose that f(2) = 2, f(8) = 9, f ′(2) = 5, and f ′(8) = 3. Furthermore, suppose that f ′′

is continuous. Find the value of

∫ 8 2

xf ′′(x)dx.

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Question 3 AA/6

Consider the following figure, along with the measurements shown (all in centimetres):

Given that all the measurements were taken 2 centimetres apart, use an appropriate method from Chapter 7 of the textbook to estimate the area of the figure.

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Question 4 AA/4

Find the average value of the function shown below over the interval [1, 8].

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Question 5 AA/6

Consider the surface xy2z3 = 2. Which points on this surface are closest to (0, 0, 0)?

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Question 6 AA/7

A clothing manufacturer specializes in jackets and sweaters. The profit from selling x jackets and y sweaters is given by

P (x, y) = −2x2 −y2 + 10x + 12y.

A total of 15 garments are to be manufactured. Use the method of Lagrange multipliers to find the number of jackets and the number of sweaters that will maximize the profit.

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Question 7 AA/8

In mathematics, we say that two variables x and y are proportional if there exists a constant k such that x = ky for all x, y.

A common model for the spread of an epidemic supposes that the rate of infection is pro- portional the product of the number of infected individuals and the number of uninfected individuals.

Suppose that a disease is spreading among the population of a small island of 10 000 in- habitants. Suppose that 200 people have the disease on January 1, and 1 600 have it on January 8. How many days does it take for 60% of the population to be infected?

Hint: let p be the number of people infected after t days – what is dp/dt in terms of p?

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Question 8 AA/6

Consider the double integral ∫ 1 0

∫ 1 √ y

√ 1 + x3dxdy.

Sketch the region of integration, then evaluate the integral.

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Question 9 AA/8

Find the general solution of the differential equation

ty′′ + 2y′ − 12t2 = 0.

Hint: recall that we have a method for solving equations of the form y′ + p(t)y = g(t). Re-write the equation above in this form by making the appropriate substitution.

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