Calculus Answers

Determine and sketech the set of pairs (x, y) in R×R that satisfy: i)|x|=|y| ii)|x|-|y|=2

Find these terms of the sequence {an}, where an =

2 · (−3)n + 5n.

a) a0

b) a1

c) a4

d) a5

(a) Evaluate∫[

x

√x

2+1

]dx.

(b) Use MATLAB to generate some typical integral curves of f(x) =

x

√x

2+1

over the

interval (−5,5).

Find the volume in the first octant bounded by x+y+z=9, and the inside cylinder 3y=27-x^3

Give an example of a function of two variables such thatf(0,0) = 0 butfis NOT continuousat (0,0). Explain why the functionfis NOT continuous at (0,0).

Give an example of a function of two variables such that f(0,0) = 0 but f is NOT continuous at (0,0). Explain why the function f is NOT continuous at (0,0).

A Manufacturer produces two products, the Klunk and the Klick. Klunk has a contribution

to profit of $3, and the Klick $4 per unit. The manufacturer wishes to establish the weekly

production plan that maximizes profit. Production of these products is limited to machine,

labor and material constraints. Each Klunk requires four hours machining, four hours labor

and one kilogram of material, where as each Klick requires two hours machining, six hours

labor and one kilogram of material. Machining and labor has a maximum of one hundred

and one hundred and eighty hours available, and total material available is forty kilograms.

Because of a trade agreement, sales of Klunk are limited to a weekly maximum of twenty

units and to honor an agreement with an old established customer at least ten units of Klick

must be sold each week.

i. Determine graphically using linear programming a suitable production mix of Klunk

and Klick. [12]

ii. What will be the company’s maximum profit?

Determine the length of the curve 𝑥 = 𝑦^2 /2 for 0 ≤ 𝑥 ≤ 1/2 . Assume 𝑦 positive.

The functions f and g are defined by f(x) =1/(1-3x) and g(x) =log_1/3(3x-2)-log₃(x) respectively

1. Write down the sets D_f (ehe domain of f) and D_g (the domain of g)

2. Solve the inequality f(x) > 2 for x"\\isin" D_f

3. Solve the inequality f(x) ≥ 2 for x"\\isin" D_g

Hint: Use the change of base formula

2. The formula h(t) = -16t² + 32t + 80 gives the height b above the ground, in feet (ft), of an object thrown, at t = 0, straight upward from the top of an 80 ft building. What is the highest point reached by the object? How long does it take the object to reach its highest point?