Calculus Answers

Given any two sets C and D, under what condition on them will C×D and D×C have the same number of elements? Give reason for your answer

A price-taking profit-maximising business, which operates at its optimal plant
size, has a TC curve give by TC = q³ − 2q² + 2q + 0.576 where TC
is measured in 1000s of £ per week and q is the rate of output measured in
1000s of units per week.
a) If the market price of output is £1.00 show that the firm will produce 1000
units per week and make a loss. Explain why the business is, at this price,
indifferent between producing and closing down. Draw a diagram to show
this.
b) If the market price of output is £3.28 what will each firm produce and what
profit is made?
c) Assuming that all firms in the industry have identical cost structures and that
there is freedom of entry, show that the long-run weekly rate of output will fall
to 1200 units.
d) What is the price of output when the industry is in equilibrium?

Is the function sin x cos x monotonic on [-π/2 , π/2] ? justify your answer

Check the continuity of f(x) = {x+1, x is less than 1}
{0, 2 is greater than 1 and x}
{2-x , x is greater than 2}

Evaluate lim x →3 [(√3x)-3] / [(√2x-4)-(√2)]

Determine the interval on which the given function is continuous
f(x)={sin1/x if x≠0, 0 if x=0

Give all the extreme points of the set
S = {(x, y) | 2x + 3y ≤ 6, − 3x + 2y ≤ 6, − 5x − 6y ≤ 30, 4x − 2y ≤ −24}.

Obtain the directional derivative for a scalar field at the poin

A cylindrical tank with radius 3 m is being filled with water at a rate of 4 m3/min. How fast is the height of the water increasing?