Calculus Answers

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = x2 - 3 and g(x) = square root of quantity three plus x

1) I=∫sec^2(πlnx)/3x dx, x>1
2) I=∫(tan(4ϴ)^-1/4 sec^2(4ϴ) dϴ
3) I=∫cos(2+ln√x)/x dx, x>1
4) ∫(1+√y)^1/4 /√y dy

using L’Hˆopital’s in the following questions

1) I = limx-0 [1/x - 2/ln(1 + 2x)]
2) limx-0 [e^x + x]^1/x
3) limx-1+ (ln x)(tan(pi*x/2))

A tent in the shape of a pyramid with a square base is to be constructed from a piece of material having a side of length 5 meters. In the base of the pyramid, let x be the distance from the center to a side (see figure below). Find a mathematical model expressing the volume of the tent as a function of x.

A rectangular box has total surface area
600 cm^2 and a square base with edge length x cm.Express the volume V of the box as a function of x. What is the volume of the box if the edge length is 5 cm?

A 20-inch wire will be cut into two parts. One part is bent to form a square and the other
into a circle. If the wire is cut x units from one end, express the sum of the areas of the figures formed
in terms of x.

A Normal window with a perimeter of 20 units has the shape of a rectangle surmounted by a
semi-circle. If its base is x units long and the height of the rectangle is y units long, write a mathematical model for its area in terms of x.

(x-1)^2 (x+1)^3
Find the maximum and minimum values

(x-1)^2 (x+1)^3