Calculus Answers

The area A of a circle is increasing at a constant rate of 2cm2s−1. If the area of the circle is given by A = πr2, what is the rate of change of the radius when the radius is 4cm?

The area A of a circle is increasing at a constant rate of 2cm2s−1. If the area of the circle is given by A = πr2, what is the rate of change of the radius when the radius is 4cm?

1. Consider the function y = x2 + 3x + 5. 2x−3

(a) Determine the domain of the function.

(b) Determine the range of the function.

(c) Determine the intercepts of the function.

(d) Find the asymptotes if they exist.

(e) Find the turning points (if they exist) and determine the type of turning points they are.

(f) Manually sketch the graph of the function.

1. Evaluate "\\intop"x²(1 + 2x³)³dx.

2. Evaluate "\\intop"xe^7x dx.

3. Find the volume of the solid of revolution when the curve y = 1 + x2 is revolved around the x-axis on [−2, 2].

Find the area of the paraboloid x2 + y2 = z inside the cylinder x2 + y2 = 9.

Calculate the area under the curve y=x3 +4x+1 from x=-3 to x=3.

Two gardens. A fence of length 100 ft is to be used to enclose two

gardens. One garden is to have a circular shape, and the other to be

square.

Determine how the fence should be cut so that the sum of the areas

inside both gardens is as large as possible.

Let f(x) = 5 + 12x -x^3. Using the differentiation techniques learnt, find

a. The intervals on which f is increasing

Find an equation of the tangent line to the curve 9x³ – y³ = 1 at the point (0, -1).

Find an equation of the tangent line to the curve x⁴ + 2y⁴ = 33 at the point (1,2).