Calculus Answers

Find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis:

Circular Disk Method:

1. R:y=√𝑥, 𝑦=4, 𝑦−𝑎𝑥𝑖𝑠 axis of revolution: 𝑦=4
2. 𝑅:𝑦=𝑥^2, 𝑥=3, 𝑥−𝑎𝑥𝑖𝑠 axis of revolution: 𝑥=3

Circular Ring Method:

1. 𝑅: 𝑦=𝑥^2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
2. 𝑅:𝑦^2=4𝑥, 𝑥^2=4𝑦 axis of revolution: 𝑥=4

Cylindrical Shell Method:

1. 𝑅:𝑦=𝑥^2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
2. 𝑅: first-quadrant region bounded by 𝑦=4𝑥 and 𝑥^3=𝑦 axis of revolution: 𝑥=2
3. An oil tank in the shape of a sphere has a diameter of 60 ft. How much oil does the tank contain if the depth of the oil is 25ft?

Find the length of the curve given by x= t^3, y= 2t^2, in 0≤ t≤2

Verify Rolle's theorem for the function,f defined by

f(x)= x(x-2)e^-x,on the interval [0,2].

The demand and total cost functions of a good are respectively and

Find expressions for TR, (profit) , MR, and MC in terms of Q.

Solve the equation

and hence determine the value of Q which maximizes profit.

Verify that, at the point of maximum profit, MR=MC.

An individual’s utility function is given by

where is the amount of leisure measured in hours per week and is income earned measured in cedis per week.

Determine the value of the marginal utilities, when = 138 and = 500. Hence estimate the change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function?

A small funnel in the shape of a cone is being emptied of fluid at the rate of 12 cubic

centimeters per second. The height of the funnel is 20 centimeters, and the radius of the top is 4 centimeters. How fast is the fluid level dropping when the level stands 5 centimeters above the vertex of the cone?

The supply and demand equations of a good are respectively given by and 

The government decides to impose a tax, t, per unit. Find the value of t (in Ghana cedis) which maximizes the governments total tax revenue on the assumption that equilibrium conditions prevail in the market.

A balloon is being inflated by pumped air at the rate of 2 cubic inches per second. How fast is the diameter of the balloon increasing when the radius is ½ inch?

The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components:

GH¢10 million for the land

GH¢1/4 million per floor

Specialized costs of GH¢10000x per floor.

How many floors should the office complex contain if the average cost per floor is to be minimized?

Create a function that is not continuous at the number 4. And explain what condition of it violates.