Calculus Answers

Let D be a convex region in R2 and let L be a line segment of length I that connects points on the boundary of D. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within D that bounds a (smaller) region R. Using the corollary to Green’s Theorem for finding area, find an expression that relates the area of R to the area of D in terms of the length I of the line segment. [You might start with some simple regions, but you must show this generally.] 

Locate the critical points and identity the stationary points of 4x^4-16x^2+17

perform this using the linear differential equation of higher order in operator form: (D2 + 3D + 2) (e^-2x + 3x²)

perform this using the linear differential equation of higher order in operator form: (D2 + 3D + 2) (e-2x + 3x²)

Suppose that a population yy grows according to the logistic model given by formula:

yy = LL

1 + AAee−kkkk .

a. At what rate is yy increasing at time tt = 0 ?

b. In words, describe how the rate of growth of yy varies with time.

c. At what time is the population growing most rapidly?

Verify that the function u(x,y,z)=1/(√x^2+y^2+z^2) is a solution of the three-dimensional Laplace equation Uxx + Uyy +Uzz = 0.

An object moves along a straight line so that after t minutes, its distance from its starting point is𝑠(𝑡)=2𝑡3+4𝑡2+6meters .

At what speed is the object moving at the end of 3minutes?

A cylindrical can is to be constructed to hold a fixed volume of liquid. The cost of the

material used for the top and bottom of the can is 3 cents per square inch, and the cost of the

material used for the curved side is 2 cents per square inch. Use calculus to derive a simple

relationship between the radius and height of the can that is the least costly to construct.

Nalini wants to open a small restaurant stand selling specialty burgers. The fixed

cost component on a daily basis is Rs 2500. For each burger made, the cost is Rs

50. In addition the daily special cost is 0.25x2 where x is the number of burgers she

will make for the day.

a) Write down the expression of the Total Cost function for a day (1 mark)

b) Write down the expression for the Average Cost function for a day

c) How many burgers should she make a day to minimize the cost per burger per

day? ( 4 marks)

d) What is the total cost per day associated with the minimized cost per burger per

day?

e) The price per burger is 250. What is the maximum daily revenue Nalini can earn

provided she decides to make at the minimum unit cost per burger?

f) What is the daily profit she would earn in e) ?

g) How many burgers would she have to sell to break even on a daily basis?