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perform this using the linear differential equation of higher order in operator form: (D2 + 3D + 2) (e^-2x + 3x²)
a body falls from rest against a resistance proportional to the cube of the speed at any instant. if the limiting speed is 3m/s, find the time required to attain a speed of 2m/s.
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?
Write down 10 uses of Surface Integration.
Suppose that a population y grows according to the logistic model given by formula: y=(L)/1+Ae^-kt . a. At what rate is y increasing at time t=0 ? b. In words, describe how the rate of growth of y varies with time. c. At what time is the population growing most rapidly?
