Calculus Answers

Give an example of a function with both a removable and a non-removable discontinuity.

In Fourier series, cos npi=(-1)^n, what is cos 2npi?

The critical value/s for y = x^3 – 3x^2 are

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 420 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one additional unit will be occupied for each 10 dollar decrease in rent. What rent should the manager charge to maximize revenue?

In all of this question assume the sales volume of a new product (in thousands of units) is given
by S =AT + 450/√A + T^2 where T is the time (in months) since the product was first introduced and
A is the amount (in hundreds of dollars) spent each month on advertising. Assume A, T > 0.

(a) Calculate the partial derivative of S with respect to time. Use that partial derivative to
predict the number of months that will elapse before sales volume begins to decrease if
the amount allocated to advertising is held fixed at $9, 000 per month.

If we consider the expression 1/2-3x
as a function on R, what will be its domain and
range? Will it have an inverse? Justify your answer.

a) Which of the following functions are 1-1 and which are onto? Justify your answer.
i) f : R - R^>=0 given by f (x) = x^2 where R^>=0 is the set {x ϵ Rjx >= 0}
ii) f : R - R given by f (x) = x^2+x+1.

A fixed Circle C1 with equation (x-1)^2 + y^2 = 1 and a shrinking circle C2 with radius r and center the origin. P is the point (0,r), Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. what happense to R as C2 shrinks, that is, as r-->0^+?

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are to be constructed from corrugated metal sheeting. Only 6 metre length sheets are available. Each of them is to be cut into two parts such that one part is used for the roof and the other is used for the front. Find the dimensions x and y of the store that will maximise the cross‐sectional area and therefore the volume. Hence determine the maximum cross‐sectional area.