If the revenue function is given by  being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

Find the area between the curve y²(4-x) = x³ and its asymptote parallel to y-axis.

Let f and g be two functions defined on R by:

i) Find the value of αfor which f is continuous at x = − .3

ii) Find all the roots of f(x) = 0.

For any two sets S and T, show that:

S∪T = (S − T ) ∪ ( S ∩ T) ∪ (T − S).

Depict this situation in the Venn diagram.

Evaluate:

Find the least value of a² sec² x + b² cosec²x, where a > 0,b>0.

Express    as a sum of partial fractions.

If z-1+2i|= 4, show that the point z+ i describes a circle. Also draw this circle.

The set R of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on R as:

Is (*) associative in R? Is (.) distributive (*) in R? Check.

Find the domain of the function f given by f(x) = 

Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.

a) The set  {S ∈R :x² - 3x + 2  = 0}is an infinite set.

b) The greatest interger function is continuous on R.

c)  

d) Every integrable function is monotonic.

e) a⊕b = a + b defines a binary operation on Q, the set of rational numbers.

A set which has no limit point

A compact set.

A set which is neither open nor closed.

A sequence which is divergent.

Evaluate  using Riemann integration.

10. a) Find the value/s of x for which the series 

is convergent.

Show that the sequence {f_n}of functions, where 

is uniformly convergent in ,0[ k],where k > .0 Show further that } { n f is not uniformly convergent in ,0[ ∞[.

Show that 

Check whether f is uniformly continuous on [− ]1,1 or not.

Find the radius of convergence of the series .