The Rule for An Even Function - Mathematics Assignment Help

Assignment Task

1. This exercise leads to the antiderivative of a certain rational function. 

(a) Find constants A, B and C such that 2x _ Ax + B C  (x2 +1)(x — 1) — x2 +1 + x —1 
(b) Find the following antiderivative, assuming that x > 1: 2x  J (x2 +1)(x — 1) dx 
(c) Find the following antiderivative, assuming now that x < 1: 2x  i (x2 +1)(x —1) dx 
 

2. This exercise explores how the properties of even and odd functions are related using differentiation and forming a particular area function. Recall that if f is function then the derivative f' may be defined by the following rule, when the limit exists:  = lim AY) — f(x)  f' (x)  y-).x y — X 

(a) Suppose that f is a differentiable function with domain lit Use the limit definition of the derivative given above to show that if f is even then f' is odd, whilst if f is odd then f' is even. (b) Suppose that f is a continuous function with domain R. Define an area function A by the following rule: A(x) = I f(t)dt .

Prove that if f is even then A is odd, whilst if f is odd then A is even.

3. Consider the function f defined by the rule 1 1 ex ± 1 2 •  f(x) — 

(a) Explain why f is an odd function. (b) Find the following definite integral: 7114 3 sine x cos x dx  f„/4 ex +1

[Hint: use part (a) and the fact that the numerator in the integrand defines the rule for an even function.] 

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