Page 178 - DMTH201_Basic Mathematics-1

Unit 6: Functions

Notes
Example
The total cost TC of producing x units of a commodity is given by TC = 2000 + 4x. If each unit is
sold at 20 per unit, find the level of output to make sure that the production breaks-even.
Solution:
We can write total revenue of producing x units as TR = 20x.

Profit p = TR - TC = 20x - 2000 - 4x = 16x - 2000
The break-even point is given by the level of output at which p = 0.
2000
Thus 16x – 2000 = 0 or x 125
16
Thus, at least 125 units should be produced to make sure that the firm does not incur losses.

6.1.4 Basic Properties

1. The only function which is both even and odd is the constant function which is identically
zero (i.e., f(x) = 0 for all x).
2. The sum of an even and odd function is neither even nor odd, unless one of the functions
is identically zero.
3. The sum of two even functions is even, and any constant multiple of an even function is
even.
4. The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
5. The product of two even functions is an even function.
6. The product of two odd functions is an even function.

7. The product of an even function and an odd function is an odd function.
8. The quotient of two even functions is an even function.
9. The quotient of two odd functions is an even function.
10. The quotient of an even function and an odd function is an odd function.

11. The derivative of an even function is odd.
12. The derivative of an odd function is even.
13. The composition of two even functions is even, and the composition of two odd functions
is odd.
14. The composition of an even function and an odd function is even.
15. The composition of any function with an even function is even (but not vice-versa).
16. The integral of an odd function from A to +A is zero (where A is finite, and the function has
no vertical asymptotes between A and A).
17. The integral of an even function from A to +A is twice the integral from 0 to +A (where A
is finite, and the function has no vertical asymptotes between A and A).
18. Series
(a) The Maclaurin series of an even function includes only even powers.

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