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Lecture 1

School

University of WindsorDepartment

Biological SciencesCourse Code

BIOL 1111Professor

Animesh SarkerLecture

1This

**preview**shows half of the first page. to view the full**3 pages of the document.**Diﬀerential Calculus

MTH 62-140

Some Topics in Functions

1. Function: A function fis a rule that assigns to each element xin a set Dexactly one

element, called f(x), in a set E. The set Dis called the domain of the function. The

range of fis the set of all possible values of f(x) as xvaries throughout the domain.

2. Four ways to represent a function: verbal, visual, numerical and algebraic.

3. Properties of a function:

(a) Domain and range

•the set of values for which the function is deﬁned

•the set of all possible values of f(x) as xvaries throughout the domain

(b) Intercepts(xand yintercepts):

•To ﬁnd xintercept(s), set y= 0 and solve for x.

•To ﬁnd yintercept(s), set x= 0 and solve for y.

(c) One-to-one:

•A function fis one-to-one if f(x1) = f(x2) implies x1=x2.

•A function is one-to-one if and only if no horizontal line intersects its graph

more than once.

(d) Symmetry:

•A function fis called an even function if f(x) = f(−x) for every x

•A function fis called an odd function if f(−x) = −f(x) for every x

•A function fis periodic if f(x) = f(x+p) for some p

(e) Intervals of increase and decrease

•A function fis called increasing on an interval Iif f(x1)< f(x2) whenever

x1< x2in I

•A function fis called decreasing on an interval Iif f(x1)> f(x2) whenever

x1< x2in I

•A function fis periodic if f(x) = f(x+p) for some p

(f) Local minimum and maximum

•A function fhas a local maximum at cif f(c)≥f(x) when xis near c.[This

means that f(c)≥f(x) for all xin some open interval containing c.]

•Afhas a local minimum at cif f(c)≤f(x) when xis near c.

(g) Absolute minimum and maximum

•A function fhas an absolute maximum at cif f(c)≥f(x) for all xin D,

where Dis the domain of f. The number f(c) is called the maximum value

of fon D.

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