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Limits and Derivatives
Concept of a Function
y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.
y = x2
Since the value of y depends on a given value of x, we call y the dependent variable and x the independent variable and of the function y = x2.
Notation for a Function : f(x)
The Idea of Limits
Consider the function
The Idea of Limits
2
4)(
2
x
xxf
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x)
Consider the function
The Idea of Limits
2
4)(
2
x
xxf
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x) 3.9 3.99 3.999 3.9999 un-defined
4.0001 4.001 4.01 4.1
Consider the function
The Idea of Limits 2)( xxg
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
g(x) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1
2)( xxg
x
y
O
2
If a function f(x) is a continuous at x0,
then . )()(lim 00
xfxfxx
4)(lim2
xfx
4)(lim2
xgx
approaches to, but not equal to
Consider the function
The Idea of Limits
x
xxh )(
x -4 -3 -2 -1 0 1 2 3 4
g(x)
Consider the function
The Idea of Limits
x
xxh )(
x -4 -3 -2 -1 0 1 2 3 4
h(x) -1 -1 -1 -1 un-defined
1 2 3 4
1)(lim0
xhx
1)(lim0
xhx
)(lim0
xhx does not
exist.
A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write
lxfxx
)(lim0
Theorems On Limits
Theorems On Limits
Theorems On Limits
Theorems On Limits
Exercise 12.1P.7
Limits at Infinity
Limits at Infinity
Consider1
1)(
2
xxf
Generalized, if
)(lim xfx
then
0)(
lim xf
kx
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Exercise 12.2P.13
Theorem
where θ is measured in radians.
All angles in calculus are measured in radians.
1sin
lim0
Exercise 12.3P.16
The Slope of the Tangent to a Curve
The Slope of the Tangent to a Curve
The slope of the tangent to a curve y = f(x) with respect to x is defined as
provided that the limit exists.
x
xfxxf
x
yAT
xx
)()(limlim of Slope
00
Exercise 12.4P.18
Increments
The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.
For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) –
f(x0).
Derivatives(A) Definition of Derivative.
The derivative of a function y = f(x) with respect to x is defined as
provided that the limit exists.
x
xfxxf
x
yxx
)()(limlim
00
The derivative of a function y = f(x) with respect to x is usually denoted by
,dx
dy),(xf
dx
d ,'y ).(' xf
The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to x exists at x = x0.
The value of the derivative of y = f(x) with respect to x at x = x0 is denoted
by or .0xxdx
dy
)(' 0xf
To obtain the derivative of a function by its definition is called differentiation of the function from first principles.
Exercise 12.5P.21