Lesson02

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2
Special Classes of Functions
• A function f is said to be an odd function if
f (−x) = −f (x) for all x.
y
O
f (x ) = x 8
3
x
g (x) = x2 sin x
◦ The graph is symmetric about the origin O .
◦ depends only on the values of f (x) for “x near a”. ◦ The limit is independent to the value of f (x) “at a”.
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Example
• Find lim
x2 − 1 . x→ 1 x − 1
h→0
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Limit of a Function
x3 − 3x2 + 4x − 4 • Consider the function: f (x) = . x−2 ◦ As x approaches 2, f (x) is getting close to 4.
•
We write lim f (x) = 4.
1
Chapter 0: Functions
Special Classes of Functions
• A function f is said to be increasing on a set I if
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x1 < x2 ⇒ f (x1 ) < f (x2 ) for any x1 , x2 ∈ I.
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y − y0 y1 − y0 = . x − x0 x1 − x0
The Tangent Line
• Consider the parabola y = x2 :
y •
2 −1 mP Q = x x −1 •
Q(x, x2 )
• • P (1, 1) O x
◦ Fix P (1, 1). Let Q be another point on the curve. ◦ Connect P and Q, and let Q approach P .
•
The resulting line is the tangent line of y = x2 at P . Slope m = lim mP Q = lim
Q→P
x2 − 1 . x→ 1 x − 1
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4
Example
• A ball is dropped from a tower 450 m above the ground. Find its instantaneous velocity after 5
A function f is said to be decreasing on a set I if
x1 < x2 ⇒ f (x1 ) > f (x2 ) for any x1 , x2 ∈ I.
y f (x ) = x 3
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x O
f ( x ) = −x 3 + 2
Special Classes of Functions
x→ a
lim f (x) = L.
We say the limit of f (x), as x approaches a, equals L.
◦ Sometimes we may simply write:
x→a
• Note. The limit lim f (x)
x→ a
⇒
f (x) → L.
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3
Chapter 1: Limits
Part I: A First Encounter with Limits
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The Straight Line
• A straight line is uniquely determined by 2 distinct points. ◦ Let A(x0 , y0 ) and B (x1 , y1 ) be distinct points on R2 .
lim f (x) f (x) = x→ a x→a g (x) lim g (x) lim
x→ a
provided that lim g (x) = 0.
x→ a
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7
Limit Laws
• The product law lim f (x)g (x) = lim f (x) lim g (x) can be generalized to the product of n
◦ Consider the graph of f (x) =
x2 − 1 : x−1
y
2
2 −1 y= x x −1
O
1
x
◦ lim
x2 − 1 = lim f (x) = 2. x→ 1 x − 1 x→ 1
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6
Limit Laws
• Let c be a real number. Then the constant function f (x) = c is not affected by the behavior of x. ◦ Let a ∈ R. x → a ⇒ c → c.
MA1102R CALCULUS Lesson 2
Wang Fei
matwf@.sg
Department of Mathematics Office: S17-06-16 Tel: 6516-2937
Chapter 0: Functions 2 Special Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 1: Limits 7 Straight line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Tangent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Direct Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
x→ 2
y
4
O
2
x
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5
Intuitive Definition of Limit
• Definition. If f (x) is arbitrarily close to L by taking x to be sufficiently close (but not equal) to
a, then we write
Consider the line ℓ passing through A and B :
y
ℓ (x, y ) B (x 1 , y 1 )
A(x0 , y0 ) O
x
Let (x, y ) be a point on ℓ. Then
•
m=
y1 − y0 is called the slope (or gradient) of ℓ. x1 − x0
odd function. Moreover, such a decomposition is unique. (Exercise!)
• In general, any function defined on R can be decomposed into the sum of an even function and an
y
11 13
x
10 12 14 16 18 20 2 4 6 8 y y= =x x
O
x
• P (x) = 3x6 + x5 − 2x4 + 7x3 + 2x2 − x − 1.