Lesson07

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3
Examples
• Let f (x) =
1−x . 2+x f ′ (x) = lim = lim = lim
h→0
f (x + h) − f (x) h 1−(x+h) −x −1 2+(x+h) 2+x h
h→0
−3h h→0 h(2 + (x + h))(2 + x) −3 = lim h→0 (2 + (x + h))(2 + x) −3 = . (2 + x)2
◦
∆s 4.9(5 + h)2 − 4.9(5)2 49h + 4.9h2 = = ∆t (5 + h) − 5 h 2 49h + 4.9h = 49 m/s. ◦ Velocity at t = 5: lim h→0 h
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Example
• The position of a particle is given by s(t) =
f ′ (a) := lim
f (a + h) − f (a) exists. h ◦ ◦
◦ f is differentiable on (a, b) if it is differentiable at every c ∈ (a, b).
What’s the relation between differentiability and continuity? What kinds of functions are differentiable? How to construct new differentiable functions? How the derivative affects the original function? 9 / 26
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a−h a a+h
O
x
Example
• Let f (x) = |x|. Find f ′ if it exists.
y
h
a
O
h
x
◦ If a = 0, f ′ (0) = lim
•
•
|h| h→0 h |h| lim h→0− h lim+
|h| − 0 does not exist. h→0 h h = lim+ = lim+ 1 = 1, h→0 h h→0 −h = lim = lim (−1) = −1. h→0− h h→0−
x→ a
x→ a
f is continuous at a ⇐⇒ lim f (x) = f (a) x→ a f (x) − f (a) lim (f (x) − f (a)) = lim · lim (x − a) x→ a x→ a x→ a x−a = L · 0 = 0. lim f (x) = lim (f (x) − f (a)) + lim f (a) = 0 + f (a) = f (a).
−h (a+h)a h→0
= lim
h −1 = lim h→0 (a + h)a 1 = − 2. a
◦ f ′ is therefore a function defined on R\{0}.
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Derivative as a Function
• The derivative of f at point x = a:
◦ Domain of f : R\{−2}; Domain of f ′ : R\{−2}.
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Examples
• Let f (x) =
√
x, (x ≥ 0).
Find f ′ .
f ′ (x) = lim
f (x + h) − f (x) h→0 h √ √ x+h− x = lim h→0 √ h √ √ √ ( x + h − x)( x + h + x) √ = lim √ h→0 h( x + h + x) h = lim √ √ h→0 h( x + h + x) 1 1 = lim √ (x > 0). √ = √ h→0 2 x x+h+ x
seconds.
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• Example. A ball is dropped from a tower 450m above the ground. Find its velocity after 5
s(5) s(5 + h)
gt2 = 4.9t2 . ◦ s(t) = 1 2
◦ v (5) = s′ (5) = 49.
• Theorem.
If f is differentiable at a, then f is continuous at a.
• Proof. Suppose f ′ (a) = L. Then
f ′ (a) = L
how?
⇐⇒ lim
f (x) − f (a) =L x→ a x−a ⇓ how? lim [f (x) − f (a)] = 0
but f is NOT differentiable at x = 0.
◦ It seems that the condition of differentiability is “stronger” than the condition of continuity.
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Differentiability Implies Continuity
The domain of f ′ may be smaller than the domain of f . 8 / 26
4
Differentiable Functions
• Definition. (We only consider the differentiability at a point or on open intervals) ◦ f is differentiable at a if
Example
• Let f (x) = |x|. Find f ′ if it exists.
y
O
◦ If a > 0,
|a + h| − |a| f ′ (a) = lim h→0 h (a + h) − a = lim h→0 h h = lim = lim 1 = 1. h→0 h h→0
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1
Chapter 3: Derivatives
Velocity
• Let s = s(t) be the position function of a particle. ◦ instantaneous velocity at time t = a: s′ (a); ◦ speed at time t = a: |s′ (a)|.
f ′ (a) = lim
• The derivative of f as a function:
f (a + h) − f (a) . h→0 h
f ′ (x) = lim
◦ f ′ (x) = y ′ =
f (x + h) − f (x) . h→0 h
dy df d = = f (x) = Dx f (x) = · · · dx dx dx dy ∆y ◦ (Leibniz, 1646–1716, German) := lim ∆x→0 ∆x dx dy ◦ f ′ (a) = . dx x=a 1 1 • Example. If f (x) = , then f ′ (x) = − 2 . x x
MA1102R CALCULUS Lesson 7
Wang Fei
matwf@.sg
Department of Mathematics Office: S17-06-16 Tel: 6516-2937
Chapter 3:Derivatives 2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Derivative as Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26