學習內容

1. 學習內容 • 3.1 Derivatives and Rates of Change • 導函數與改變率 • 3.2 The Derivative as a Function • 視導函數為函數 • 3.3 Differentiation Formulas • 微分公式 • 3.4 Derivatives of Trigonometric Functions • 三角函數的導函數 • 3.5 The Chain Rule • 鎖鏈法則 • 3.6 Implicit Differentiation • 隱函數微分 • 3.9 Linear Approximations and Differentials • 線性估計與微變量

2. 3.9 Linear Approximations and Differentials 線性估計與微變量

3. 學習重點 • 函數之線性估計式 • 用微變量估計改變量

4. 線性估計的發想 曲線幾乎近似於直線 f (x) ~ L(x) Figure 3.9.1, p. 189

5. 曲線函數 y = f(x) • 切線函數y = f(a) + f’(a)(x – a) y = f (x) ≈ L(x) = f(a) + f’(a)(x –a) L(x) is a Linearization of f (x) near x = a

6. Example 1: Find the linearization of at a = 1. Approximate . y = f(a) + f’(a)(x – a) = 2 1 1 1 1 1 = 1/4 1 1 Linearization

7. Approximate . 誤差 0.00000627 0.98 1.05 0.98 1.05 誤差 0.0003883

8. Example 2 • For what values of x is the linear approximation accurate to within 0.5? |Error| =

10. 0.5

11. 改變量與微變量 • 改變量 ( increment) • Δx= (x + Δx) - x • Δy = f (x + Δx) - f (x) • 微變量 (differential) • dx • .

12. 微變量~微變量 • 當dx ~ Δx • 則 dy ~ Δy

13. Example 3 • Compare the values of ∆y and dyif y = f(x) = x3 + x2 – 2x + 1 and x changes from: • 2 to 2.05 • 2 to 2.01 f’(x) = 3x2 + 2x – 2 Figure 3.9.6, p. 192

14. Example 3(a) x changes from 2 to 2.05 改變量 • ∆y = f(2.05) – f(2) = 0.717625 • f(2) = 23 + 22 – 2(2) + 1 = 9 • f(2.05) = (2.05)3 + (2.05)2 – 2(2.05) + 1 =9.717625 • dy = f’(x)dx = (3x2 + 2x – 2) dx • When x = 2 and dx = ∆x = 0.05 • dy = [3(2)2 + 2(2) – 2]0.05 = 0.7 誤差0.017625 微變量

15. Example 3(a) x changes from 2 to 2.05 改變量 • ∆y = f(2.01) – f(2) = 0.140701 • f(2) = 23 + 22 – 2(2) + 1 = 9 • f(2.01) = (2.01)3 + (2.01)2 – 2(2.01) + 1 =9.140701 • dy = f’(x)dx = (3x2 + 2x – 2) dx • When x = 2 and dx = ∆x = 0.01 • dy = [3(2)2 + 2(2) – 2]0.01 = 0.14 誤差0.000701 微變量