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Chap1 - Chapter 1 Nakanishi Office Hours for Week 2 only...

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Unformatted text preview: Chapter 1 Nakanishi Office Hours for Week 2 only: PHYS Rm. 264, M (8/30) and F (9/3), 11:30 am – 12:20 pm From Week 3 onward: Grader (Mr. Shen) Office Hour: PHYS Rm. 105, Th. 3 – 4 pm Nakanishi Office Hour: PHYS Rm. 264, F 11:30 am – 12:20 pm f '( x) ≡ lim ∆x → 0 Derivative of f(x): f ( x + ∆x) − f ( x) ∆x • Simple and explicitly given f(x) • Not-so-simple, but explicitly given f(x) • Composite function f(u(x)) • Implicitly given f(x), e.g., x=g(y), g(x)+h(y)=0, or g(x,y)=0 • Parametrically given function: x=g(t), y=h(t) logarithm and exponential: y = ln x and y = ex Chapter 1 More general power laws: y = x−n , y = x p / q , y = xa Taylor series for f(x): x2 x3 f ( x) = f (0) + f '(0) x + f ''(0) + f '''(0) + ... 2! 3! ∞ For example, xn ex = ∑ n=0 n ! Trigonometric and Hyperbolic Functions: ∞ ∞ (−1) n x 2 n (−1) n x 2 n +1 cos x = ∑ , sin x = ∑ (2n)! n=0 n = 0 (2n + 1)! e x + e− x e x − e− x cosh x = , sinh x = 2 2 Chapter 1 L’Hôspital’s Rule: • If [f(x)→∞ and g(x) →∞] or [f(x)→0 and g(x) →0] as x→a, then f ( x) f '( x) lim x →a g ( x) x→a ≡ lim g '( x) • If f’(x) and g’(x) still tends to infinity or zero, keep taking the derivatives until they will have a definite limiting ratio. Differential of f(x) at x=x0: df ≡ df dx dx x0 df dx ...
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Chap1 - Chapter 1 Nakanishi Office Hours for Week 2 only...

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