Calculus Week 4: Lecture Note

Lecture Notes: Advanced Calculus 1. Limits and Continuity Definition of a Limit Let f (x) be defined on an open interval around c, except possibly at c. We say: lim f (x) = L x→c if for every ϵ > 0, there exists a δ > 0 such that 0 < |x − c| < δ implies |f (x) − L| < ϵ. Properties of Limits 1. Sum Rule: limx→c [f (x) + g(x)] = limx→c f (x) + limx→c g(x). 2. Product Rule: limx→c [f (x)g(x)] = limx→c f (x) · limx→c g(x). 3. Quotient Rule: limx→c f (x) g(x) = limx→c f (x) , limx→c g(x) Continuity A function f (x) is continuous at c if: 1. f (c) is defined. 2. limx→c f (x) exists. 3. limx→c f (x) = f (c). 1 provided limx→c g(x) ̸= 0. 2. Differentiation Definition of the Derivative The derivative of f (x) at x = c is: f (c + h) − f (c) . h→0 h f ′ (c) = lim Rules of Differentiation 1. Power Rule: If f (x) = xn , then f ′ (x) = nxn−1 . 2. Product Rule: (uv)′ = u′ v + uv ′ .
′ ′ ′ 3. Quotient Rule: uv = u v−uv . v2 4. Chain Rule: If y = f (g(x)), then y ′ = f ′ (g(x))g ′ (x). Higher-Order Derivatives The n-th derivative of f (x) is denoted as: f (n) (x) = dn f (x). dxn 3. Integration Definition of the Definite Integral The definite integral of f (x) on [a, b] is: Z b f (x) dx = lim a where ∆x = b−a n n→∞ n X f (x∗i )∆x, i=1 and x∗i is a sample point in [xi−1 , xi ]. 2 Fundamental Theorem of Calculus 1. If F (x) is an antiderivative of f (x), then: Z b f (x) dx = F (b) − F (a). a 2. If f (x) is continuous on [a, b], then: Z x d f (t) dt = f (x). dx a Techniques of Integration R R 1. Substitution: Let u = g(x), then f (g(x))g ′ (x) dx = f (u) du. R R 2. Integration by Parts: u dv = uv − v du. 3. Partial Fractions: Decompose P (x) Q(x) into simpler fractions. 4. Applications of Calculus Optimization To find local maxima or minima of f (x): 1. Compute f ′ (x) and solve f ′ (x) = 0 for critical points. 2. Use the second derivative test: f ′′ (x) > 0 =⇒ local minimum, f ′′ (x) < 0 =⇒ local maximum. Area and Volume 1. Area under a curve: Z Area = b f (x) dx. a 2. Volume of revolution: Using the disk method: Z b V =π [f (x)]2 dx. a 3