Limits: The limit of a function f(x) as x approaches a particular value, c, is denoted as lim(x→c) f(x). The limit indicates the behavior of the function as x gets closer and closer to c. The limit exists if the function approaches a single value as x approaches c. Continuity: A function f(x) is continuous at a point c if the limit of f(x) as x approaches c equals f(c). A function is continuous on an interval if it is continuous at every point within the interval. Derivatives: The derivative of a function f(x) at a point x = a, denoted as f'(a), represents the instantaneous rate of change of f(x) at that point. The derivative is defined as the limit of the difference quotient as the change in x approaches 0: f'(a) = lim(h→0) [f(a+h) - f(a)] / h. The derivative measures the slope of the tangent line to the curve at a specific point. Rules of Differentiation: Power rule: d/dx [x^n] = nx^(n-1) Sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x) Product rule: d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x) Quotient rule: d/dx [f(x) / g(x)] = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2 Chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x) Applications of Derivatives: Local extrema: Points where a function has a maximum or minimum value within an interval. Critical points: Points where the derivative is zero or undefined. First derivative test: Determines intervals of increase or decrease. Second derivative test: Determines concavity and inflection points. Related Rates:
Related rates problems involve finding how the rates of two changing quantities are related. Strategy: Identify the relevant variables, write an equation relating them, and then differentiate both sides with respect to time. Integration: The integral of a function f(x) represents the accumulated area under the curve. Definite integral: ∫[a, b] f(x) dx calculates the net area between f(x) and the x-axis from a to b. Indefinite integral: ∫ f(x) dx represents the antiderivative of f(x). The integral of a derivative is the original function: ∫ f'(x) dx = f(x) + C. Fundamental Theorem of Calculus: Part 1: If F(x) is the antiderivative of f(x), then ∫[a, b] f(x) dx = F(b) - F(a). Part 2: If F(x) is continuous on [a, b], then d/dx ∫[a, x] f(t) dt = f(x). Techniques of Integration: Substitution: Choosing a substitution to simplify an integral. Integration by parts: ∫ u dv = uv - ∫ v du. Trigonometric substitution: Substituting trigonometric identities to solve integrals. Partial fractions: Decomposing rational functions into simpler fractions. Applications of Integration: Area under a curve: Integrating to find the area enclosed by the curve and the x-axis. Volume of revolution: Using integration to calculate the volume formed by revolving a region around an axis. Arc length: Integrating to find the length of a curve.
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Calculus Fundamentals: Limits, Derivatives, and Integrals