Overview

This foundational topic introduces essential calculus formulas and rules used throughout the subject. Whether you're evaluating limits, computing derivatives, or applying integrals, these core concepts form the building blocks for understanding and mastering calculus.

Key Concepts and Formulas

• Algebra and Trigonometry Refresher:
  • (a + b)^2 = a^2 + 2ab + b^2; (a - b)^2 = a^2 - 2ab + b^2
  • a^2 - b^2 = (a - b)(a + b)
  • sin^2 x + cos^2 x = 1; tan^2 x + 1 = sec^2 x
  • ln(ab) = ln a + ln b; ln(a^n) = n ln a
• Limits:
  • \lim_{x \to a} c = c; \lim_{x \to a} x = a
  • Use properties to simplify: sums, products, quotients
  • Know indeterminate forms: 0/0, ∞/∞
• Basic Derivative Rules:
  • \frac{d}{dx}(x^n) = nx^{n-1}
  • \frac{d}{dx}(\sin x) = \cos x; \frac{d}{dx}(\cos x) = -\sin x
  • \frac{d}{dx}(e^x) = e^x; \frac{d}{dx}(\ln x) = 1/x
  • Product rule: (fg)' = f'g + fg'
  • Quotient rule: (f/g)' = (f'g - fg') / g^2
  • Chain rule: \frac{d}{dx} f(g(x)) = f'(g(x))g'(x)
• Antiderivatives and Integration:
  • \int x^n dx = \frac{x^{n+1}}{n+1} + C
  • \int \frac{1}{x} dx = \ln|x| + C
  • \int e^x dx = e^x + C; \int \sin x dx = -\cos x + C
  • Use u-substitution to reverse chain rule patterns
• The Fundamental Theorem of Calculus:
  • Part 1: \int_a^b f'(x) dx = f(b) - f(a)
  • Part 2: \frac{d}{dx} \int_a^x f(t) dt = f(x)
• Essential Integrals and Applications:
  • Average value: \frac{1}{b-a} \int_a^b f(x) dx
  • Area between curves: \int_a^b [f(x) - g(x)] dx
  • Displacement and total distance from velocity: integrate |v(t)|

Quick Tip

Practice Problems

1. Differentiate the function: f(x) = 3x^2 + 5x - 7
   Show Solution

   Step 1: Use the power rule: \frac{d}{dx}(x^n) = nx^{n-1}

   Step 2: Differentiate each term:

   • \frac{d}{dx}(3x^2) = 6x
   • \frac{d}{dx}(5x) = 5
   • \frac{d}{dx}(-7) = 0

   Final Answer: f'(x) = 6x + 5

2. Evaluate the definite integral: \int_1^3 (2x) dx
   Show Solution

   Step 1: Find the antiderivative of 2x: \int 2x dx = x^2 + C

   Step 2: Apply the limits: F(3) - F(1) = 9 - 1 = 8

   Final Answer: 8