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Fundamentals of Calculus Concepts

Sep 4, 2024

Calculus Lecture Notes

Overview

  • Newton's Quotient
    • Fundamental to understanding derivatives from first principles.
    • Used to find the slope or instantaneous rate of change of a function at a specific point.
    • Calculate derivative by evaluating the limit as h approaches zero.
    • Example: Differentiating ( f(x) = x^2 - 2x + 4 ) and evaluating ( f'(4) ).

Derivative Rules

  • Power Rule: ( \frac{d}{dx} x^n = nx^{n-1} )
  • Product Rule: ( \frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) )
  • Quotient Rule: ( \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} )
  • Chain Rule: ( \frac{d}{dx} f(g(x)) = f'(g(x))g'(x) )

Derivatives of Special Functions

  • Exponential Functions: ( \frac{d}{dx} b^x = b^x \ln(b) )
  • Logarithmic Functions: ( \frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)} )
  • Trigonometric Functions:
    • ( \frac{d}{dx} \sin(x) = \cos(x) )
    • ( \frac{d}{dx} \cos(x) = -\sin(x) )
    • ( \frac{d}{dx} \tan(x) = \sec^2(x) )

First Derivative Test

  • Critical Numbers: Points where the first derivative is zero or undefined.
  • Test: Classify critical points as local maxima, minima, or neither by analyzing sign changes of the derivative.

Second Derivative Test

  • Concavity:
    • Positive second derivative indicates concave up.
    • Negative second derivative indicates concave down.
  • Test: Determine local extrema using the concavity and critical points.

Curve Sketching

  • Use derivatives to analyze function behavior:
    • Identify intervals of increase/decrease.
    • Determine concavity changes and points of inflection.
    • Utilize critical points to indicate maxima/minima.

Optimization

  • Problem Solving:
    • Define objective and constraint functions.
    • Utilize first or second derivative tests to find minimum or maximum values.
    • Example: Minimize surface area of an open box given volume constraints.

Antiderivatives

  • Notation: ( \int f(x) , dx = F(x) + C )
  • Methods: Use power rule for integration and substitution techniques.

Definite Integrals

  • Applications: Calculate areas under curves.
  • Fundamental Theorem of Calculus: Evaluate antiderivative at endpoints.

Calculating Volumes

  • Disk Method: Calculate volume of solids of revolution.
    • Use definite integrals to sum volumes of infinitesimal disks.

This lecture covered foundational concepts and rules essential to calculus, focusing on differentiation and integration techniques and their applications in problem-solving scenarios.

Full transcript