Derivatives - Rate of change of quantities

  • Introduction to derivatives
  • Meaning of rate of change
  • Definition of derivative
  • Notation for derivatives
  • Finding derivatives using limits Example:
    • Find the derivative of the function f(x) = 3x^2 at x = 2.

Equation:

  • f’(x) = lim(h->0) [f(x+h) - f(x)] / h

Derivatives - Differentiation rules

  • Power rule
  • Constant rule
  • Sum and difference rule
  • Product rule
  • Quotient rule Example:
    • Find the derivative of the function f(x) = 4x^3 - 5x^2 + 2x - 7. Equations:
    • (d/dx) [x^n] = nx^(n-1)
    • (d/dx) [c] = 0
    • (d/dx) [f(x) + g(x)] = f’(x) + g’(x)
    • (d/dx) [f(x) - g(x)] = f’(x) - g’(x)
    • (d/dx) [f(x) * g(x)] = f’(x) * g(x) + f(x) * g’(x)
    • (d/dx) [f(x) / g(x)] = [f’(x) * g(x) - f(x) * g’(x)] / (g(x))^2

Derivatives - Chain rule

  • Statement of chain rule
  • Applying chain rule to composite functions
  • Derivatives of inverse functions Example:
    • Find the derivative of the function f(x) = sin(3x^2). Equation:
    • (d/dx) [f(g(x))] = f’(g(x)) * g’(x)

Derivatives - Implicit differentiation

  • Implicit differentiation explained
  • Applying implicit differentiation to equations
  • Finding second derivatives implicitly Example:
    • Find the derivative of the equation x^2 + y^2 = 25. Equation:
    • (d/dx) [y] = - (dy/dx) / (dx/dy)

Derivatives - Related rates

  • Understanding related rates problems
  • Process of solving related rates problems
  • Finding rates of change using derivatives Example:
    • A rectangular prism has a volume of 500 cubic units. The length, width, and height are changing at rates of 2, 3, and -4 units per second respectively. Find the rate of change of the volume when the length is 10 units, width is 5 units, and height is 4 units. Equation:
    • V = lwh

Derivatives - Optimization

  • Introduction to optimization problems
  • Process of solving optimization problems
  • Finding maximum and minimum values using derivatives Example:
    • A farmer has 200 feet of fencing and wants to enclose a rectangular area. What dimensions will maximize the enclosed area? Equation:
    • A = l * w

Derivatives - L’Hopital’s rule

  • Statement of L’Hopital’s rule
  • Using L’Hopital’s rule to evaluate limits
  • Applying L’Hopital’s rule to indeterminate forms Example:
    • Evaluate the limit (sin(x) / x) as x approaches 0. Equation:
    • lim(x->a) [f(x) / g(x)] = lim(x->a) [f’(x) / g’(x)]

Derivatives - Euler’s method

  • Introduction to Euler’s method
  • Approximating solutions using derivatives
  • Step-by-step process of using Euler’s method Example:
    • Use Euler’s method to approximate the value of y at x = 1 for the differential equation dy/dx = x + y, given y(0) = 1. Equation:
    • y_n+1 = y_n + h * f(x_n, y_n)

Derivatives - Applications in physics

  • Velocity and acceleration
  • Tangent line approximation
  • Applications of derivatives in motion Example:
    • An object is thrown vertically upward with an initial velocity of 25 m/s. Find the maximum height reached by the object. Equation:
    • v(t) = v0 - 9.8t

Slide 11

  • Derivatives - Rate of change of quantities continued
  • Second derivative: rate of change of rate of change Example:
    • Find the second derivative of the function f(x) = 3x^2. Equation:
    • f’’(x) = (d^2/dx^2) [f(x)]

Slide 12

  • Derivatives - Rate of change of quantities continued
  • Higher order derivatives: rate of change of higher order Example:
    • Find the third derivative of the function f(x) = 4x^3 - 5x^2 + 2x - 7. Equation:
    • f’’’(x) = (d^3/dx^3) [f(x)]

Slide 13

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in economics and business Examples:
    • Determining marginal cost and revenue
    • Maximizing profit and minimizing cost
    • Optimization in production and pricing Equation:
    • Marginal cost = (dC/dx)
    • Marginal revenue = (dR/dx)
    • Profit = Revenue - Cost

Slide 14

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in biology and medicine Examples:
    • Modeling population growth and decay
    • Analyzing enzyme kinetics
    • Understanding drug absorption rates Equation:
    • Population growth model: dP/dt = rP - cP
    • Enzyme kinetics: v = (Vmax * [S]) / (Km + [S])
    • Drug absorption: dD/dt = -kD

Slide 15

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in physics Examples:
    • Velocity and acceleration
    • Projectile motion
    • Harmonic motion Equation:
    • Velocity: v = dx/dt
    • Acceleration: a = dv/dt
    • Projectile motion: y = (v0sinθ)t - (1/2)gt^2
    • Harmonic motion: x = Acos(ωt + φ)

Slide 16

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in engineering Examples:
    • Calculating stress and strain in materials
    • Optimizing designs for efficient structures
    • Analyzing fluid flow in pipes and channels Equation:
    • Stress: σ = F/A
    • Strain: ε = ΔL/L
    • Fluid flow: Q = A * v

Slide 17

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in computer science Examples:
    • Analyzing algorithms and time complexity
    • Image processing and edge detection
    • Data analysis and machine learning Equation:
    • Time complexity: O(n)
    • Edge detection: ∇f = |∂f/∂x| + |∂f/∂y|
    • Machine learning: ∂L/∂W = X^T(Y - Ŷ)

Slide 18

  • Derivatives - Rate of change of quantities continued
  • Applications of derivatives in finance and investment Examples:
    • Calculating interest rates and yields
    • Portfolio optimization
    • Option pricing and risk analysis Equation:
    • Interest rate: r = (F - P)/P
    • Portfolio return: R = Σ(wi * Ri)
    • Option pricing: C = S - Ke^(-rt)

Slide 19

  • Derivatives - Rate of change of quantities continued
  • Common mistakes to avoid when using derivatives Examples:
    • Forgetting to simplify the derivative
    • Misapplying the product or chain rule
    • Incorrectly interpreting the derivative in context Equation:
    • Simplify the derivative before evaluating
    • Double-check your application of rules
    • Always consider the context of the problem

Slide 20

  • Derivatives - Rate of change of quantities continued
  • Summary of key concepts and equations Equations:
    • f’(x) = lim(h->0) [f(x+h) - f(x)] / h
    • (d/dx) [x^n] = nx^(n-1)
    • (d/dx) [c] = 0
    • (d/dx) [f(x) + g(x)] = f’(x) + g’(x)
    • (d/dx) [f(x) * g(x)] = f’(x) * g(x) + f(x) * g’(x)

Slide 21

  • Derivatives - Mean Value Theorem
  • Statement of the Mean Value Theorem
  • Understanding the conditions for applying the theorem
  • Finding the c-value that satisfies the theorem Example:
    • Find the c-value for the function f(x) = 3x^2 - 2x + 1 on the interval [1, 2]. Equation:
    • If f’(c) = (f(b) - f(a)) / (b - a), then there exists a c in (a, b) such that f’(c) = slope of the secant line between (a, f(a)) and (b, f(b)).

Slide 22

  • Derivatives - Rolle’s Theorem
  • Statement of Rolle’s Theorem
  • Understanding the conditions for applying the theorem
  • Consequences of Rolle’s Theorem Example:
    • Find all values of c for the function f(x) = x^3 - 3x^2 - 9x + 1 on the interval [-3, 3]. Equation:
    • If f(a) = f(b), then there exists a c in (a, b) such that f’(c) = 0.

Slide 23

  • Derivatives - Techniques of Integration
  • Introduction to integration as reverse process of differentiation
  • Basic integration formulas
  • Integration by substitution
  • Integration by parts Example:
    • Evaluate the integral ∫ 3x^2 + 2x + 1 dx. Equation:
    • ∫ a^n dx = (a^(n+1))/(n+1) + C

Slide 24

  • Derivatives - Techniques of Integration (continued)
  • Integration by partial fractions
  • Trigonometric substitutions in integration
  • Integration of rational functions
  • Improper integrals Example:
    • Evaluate the integral ∫ (x + 1)/(x^2 + 3x + 2) dx. Equation:
    • Partial fraction decomposition: (x + 1)/(x^2 + 3x + 2) = A/(x + 1) + B/(x + 2)

Slide 25

  • Derivatives - Techniques of Integration (continued)
  • Integration with trigonometric functions
  • Integration of exponential and logarithmic functions
  • Integration involving inverse trigonometric functions Example:
    • Evaluate the integral ∫ sin^2 x cos x dx. Equation:
    • ∫ sin^2 x cos x dx = ∫ (1 - cos^2 x) cos x dx

Slide 26

  • Derivatives - Techniques of Integration (continued)
  • Applications of integration in physics
  • Finding areas under curves
  • Calculating volumes of solids of revolution Example:
    • Find the area bounded by the curve y = x^2 and the x-axis over the interval [1, 3]. Equation:
    • Area = ∫ (top curve - bottom curve) dx

Slide 27

  • Derivatives - Techniques of Integration (continued)
  • Numerical methods of integration
  • Riemann sums and the Trapezoidal Rule
  • Simpson’s Rule for approximating definite integrals Example:
    • Approximate the integral ∫ (1/x) dx using the Trapezoidal Rule with 4 subintervals. Equation:
    • Trapezoidal Rule: ∫ f(x) dx ≈ h/2 * (f(a) + 2∑(f(x_i)) + f(b))

Slide 28

  • Derivatives - Techniques of Integration (continued)
  • Improper integrals
  • Understanding infinite limits of integration
  • Evaluating improper integrals with vertical asymptotes Example:
    • Evaluate the integral ∫ (1/x) dx over the interval [1, ∞]. Equation:
    • ∫ (1/x) dx = lim(b->∞) ∫(1, b) (1/x) dx

Slide 29

  • Derivatives - Applications in Engineering
  • Applications of derivatives in engineering
  • Optimization in engineering design
  • Estimating error in engineering measurements Example:
    • Optimize the dimensions of a rectangular box to minimize the cost of materials, given the volume must be 100 cubic units. Equation:
    • Cost function: C = lwh + 2lw + 2lh + 2wh

Slide 30

  • Derivatives - Review and Exam Tips
  • Review of key concepts and formulas in derivatives
  • Tips for studying and preparing for exams Equations:
    • Power rule: (d/dx) [x^n] = nx^(n-1)
    • Chain rule: (d/dx) [f(g(x))] = f’(g(x)) * g’(x)
    • Mean Value Theorem: If f’(c) = (f(b) - f(a)) / (b - a), then there exists a c in (a, b) such that f’(c) = slope of the secant line between (a, f(a)) and (b, f(b))
    • Rolle’s Theorem: If f(a) = f(b), then there exists a c in (a, b) such that f’(c) = 0