• Before Calculus

New functions from old

Exercise

• Limits and Continuity
  • ex1, ex2

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• The Derivating in Graphing and Applications

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• Rolle's Theorem; Mean Value Theorem

An example

• Integration
  • The |Indefinite Integral
    • Two equal integrals
  • Numerical Approximation of Area /area under the curve/
  • A 2nd example of midpoint approximation
  • 3rd Example midpoint approximation of the integral using n = 20 subintervals
• Graphing Functions defined by Integrals (An example of a piecewise function)
• Principles of Integral Evaluation
  • Numerical Integration; Simpson’s Rule
    • Example of Trapezoidal Approximation, pic1 (Geogebra), pic2 (Excel)
    • Example of Simpson Rule

• Applications of the Definite Integral in Geometry, Science, and Engineering
  • Areas and Volumes known shapes and solids
  • Solids Volume
• Polar coordinates
  • A four petal rose
  • r(θ) = 1+cos(θ)
  • Area Bound by  r² = 4^.cos(θ)
  • Area bound by r(θ) = 1+cos(θ) (a cardioid)
  • Area between by r(θ) = 2+2^.cos(θ) (a cardioid) and circle

• Infinite Series

• SEQUENCES DEFINED RECURSIVELY

  • Example pages 1963, 1964

• Integral Test: example 1

• Maclaurin and Taylor Polynomials

  • Recall Local Linear Approximation

    • Example of Linear Approximation - Use the local linear approximation to approximate  f(x) = x² + 1, at x₀=1

• Find the Maclaurin polynomials  p₁, p₂, p₃, p₄, for f(x)=e^x