Calculus

Lenore Horner, Jan 25, 2017

1D calculus

Table of Contents

  1. Limits
    1. delta epsilon limits
  2. Numerical Methods
    1. Newton's Method
    2. Stair-step approximation of function
    3. Sums of Rectangles
    4. Riemann sums for definite integrals
    5. Riemann, Trapezoid, and Simpson's methods for numerical integration
    6. Simpson's Rule with Cubic Functions
    7. Numerical Integration
    8. Euler's Method
    9. Slope Fields and Solution Curves
    10. Slope Field
    11. Safer Slope Field
  3. Differentiation
    1. Constructing a Derivative
    2. Curve Sketch
    3. Secant -> Tangent -> trace for derivative
    4. Rates of change on a parabola
    5. Chain Rule Illustrated
    6. Estimating Change
    7. Proving Mean Value Thm
    8. Cauchy MVT
  4. Integration
    1. Defining Definite Integral
    2. FTC: rate of change of area
    3. FTC: area ''under'' curve
    4. Improper Integral with Diverging Integrand
  5. Volumes and Surface Areas
    1. Washer Method
    2. illustration of volume as a set of nested cylinders
    3. exponentials volume of revolution
    4. Surfaces of Revolution
    5. Solids with regular-polygon cross-sections
    6. Circles between Curves
    7. Equilateral Triangles based between Curves
    8. Isosceles Triangles between Curves
    9. Squares with diagonal bounded by Curves
    10. Revolving 2 exponentials
    11. Revolving exponential and polynomial
    12. Stepping through Volume-by-Shells method
    13. Arc Length for y(x)
  6. Series
    1. Linearization improvement leads to Maclaurin series
    2. Substituting new arguments into Taylor Series
    3. Zeno
    4. Conditional Convergence
  7. Fourier Series
    1. Fourier Example
    2. Fourier Products
    3. Fourier Series
    4. Fourier Products
  8. misc
    1. Hyperbolic Functions

1.

Limits

  • 1. delta epsilon limits

1.1.

delta epsilon limits

This applet illustrates the formal definition of a limit. The limit as x approaches A of f(x) is C if for any epsilon (illustrated by the vertical red bar) you can find a delta (illustrated by the width of the blue band at the x-axis) such that all values D within that blue band yield y values within epsilon of the limit (within the red band).[br][br]The arrow will turn on the animation of the move slider so that D moves from A-delta to A+delta. Check that the resulting y-values are always between C-epsilon and C+epsilon.
delta epsilon limits
– Lenore Horner

2.

Numerical Methods

  • 1. Newton's Method
  • 2. Stair-step approximation of function
  • 3. Sums of Rectangles
  • 4. Riemann sums for definite integrals
  • 5. Riemann, Trapezoid, and Simpson's methods for numerical integration
  • 6. Simpson's Rule with Cubic Functions
  • 7. Numerical Integration
  • 8. Euler's Method
  • 9. Slope Fields and Solution Curves
  • 10. Slope Field
  • 11. Safer Slope Field

2.1.

Newton's Method

Illustration of Newton's method for finding zeros of functions.
Newton's Method
– Lenore Horner

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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2.11.

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3.

Differentiation

  • 1. Constructing a Derivative
  • 2. Curve Sketch
  • 3. Secant -> Tangent -> trace for derivative
  • 4. Rates of change on a parabola
  • 5. Chain Rule Illustrated
  • 6. Estimating Change
  • 7. Proving Mean Value Thm
  • 8. Cauchy MVT

3.1.

Constructing a Derivative

– Lenore Horner

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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4.

Integration

  • 1. Defining Definite Integral
  • 2. FTC: rate of change of area
  • 3. FTC: area ''under'' curve
  • 4. Improper Integral with Diverging Integrand

4.1.

Defining Definite Integral

– Lenore Horner

4.2.

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4.3.

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4.4.

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5.

Volumes and Surface Areas

Shapes formed by polygonal slices and by revolution

  • 1. Washer Method
  • 2. illustration of volume as a set of nested cylinders
  • 3. exponentials volume of revolution
  • 4. Surfaces of Revolution
  • 5. Solids with regular-polygon cross-sections
  • 6. Circles between Curves
  • 7. Equilateral Triangles based between Curves
  • 8. Isosceles Triangles between Curves
  • 9. Squares with diagonal bounded by Curves
  • 10. Revolving 2 exponentials
  • 11. Revolving exponential and polynomial
  • 12. Stepping through Volume-by-Shells method
  • 13. Arc Length for y(x)

5.1.

Washer Method

This is an example of a volume of revolution by the washer method - i.e. there is hollow space in the middle that must be subtracted away.
– Lenore Horner

5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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5.10.

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5.11.

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5.12.

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5.13.

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6.

Series

collection of my applets on series concepts from calculus

  • 1. Linearization improvement leads to Maclaurin series
  • 2. Substituting new arguments into Taylor Series
  • 3. Zeno
  • 4. Conditional Convergence

6.1.

Linearization improvement leads to Maclaurin series

Extending linearization to the Maclaurin series.
Linearization improvement leads to Maclaurin series
– Lenore Horner

6.2.

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6.3.

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6.4.

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7.

Fourier Series

Illustrations of fourier series and their derivation.

  • 1. Fourier Example
  • 2. Fourier Products
  • 3. Fourier Series
  • 4. Fourier Products

7.1.

Fourier Example

Fourier Example
– Lenore Horner

7.2.

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7.3.

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7.4.

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8.

misc

  • 1. Hyperbolic Functions

8.1.

Hyperbolic Functions

For the moment, just work with the displayed function and the four sliders in the bottom left corner. How do the fours sliders affect the graph?[br][br]Since the circle isn't a function, it has been graphed parametrically. Which trig function goes with the $ x $-coordinate of a given point and which one goes with the $ y $-coordinate? [br][br]Now click on the empty circle by the second parametric equation to display it. How do the sliders affect this shape? [br][br]What names does this graph suggest for our two new functions?
– Lenore Horner

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