applets for E15

Judah L Schwartz, Mar 22, 2015

This collection of applets were written for incorporation into a new version of E15 - Introduction to the Calculus A, a course in the Mathematics for Teaching program at the Extension School of Harvard University.

Table of Contents

  1. week 1
    1. Graphs of Exponential Functions
    2. A Compound Interest Problem
    3. The Effect of k on the Graph of f(x)+k
    4. The Effect of k on the Graph of kf(x)
    5. The Effect of k on the Graph of f(x+k)
    6. The Effect of k on the Graph of f(kx)
    7. How are the graphs of F & Finverse related?
    8. CHALLENGE - tools for learning about functions as objects
    9. Families of Exponential Functions
    10. CHALLENGE -How are the graphs of f(x) & its inverse related?
  2. week 2
    1. Exponential functions & their inverses
    2. power functions and limits
    3. transformations of trigonometric functions
    4. CHALLENGE - the sine & cosine challenge
  3. week 3
    1. Exploring Average Rate of Change Numerically & Graphically
    2. Exploring the slope of two functions
    3. The Derivative Function
    4. CHALLENGE - an imposter function
  4. week 4
    1. Sketching f' given f
    2. Sketching f given f'
    3. projectile motion
    4. CHALLENGE - The Princess and the Key
  5. week 5
    1. Exploring (2^h-1)/h as h approaches 0
    2. Exploring (b^h-1)/h as h approaches 0
    3. If f(x) = b^x, for what value of b is f=f' ?
    4. The Constant Multiple Rule
    5. CHALLENGE - How can f(x) = f'(x) ?
  6. week 6
    1. A curious property of the log function
    2. Why radians?
    3. Why do we use radian measure in calculus?
    4. Discovering the Chain Rule
    5. Discovering the Chain Rule - a more general tool
    6. CHALLENGE - a 3 dimensional protractor ?
  7. week 7
    1. The Folium of Descartes
    2. The Lemniscate of Bernoulli
    3. Local linearization - I
    4. Local linearization - II
    5. indeterminate form 0/0
    6. Indeterminate form inf/inf
    7. Tautochrone
    8. Rational F'ns & Asymptotic Behavior
  8. week 8
    1. The Family of Curves ax^3-bx^2
    2. The Family of Curves f(x) =x^m e^(px)
    3. The most room for your animals
    4. The most efficient can of vegetables
    5. The optimal cone -
    6. Calculus Saves Lives
    7. Calculus Saves Even More Lives
  9. week 9
    1. Exploring Riemann Sums
    2. Looking at RHS(n) - LHS(n)
    3. The Area Function - Part I
    4. The Area Function - Part II
    5. Exploring Riemann Sums - Part II
  10. week 10
    1. A Graphical Look at u-substitution
    2. Areas of Regions of Infinite Length
    3. Area Between Two Curves
    4. A Graphical Look at Average Value
    5. Integrating g'f(g)
  11. week 11
    1. Sketching Antiderivatives
    2. A Graphical Look at Our Game
    3. The Hat Check Problem
    4. A Leaky Bathtub - competing rates & equilibrium
    5. Euler's Formula
    6. Retirement & leaky bathtubs; 2 metaphors - 1model

1.

week 1

  • 1. Graphs of Exponential Functions
  • 2. A Compound Interest Problem
  • 3. The Effect of k on the Graph of f(x)+k
  • 4. The Effect of k on the Graph of kf(x)
  • 5. The Effect of k on the Graph of f(x+k)
  • 6. The Effect of k on the Graph of f(kx)
  • 7. How are the graphs of F & Finverse related?
  • 8. CHALLENGE - tools for learning about functions as objects
  • 9. Families of Exponential Functions
  • 10. CHALLENGE -How are the graphs of f(x) & its inverse related?

1.1.

–

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

2.

week 2

  • 1. Exponential functions & their inverses
  • 2. power functions and limits
  • 3. transformations of trigonometric functions
  • 4. CHALLENGE - the sine & cosine challenge

2.1.

–

2.2.

–

2.3.

–

2.4.

–

3.

week 3

  • 1. Exploring Average Rate of Change Numerically & Graphically
  • 2. Exploring the slope of two functions
  • 3. The Derivative Function
  • 4. CHALLENGE - an imposter function

3.1.

–

3.2.

–

3.3.

–

3.4.

–

4.

week 4

  • 1. Sketching f' given f
  • 2. Sketching f given f'
  • 3. projectile motion
  • 4. CHALLENGE - The Princess and the Key

4.1.

–

4.2.

–

4.3.

–

4.4.

–

5.

week 5

  • 1. Exploring (2^h-1)/h as h approaches 0
  • 2. Exploring (b^h-1)/h as h approaches 0
  • 3. If f(x) = b^x, for what value of b is f=f' ?
  • 4. The Constant Multiple Rule
  • 5. CHALLENGE - How can f(x) = f'(x) ?

5.1.

–

5.2.

–

5.3.

–

5.4.

–

5.5.

–

6.

week 6

  • 1. A curious property of the log function
  • 2. Why radians?
  • 3. Why do we use radian measure in calculus?
  • 4. Discovering the Chain Rule
  • 5. Discovering the Chain Rule - a more general tool
  • 6. CHALLENGE - a 3 dimensional protractor ?

6.1.

–

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

7.

week 7

  • 1. The Folium of Descartes
  • 2. The Lemniscate of Bernoulli
  • 3. Local linearization - I
  • 4. Local linearization - II
  • 5. indeterminate form 0/0
  • 6. Indeterminate form inf/inf
  • 7. Tautochrone
  • 8. Rational F'ns & Asymptotic Behavior

7.1.

–

7.2.

–

7.3.

–

7.4.

–

7.5.

–

7.6.

–

7.7.

–

7.8.

–

8.

week 8

  • 1. The Family of Curves ax^3-bx^2
  • 2. The Family of Curves f(x) =x^m e^(px)
  • 3. The most room for your animals
  • 4. The most efficient can of vegetables
  • 5. The optimal cone -
  • 6. Calculus Saves Lives
  • 7. Calculus Saves Even More Lives

8.1.

–

8.2.

–

8.3.

–

8.4.

–

8.5.

–

8.6.

–

8.7.

–

9.

week 9

  • 1. Exploring Riemann Sums
  • 2. Looking at RHS(n) - LHS(n)
  • 3. The Area Function - Part I
  • 4. The Area Function - Part II
  • 5. Exploring Riemann Sums - Part II

9.1.

–

9.2.

–

9.3.

–

9.4.

–

9.5.

–

10.

week 10

  • 1. A Graphical Look at u-substitution
  • 2. Areas of Regions of Infinite Length
  • 3. Area Between Two Curves
  • 4. A Graphical Look at Average Value
  • 5. Integrating g'f(g)

10.1.

–

10.2.

–

10.3.

–

10.4.

–

10.5.

–

11.

week 11

  • 1. Sketching Antiderivatives
  • 2. A Graphical Look at Our Game
  • 3. The Hat Check Problem
  • 4. A Leaky Bathtub - competing rates & equilibrium
  • 5. Euler's Formula
  • 6. Retirement & leaky bathtubs; 2 metaphors - 1model

11.1.

–

11.2.

–

11.3.

–

11.4.

–

11.5.

–

11.6.

–

Information