IB Mathematics HL

Sandi Yoder, Jun 9, 2016

Table of Contents

  1. Limits and Continuity
    1. Important Trig Limits
    2. One Special Limit
    3. IVT Dynamic Illustrator
    4. Continuity Questions
    5. Vertical Asymptote (Defined as a Limit)
    6. Horizontal Asymptote (Defined as a Limit)
  2. Derivatives - definitions and visualizations
    1. Definition of a Derivative
    2. Average Velocity
    3. Average to Instantaneous (Intro)
    4. Derivatives (Intro)-1
    5. Derivatives (Intro)-2
    6. Workshop #3: Introducing Derivatives - The Slope Function
    7. Derivative of Sine Function (Intuitive Discovery)
    8. Derivative of Cosine Function (Intuitive Discovery)
    9. Sketch the Derivative (Warm Up)
    10. 3-Way Color-Changing Derivative Grapher
    11. Graphing The Derivative of a Function
  3. Techniques of differentiation
    1. Derivative of a Product
    2. Implicit Differentiation: Intro
    3. Derivatives of Inverse Functions
    4. Derivative of Arcsine Function
    5. Derivative of Arccosine Function
    6. Derivative of Arctangent Function
    7. Derivative of Natural Log Function
  4. Related Rates and Optimization
    1. Related Rates (1)
    2. Related Rates (2)
    3. Optimization Challenge 1
    4. Falling Ladder !!!
    5. Related Rates (Rolling Carts Problem)
    6. Related Rates (3)
    7. Oil Pipeline Optimization Problem
    8. Optimization: Maximizing Area of a Rectangle Inside a Right Triangle
    9. Cylinder Inscribed In Cone
  5. Integral Calculus
    1. Definite Integral Illustrator (I)
    2. Riemann Sums - Rectangular Approximation (LRAM, RRAM, MRAM)
    3. fundamental theorem of calculus
    4. The Fundamental Theorem of Calculus
    5. Definite Integrals: Sum
    6. Integral as area
    7. Rules of the Definite Integral
    8. Areas Between Curves (Intro)
    9. Area Between Curves
    10. Mean Value Theorem
    11. U-Substitution (Definite Integrals)
    12. Solids of Revolution (Discs)
    13. Parabola Around y-Axis
    14. Parabola Rotated Around y-Axis Wireframe
    15. Parabola Rotated Around x-Axis
    16. Solids Of Revolution: Disk Method
    17. Riemann Sums
  6. Calculus Option
    1. Mean Value Theorem (Without Words)
  7. Series
    1. Taylor Polynomials
    2. Taylor Polynomial of f(x) centered at point a
    3. Convergent or Divergent Series
    4. Converging series
    5. Integral Test for Series Convergence
    6. Taylor Series
    7. Properties of Power Series
    8. Alternating Series
    9. Partial Sums and Series
    10. The Divergence and Integral Tests
    11. The Taylor and Maclaurin series.
    12. Power Series Representations of Functions
    13. Maclaurin series
  8. Slope Fields
    1. Slope field plotter
    2. Slope Field
    3. Slope Fields.
    4. Slope field
    5. Slope Field "window"
    6. Slope Field Generator

1.

Limits and Continuity

  • 1. Important Trig Limits
  • 2. One Special Limit
  • 3. IVT Dynamic Illustrator
  • 4. Continuity Questions
  • 5. Vertical Asymptote (Defined as a Limit)
  • 6. Horizontal Asymptote (Defined as a Limit)

1.1.

Important Trig Limits

Use the results of your obseravations here in this worksheet to complete the questions found in the [url=https://www.geogebra.org/m/fVSkvdcu]Derivative of Sine & Cosine Functions worksheet[/url]. [br][br](In both graphs, what happens to the [b][color=#38761d]BIG GREEN POINT[/color][/b] if you let its x-coordinate become zero?)
– Tim Brzezinski

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Derivatives - definitions and visualizations

[color=#1551b5][/color]

  • 1. Definition of a Derivative
  • 2. Average Velocity
  • 3. Average to Instantaneous (Intro)
  • 4. Derivatives (Intro)-1
  • 5. Derivatives (Intro)-2
  • 6. Workshop #3: Introducing Derivatives - The Slope Function
  • 7. Derivative of Sine Function (Intuitive Discovery)
  • 8. Derivative of Cosine Function (Intuitive Discovery)
  • 9. Sketch the Derivative (Warm Up)
  • 10. 3-Way Color-Changing Derivative Grapher
  • 11. Graphing The Derivative of a Function

2.1.

Definition of a Derivative

This page demonstration where the definition of a derivative comes from. Notice the variable "h" is the space between the two points (horizontally). As h gets closer and closer to zero the slope between the two points (x,f(x)) and (x+h,f(x+h)) becomes closer and closer to the slope of the tangent line at x. This is where the limit as h approaches zero comes from.
Definition of a Derivative
– Phillip Clontz

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2.7.

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2.9.

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2.10.

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2.11.

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3.

Techniques of differentiation

  • 1. Derivative of a Product
  • 2. Implicit Differentiation: Intro
  • 3. Derivatives of Inverse Functions
  • 4. Derivative of Arcsine Function
  • 5. Derivative of Arccosine Function
  • 6. Derivative of Arctangent Function
  • 7. Derivative of Natural Log Function

3.1.

Derivative of a Product

Provided the described limits exist, we've already discovered that [br][br]1) The derivative of a sum of two functions = the sum of the derivatives of these functions. [br]2) The derivative of a difference of two functions = the difference of the derivatives of these functions.[br][br]Yet what about replacing the word "sum" or "difference" with the word "product"? [br][color=#ff00ff][b]That is, is the derivative of the product of two functions evaluated at any input = the product of the derivatives of these two functions (evaluated at this same input)? [/b][/color][br][br]Interact with the applet below for a minute. Study it carefully. [br]Move any 1 of the white points around if you need to. [br][br][color=#ff00ff][b]Then answer the question in pink.[/b][/color]
– Tim Brzezinski

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4.

Related Rates and Optimization

Applications for differentiation

  • 1. Related Rates (1)
  • 2. Related Rates (2)
  • 3. Optimization Challenge 1
  • 4. Falling Ladder !!!
  • 5. Related Rates (Rolling Carts Problem)
  • 6. Related Rates (3)
  • 7. Oil Pipeline Optimization Problem
  • 8. Optimization: Maximizing Area of a Rectangle Inside a Right Triangle
  • 9. Cylinder Inscribed In Cone

4.1.

Related Rates (1)

Suppose [color=#0000ff][b]particle A [/b][/color]and [color=#cc0000][b]particle B[/b][/color] start at the same spot. [br][br]Suppose [color=#0000ff][b]particle A[/b][/color] [color=#0000ff]starts to head east at a speed of 10 m/s.[/color] [br]At the same time [b][color=#0000ff]A[/color][/b] leaves its initial position, [color=#cc0000][b]particle B[/b][/color] [color=#cc0000]starts heading north at a speed of 5 m/s[/color]. [br][br]Play to animate, then answer the questions that follow. [br]([color=#ff00ff][b]Please don't click the checkbox yet. Do so when prompted to.) [/b][/color]
1.
At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 3 sec?
2.
At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 5 sec?
3.
At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 6 sec?
– Tim Brzezinski

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4.7.

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5.

Integral Calculus

  • 1. Definite Integral Illustrator (I)
  • 2. Riemann Sums - Rectangular Approximation (LRAM, RRAM, MRAM)
  • 3. fundamental theorem of calculus
  • 4. The Fundamental Theorem of Calculus
  • 5. Definite Integrals: Sum
  • 6. Integral as area
  • 7. Rules of the Definite Integral
  • 8. Areas Between Curves (Intro)
  • 9. Area Between Curves
  • 10. Mean Value Theorem
  • 11. U-Substitution (Definite Integrals)
  • 12. Solids of Revolution (Discs)
  • 13. Parabola Around y-Axis
  • 14. Parabola Rotated Around y-Axis Wireframe
  • 15. Parabola Rotated Around x-Axis
  • 16. Solids Of Revolution: Disk Method
  • 17. Riemann Sums

5.1.

Definite Integral Illustrator (I)

[b]Note: [br][/b][color=#0000ff]This applet only generates nonnegative areas. [br]Thus, incorrect values will be displayed for any areas underneath the [i]x[/i]-axis.[br][br][/color]For an applet that also includes [b][color=#ff7700]trapezoidal approximations[/color][/b], go to [url=https://www.geogebra.org/m/yUpHvBYc]Definite Integral Illustrator (II)[/url].
– Tim Brzezinski

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5.11.

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5.12.

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6.

Calculus Option

Topics from Option 9 IB Mathematics HL

  • 1. Mean Value Theorem (Without Words)

6.1.

Mean Value Theorem (Without Words)

Is there a function [i]f[/i] you can input for which the Mean Value Theorem would NOT apply for a specified interval [a,b] within its domain? If so, provide such an example and clearly illustrate why the MVT would not apply in this situation.
– Tim Brzezinski

7.

Series

  • 1. Taylor Polynomials
  • 2. Taylor Polynomial of f(x) centered at point a
  • 3. Convergent or Divergent Series
  • 4. Converging series
  • 5. Integral Test for Series Convergence
  • 6. Taylor Series
  • 7. Properties of Power Series
  • 8. Alternating Series
  • 9. Partial Sums and Series
  • 10. The Divergence and Integral Tests
  • 11. The Taylor and Maclaurin series.
  • 12. Power Series Representations of Functions
  • 13. Maclaurin series

7.1.

Taylor Polynomials

[color=#000000]Reference applet for [/color][color=#9900ff]Taylor Polynomials[/color][color=#000000] and [/color][color=#9900ff]Maclaurin Polynomials [/color][color=#000000](n = 0 to n = 40) centered at [i]x[/i] = [i]a[/i]. [/color][color=#9900ff][br][/color][color=#000000][i]Recall a Maclaurin Series is simply a Taylor Series centered at a = 0. [/i][/color]
– Tim Brzezinski

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7.7.

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7.8.

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7.9.

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7.10.

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7.11.

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7.12.

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7.13.

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8.

Slope Fields

  • 1. Slope field plotter
  • 2. Slope Field
  • 3. Slope Fields.
  • 4. Slope field
  • 5. Slope Field "window"
  • 6. Slope Field Generator

8.1.

Slope field plotter

A direction field (or slope field / vector field) is a picture of the general solution to a first order differential equation with the form [math]\frac{dy}{dx} = f(x,y)[/math]
[list][br][*]Edit the gradient function [math]f^{\prime}(x,y)[/math] in the input box at the top. The function you input will be shown in blue underneath as [math]dy/dx=f(x,y)[/math][br][*]The Density slider controls the number of vector lines.[br][*]The Length slider controls the length of the vector lines.[br][*]Adjust [math]x_{min}, x_{max}, y_{min}[/math] and [math]y_{max}[/math] to define the limits of the slope field.[br][*]Check the Solution boxes to draw curves representing numerical solutions to the differential equation.[br][*]Click and drag the points A, B, C and D to see how the solution changes across the field.[br][*]Change the Step size to improve or reduce the accuracy of solutions (0.1 is usually fine but 0.01 is better). [br]If anything messes up....hit the reset button to restore things to default.[br][/list]
– Dr Adrian Jannetta

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Information