MCV4U Companion Activities

Kara Babcock, Apr 16, 2017

This is a collection of activities that are used in an online version of Grade 12 Calculus and Vectors (MCV4U), originally designed at Lakehead Adult Education Centre.

Table of Contents

  1. Vectors
    1. Vector Addition with Parallelograms
    2. Vector Addition with Triangles
    3. Components of a Vector
    4. Vectors in 3D Basics
    5. Dot Product Visualization
    6. Vector Projection Visualization
    7. Cross Product and Area Visualization
    8. Cross Product Magnitude as Area of Parallelogram
  2. Lines and Planes in 3-space
    1. Example of a Plane through 3 Points
    2. Coincident Planes
    3. Parallel, Distinct Planes
    4. Two Planes Intersecting in a Line
    5. 3 Coincident Planes
    6. 2 Coincident Planes & 1 Parallel Plane
    7. 3 Parallel Planes
    8. 3 Planes Intersecting Along Distinct Lines
    9. 3 Planes Intersecting Along 1 Line
    10. 3 Planes Intersecting at a Point
    11. Line Lying within a Plane
    12. Line Parallel to a Plane
    13. Line Intersecting a Plane at a Point
    14. Coincident and Parallel Lines
    15. Lines Intersecting at a Point
    16. Skew Lines
  3. Rates of Change
    1. Slope of a Line
    2. Slope of Secant versus Tangent
    3. Tangents at a point
    4. Tangent to Exponential Functions
    5. Calculus - Derivative of exponential functions f(x)=b^x
  4. Applications
    1. Effect of Translations on the Derivative
    2. Effect of Reflections on the Derivative
    3. Effect of Stretches on the Derivative
    4. Minimizing Trail Length Problem

1.

Vectors

Basics of vectors in 2 and 3 dimensions.

  • 1. Vector Addition with Parallelograms
  • 2. Vector Addition with Triangles
  • 3. Components of a Vector
  • 4. Vectors in 3D Basics
  • 5. Dot Product Visualization
  • 6. Vector Projection Visualization
  • 7. Cross Product and Area Visualization
  • 8. Cross Product Magnitude as Area of Parallelogram

1.1.

Vector Addition with Parallelograms

Drag vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] to see how the resultant [math]\vec{r}[/math] changes to visualize the Parallelogram Law for vector addition.
– Kara Babcock

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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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1.7.

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1.8.

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2.

Lines and Planes in 3-space

We investigate the various possibilities for the interactions between and among lines and planes in 3-dimensional space.

  • 1. Example of a Plane through 3 Points
  • 2. Coincident Planes
  • 3. Parallel, Distinct Planes
  • 4. Two Planes Intersecting in a Line
  • 5. 3 Coincident Planes
  • 6. 2 Coincident Planes & 1 Parallel Plane
  • 7. 3 Parallel Planes
  • 8. 3 Planes Intersecting Along Distinct Lines
  • 9. 3 Planes Intersecting Along 1 Line
  • 10. 3 Planes Intersecting at a Point
  • 11. Line Lying within a Plane
  • 12. Line Parallel to a Plane
  • 13. Line Intersecting a Plane at a Point
  • 14. Coincident and Parallel Lines
  • 15. Lines Intersecting at a Point
  • 16. Skew Lines

2.1.

Example of a Plane through 3 Points

Click and drag to rotate. Use the scroll wheel (or pinch and pull your fingers in/out) to zoom in/out.
– Kara Babcock

2.2.

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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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2.12.

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2.13.

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2.14.

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2.15.

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2.16.

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3.

Rates of Change

Moving now into calculus, we look at the problem of finding the rate of change of a function. From there, we investigate the definition of the derivative and how it relates to rate of change.

  • 1. Slope of a Line
  • 2. Slope of Secant versus Tangent
  • 3. Tangents at a point
  • 4. Tangent to Exponential Functions
  • 5. Calculus - Derivative of exponential functions f(x)=b^x

3.1.

Slope of a Line

Slope of a Line
– Irina Boyadzhiev

3.2.

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3.3.

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3.4.

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3.5.

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4.

Applications

Now that we have somewhat of an understanding how to find and use derivatives, let's look in more detail at the applications of the derivative.

  • 1. Effect of Translations on the Derivative
  • 2. Effect of Reflections on the Derivative
  • 3. Effect of Stretches on the Derivative
  • 4. Minimizing Trail Length Problem

4.1.

Effect of Translations on the Derivative

As we translate a function, what happens to its derivative?[br][br]Use the sliders to investigate this below. A quadratic function and its derivative are visible by default. Move slider [i]p[/i] to translate the function horizontally and [i]q[/i] to translate it vertically, then move both sliders at the same time. What do you notice?[br][br]Use the circles next to the function definitions to make the quadratic functions invisible, then reveal the exponential function and its derivative. Repeat your investigation.[br][br]Do this one more time for the sinusoidal function and its derivative.[br][br]What can you conclude?
Which of the following best describes how translations affect the derivative?
– Kara Babcock

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4.4.

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