Teaching and Learning Calculus using GeoGebra

Dr.Abraham Varghese, Nov 8, 2017

[color=#b20ea8][b][i]"The mediocre teacher tells The good teacher explains The great teacher inspires The superior teacher demonstrates"[/b][/color][/i]

Table of Contents

  1. Why Calculus?
    1. Why calculus
    2. slope
    3. Slope of perpendicular
  2. Limit and Continuity
    1. trigocircular
    2. sequences
    3. Limit
    4. Limit Laws
    5. Continuity
    6. Limit at Infinity
    7. Asymptote
    8. Limit Quiz
  3. Differentiation
    1. derivative
    2. Derivative of a inverse function
  4. Application of Derivatives
    1. increasing-decreasing functions
    2. Tangents and Normals
    3. Maximum and Minimum of a function
    4. Extrema Quiz
  5. Integration
    1. Integral
    2. Indefinite Integral- Quiz

1.

Why Calculus?

  • 1. Why calculus
  • 2. slope
  • 3. Slope of perpendicular

1.1.

Why calculus

Calculus is the study of how functions [i]change[/i]. In Physics, you learned that an object whose position is changing is moving, and therefore has velocity (speed with direction). If it moves farther in a given amount of time, it is going faster than if it moves less in the same amount of time. Conversely, we know that if an object is moving at a certain speed for a given amount of time, we can calculate the change in its position (distance traveled). So not only can we relate [i]change in position[/i] to [math]speed[/math], we can reverse the process and find out about [math]position[/math] when we know [math]speed[/math]. Calculus ties these two things together and gives us tools to analyze them in both "directions
The [color=#c51414][b]red[/b][/color] graph represents the object's position during the time interval from 0 to 10 seconds. The [b][color=#1551b5]blue[/color][/b] graph represents the velocity of the object. Remember that "velocity" is the same as the [i]rate of change[/i] of the position: an object moving at a [i]speed[/i] of 3 m/s is [i]changing its position[/i] at a [i]rate[/i] of 3 m every second
First, note how the slope of the position graph always matches the height of the velocity graph. Then notice how the area under the velocity graph always matches the position of the ball (the ball's initial position is zero). When the ball rises,both its position and its velocity are positive. When the ball falls, its position is still positive (still above ground), but its velocity (slope of position) is negative (since it is moving down).
– Dr.Abraham Varghese

1.2.

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1.3.

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2.

Limit and Continuity

Let [math]f(x)[/math] be a function of [math]x[/math]. If the values of [math]f(x)[/math] can be made as close as possible to L by making [math]x[/math] close enough to [math]a[/math]. It is denoted by [math]lim_{x\rightarrow a}f(x))= L [/math]

  • 1. trigocircular
  • 2. sequences
  • 3. Limit
  • 4. Limit Laws
  • 5. Continuity
  • 6. Limit at Infinity
  • 7. Asymptote
  • 8. Limit Quiz

2.1.

trigocircular

– Dr.Abraham Varghese

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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3.

Differentiation

  • 1. derivative
  • 2. Derivative of a inverse function

3.1.

derivative

– Dr.Abraham Varghese

3.2.

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4.

Application of Derivatives

A function is said to be increasing in the region where the value of the function(y) increases as we increase the value of x. A function is said to be decreasing in the region where the value of the function(y) decreases as we increase the value of x. We can clearly see the blue graph, it is that of the locus of the function f(x)=y= ax^3+bx^2+cx+d . We can change the values of a, b, c and d using their respective slide bars. Point A lies on the function f(x) and a tangent is drawn at this point. We can see that an angle α is also shown between the positive x axis and this tangent. We know that tan⁡α is positive when α<90˚ and tan⁡α is negative when α>90˚. α is the angle between the x axis and the tangent at point A so tan⁡α gives the slope of the tangent. We also know that the value of df(x)/dx at the point A is equal to the slope of the tangent at that point. Let us change the values of a, b, c and d and move the point A and now observe how the angle α gets changed.[/b]

  • 1. increasing-decreasing functions
  • 2. Tangents and Normals
  • 3. Maximum and Minimum of a function
  • 4. Extrema Quiz

4.1.

increasing-decreasing functions

– Dr.Abraham Varghese

4.2.

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4.3.

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4.4.

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5.

Integration

  • 1. Integral
  • 2. Indefinite Integral- Quiz

5.1.

Integral

– Dr.Abraham Varghese

5.2.

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