Class-12 Math Lab Activity

PRAVEEN KUMAR SAHU, Jan 13, 2021

Math Lab Activity 12 CBSE, Prepared By Praveen Sir.

Table of Contents

  1. 1. To demonstrate a function which is not one-one but is onto.
    1. Horizontal line test
    2. Function: many-to-one
    3. Is y = cos(x) one-to-one?
    4. Is y = sin(x) one-to-one?
  2. 2. To explore the principal value of the function sin^-1x using a unit circle.
    1. Restricting the Range of arcsine and arccosine
    2. The Unit Circle
  3. 3. To understand the concepts of decreasing and increasing functions.
    1. Increasing/Decreasing function
  4. 4. To understand the concepts of local maxima, local minima and point of inflection.
    1. Maxima and minima
  5. 5. To construct an open box of maximum volume from a given rectangular sheet by cutting equal squares from each corner.
    1. Maximizing Trapezoid Area
    2. Maximizing Area
  6. 6. To verify geometrically that c×(a +b)=c×a +c×b
    1. Cross-product
    2. Geometry of Cross Product
    3. Cross Product Magnitude as Area of Parallelogram
    4. Cross Product and Area Visualization
  7. 7. To demonstrate the equation of a plane in normal form.
    1. Geometric Interpretation of the Cross Product
    2. Cross Product: Introduction
    3. Cross product and direction
  8. 8. To measure the shortest distance between two skew lines and verify it analytically.
    1. Skew lines: finding the distance (dot product)
  9. 9. To explain the computation of conditional probability of a given event A, when event B has already occurred, through an example of throwing a pair of dice.
    1. Conditional probability
    2. Probability: Selecting Balls
    3. conditional probability interation 4
    4. Conditional probability

1.

1. To demonstrate a function which is not one-one but is onto.

  • 1. Horizontal line test
  • 2. Function: many-to-one
  • 3. Is y = cos(x) one-to-one?
  • 4. Is y = sin(x) one-to-one?

1.1.

Horizontal line test

– Taeil Yi

1.2.

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1.3.

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1.4.

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2.

2. To explore the principal value of the function sin^-1x using a unit circle.

  • 1. Restricting the Range of arcsine and arccosine
  • 2. The Unit Circle

2.1.

Restricting the Range of arcsine and arccosine

Why do we need to restrict the range of the inverse trig functions?[br][br]Move the sliders to demonstrate that there are two angles that yield the same value of sine, or the same value of cosine. Check "sine" or "cosine" to illustrate each separately.
Restricting the Range of arcsine and arccosine
– Ken Schwartz

2.2.

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3.

3. To understand the concepts of decreasing and increasing functions.

  • 1. Increasing/Decreasing function

3.1.

Increasing/Decreasing function

– Juan Carlos Ponce Campuzano

4.

4. To understand the concepts of local maxima, local minima and point of inflection.

  • 1. Maxima and minima

4.1.

Maxima and minima

We can clearly see the blue graph, it is the locus of a 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 (pink line) is drawn at this point. We can move the point A. m is the slope of the tangent
Maxima and minima
Question/s to think about.[br][br]1. What is the case when m=0?[br]2. What is the value of m when A is at the local maxima or local minima of the function?[br]3. Set the values of a, b, c and d as a=0.6, b=1, c=0.5 and d=2. What do we observe?
– Mathguru

5.

5. To construct an open box of maximum volume from a given rectangular sheet by cutting equal squares from each corner.

  • 1. Maximizing Trapezoid Area
  • 2. Maximizing Area

5.1.

Maximizing Trapezoid Area

[color=#000000]Suppose both legs and one base of the isosceles trapezoid below each have length 1. [br](Drag black slider.) [br][br]If this is so, determine the value of [/color][math]\theta[/math][color=#000000] that maximizes the area of this isosceles trapezoid.[br]Be sure to solve this problem first! How do your results compare with what this applet suggests?[/color]
– Tim Brzezinski

5.2.

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6.

6. To verify geometrically that c×(a +b)=c×a +c×b

  • 1. Cross-product
  • 2. Geometry of Cross Product
  • 3. Cross Product Magnitude as Area of Parallelogram
  • 4. Cross Product and Area Visualization

6.1.

Cross-product

illustration of computing the cross product from vector components
Cross-product
– Lenore Horner

6.2.

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6.3.

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6.4.

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7.

7. To demonstrate the equation of a plane in normal form.

  • 1. Geometric Interpretation of the Cross Product
  • 2. Cross Product: Introduction
  • 3. Cross product and direction

7.1.

Geometric Interpretation of the Cross Product

This worksheet illustrates the geometric significance of the cross product. Move the yellow points to adjust the vectors.
What does the vector along the [math]z[/math]-axis represent?
Note that [math]\vec{u}[/math] and [math]\vec{v}[/math] are confined to the [math]xy[/math]-plane in this worksheet. How can we be sure that [math]\vec{u}\times\vec{v}[/math] will always point along the [math]z[/math]-axis?
What does the equality of the two numerical values signify?
Adjust [math]\vec{u}[/math] and [math]\vec{v}[/math] until their cross product points downward. Now, how big is the [i]smallest positive[/i] angle from [math]\vec{u}[/math] to [math]\vec{v}[/math]?
– Kristen Beck

7.2.

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7.3.

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8.

8. To measure the shortest distance between two skew lines and verify it analytically.

  • 1. Skew lines: finding the distance (dot product)

8.1.

Skew lines: finding the distance (dot product)

Use the sliders to find the values of [math]\lambda[/math] and [math]\mu[/math] such that the vector PQ is perpendicular to both lines.
– GeoGebra Institute of MEI

9.

9. To explain the computation of conditional probability of a given event A, when event B has already occurred, through an example of throwing a pair of dice.

  • 1. Conditional probability
  • 2. Probability: Selecting Balls
  • 3. conditional probability interation 4
  • 4. Conditional probability

9.1.

Conditional probability

– Steve Phelps, loreang98

9.2.

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9.3.

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9.4.

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