MA441

Whan Ki Lee, Jan 30, 2018

Table of Contents

  1. Limits
    1. Secant and Tangent Lines
    2. The Limit of sin x / x as x goes to 0
    3. f(x) = sin (pi/x)
    4. Continuous or not at 2?
  2. Derivatives
    1. Secant and Tangent lines
    2. lim [(sin x)/ x] as x ->0
    3. Tangent Lines and Approximations
    4. Linear Approximation Error
  3. Applications of Differentiation
  4. Integrals
    1. Upper and lower Riemann Sums
    2. Riemann Sums
    3. Riemann Sums
    4. Definite Integral
  5. Applications of Integration

1.

Limits

  • 1. Secant and Tangent Lines
  • 2. The Limit of sin x / x as x goes to 0
  • 3. f(x) = sin (pi/x)
  • 4. Continuous or not at 2?

1.1.

Secant and Tangent Lines

– Whan Ki Lee

1.2.

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1.3.

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1.4.

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2.

Derivatives

  • 1. Secant and Tangent lines
  • 2. lim [(sin x)/ x] as x ->0
  • 3. Tangent Lines and Approximations
  • 4. Linear Approximation Error

2.1.

Secant and Tangent lines

Drag B towards A. Notice that the secant line (green line) converges toward the tangent line (blue line). [br]Move A to a different place. Can you move A anywhere so that this doesn't work?
Try it with different functions.[br][br]Here are some suggestions:[br]f(x)=sin(x)[br]f(x)=x^(2/3)[br]f(x)=x^(1/3)[br][br]Pay particular attention to what happens to the secant line when A is at the origin.
– mckaysm

2.2.

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2.3.

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2.4.

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3.

Applications of Differentiation


    This chapter does not contain any resources yet.

4.

Integrals

  • 1. Upper and lower Riemann Sums
  • 2. Riemann Sums
  • 3. Riemann Sums
  • 4. Definite Integral

4.1.

Upper and lower Riemann Sums

This applet shows how upper and lower Riemann sums can approximate an integral [math]\int^b_a f(x) \, \textrm{d}x[/math][br]Further, they show that as the number of strips [math]n[/math] increases, the Riemann sums converge to true value of the definite integral.[br]Input your own function into the textbox and set the limits to different values.
– Dr Adrian Jannetta

4.2.

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4.3.

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4.4.

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5.

Applications of Integration


    This chapter does not contain any resources yet.

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