Calculus 1 Graphical Illustrations
1. Limits and Derivatives
1. Section 2.1 - The Tangent Line
2. Section 2.1: Approximating the Slope of a Tangent Line
3. Section 2.2 - Intuitive Definition of Limits of a Function
4. Section 2.2: Intuitive Definition of Limits of a Function
5. Section 2.2: Limit Does Not Exist
6. Section 2.2 Infinite Limits
7. Section 2.3: One-sided Limits
8. Section 2.6: Vertical Asymptotes
9. Section 2.5: Intermediate Value Theorem
10. Section 2.6: Horizontal Asymptote
11. Section 2.6 Horizontal and Vertical Asymptote
12. Section 2.7: Differentiability
13. Section 2.8: Derivative as a Function
14. Section 2.8: Higher Derivatives
2. Differentiation Rules
1. Section 3.2: Tangent Line to a Curve (Quotient Rule)
2. Section 3.3 - Trig Limits - Squeeze Theorem
3. Section 3.3: Derivative of Sine Function
4. Section 3.3: Derivative Application
5. Section 3.4: Motion of a Spring
6. Section 3.5: Implicit Differentiation - Descartes' Leaf
7. Section 3.5: Implicit Differentiation (Cramer, 1750)
8. Section 3.5: Inverse Tangent Function
9. Section 3.10: Linear Approximation
10. Section 3.11 Hyperbolic Functions
11. St Louis Gateway Arch
3. Derivative Applications
1. C4.1 Absolute Maximum and Absolute Minimum
2. C4.1 Absolute Maximum and Absolute Minimum Ex 2
3. Section 4.2: The Mean Value Theorem Illustrated
4. Integrals
1. Upper and lower Riemann Sums

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Calculus 1 Graphical Illustrations

Junvie Pailden, Jan 15, 2015

The curve [math]x^3+y^3=6xy[/math] is called the "folium of Descartes" ("folium" is Latin for "leaf"). This equation first came out in a letter from Descartes to Fermat (Fermat was a lawyer by profession, but did mathematics as a hobby). Fermat claimed to have a method to find tangent lines to any curve, and Descartes invented this curve as a challenge to Fermat. Fermat was successful in finding tangent lines to the curve. Today, however, the problem of finding tangent lines is straightforward, due to the invention of calculus sometime later. This event was notable enough in the history of calculus that it is memorialized in the Albanian postage stamp above. Source: http://goo.gl/JhLTgE

Limits and Derivatives
1. Section 2.1 - The Tangent Line
2. Section 2.1: Approximating the Slope of a Tangent Line
3. Section 2.2 - Intuitive Definition of Limits of a Function
4. Section 2.2: Intuitive Definition of Limits of a Function
5. Section 2.2: Limit Does Not Exist
6. Section 2.2 Infinite Limits
7. Section 2.3: One-sided Limits
8. Section 2.6: Vertical Asymptotes
9. Section 2.5: Intermediate Value Theorem
10. Section 2.6: Horizontal Asymptote
11. Section 2.6 Horizontal and Vertical Asymptote
12. Section 2.7: Differentiability
13. Section 2.8: Derivative as a Function
14. Section 2.8: Higher Derivatives
Differentiation Rules
1. Section 3.2: Tangent Line to a Curve (Quotient Rule)
2. Section 3.3 - Trig Limits - Squeeze Theorem
3. Section 3.3: Derivative of Sine Function
4. Section 3.3: Derivative Application
5. Section 3.4: Motion of a Spring
6. Section 3.5: Implicit Differentiation - Descartes' Leaf
7. Section 3.5: Implicit Differentiation (Cramer, 1750)
8. Section 3.5: Inverse Tangent Function
9. Section 3.10: Linear Approximation
10. Section 3.11 Hyperbolic Functions
11. St Louis Gateway Arch
Derivative Applications
1. C4.1 Absolute Maximum and Absolute Minimum
2. C4.1 Absolute Maximum and Absolute Minimum Ex 2
3. Section 4.2: The Mean Value Theorem Illustrated
Integrals
1. Upper and lower Riemann Sums

1.

Limits and Derivatives

This chapter contains intuitive notions of limits and derivatives.

1. Section 2.1 - The Tangent Line
2. Section 2.1: Approximating the Slope of a Tangent Line
3. Section 2.2 - Intuitive Definition of Limits of a Function
4. Section 2.2: Intuitive Definition of Limits of a Function
5. Section 2.2: Limit Does Not Exist
6. Section 2.2 Infinite Limits
7. Section 2.3: One-sided Limits
8. Section 2.6: Vertical Asymptotes
9. Section 2.5: Intermediate Value Theorem
10. Section 2.6: Horizontal Asymptote
11. Section 2.6 Horizontal and Vertical Asymptote
12. Section 2.7: Differentiability
13. Section 2.8: Derivative as a Function
14. Section 2.8: Higher Derivatives

1.1.

Section 2.1 - The Tangent Line

1.2.

1.3.

1.4.

1.5.

1.6.

1.7.

1.8.

1.9.

1.10.

1.11.

1.12.

1.13.

1.14.

2.

Differentiation Rules

1. Section 3.2: Tangent Line to a Curve (Quotient Rule)
2. Section 3.3 - Trig Limits - Squeeze Theorem
3. Section 3.3: Derivative of Sine Function
4. Section 3.3: Derivative Application
5. Section 3.4: Motion of a Spring
6. Section 3.5: Implicit Differentiation - Descartes' Leaf
7. Section 3.5: Implicit Differentiation (Cramer, 1750)
8. Section 3.5: Inverse Tangent Function
9. Section 3.10: Linear Approximation
10. Section 3.11 Hyperbolic Functions
11. St Louis Gateway Arch

2.1.

Section 3.2: Tangent Line to a Curve (Quotient Rule)

Drag the red point P along the curve [math]f(x)=\frac{1}{1+x^2}[/math] and notice how the tangent line changes. Note the relationship between the slope of the tangent line and the curve of the derivative function.

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

2.10.

2.11.

3.

Derivative Applications

1. C4.1 Absolute Maximum and Absolute Minimum
2. C4.1 Absolute Maximum and Absolute Minimum Ex 2
3. Section 4.2: The Mean Value Theorem Illustrated

3.1.

C4.1 Absolute Maximum and Absolute Minimum

Extreme Value Theorem: If function [math]f[/math] is continuous on a closed interval [math][a,b][/math], then [math]f[/math] attains absolute maximum value [math]f(c)[/math] and an absolute minimum value [math]f(d)[/math] at some numbers [math]c[/math] and [math]d[/math] in [math][a,b][/math].

3.2.

3.3.

4.

Integrals

1. Upper and lower Riemann Sums

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.

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