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AP Calculus

Tim Brzezinski, Theresa Montenarello, Jan 18, 2017

Contains applets that can be used to foster discovery learning, meaningful remediation, differentiation, and assessment within an AP Calculus-AB or Ap Calculus-BC class (Calculus 1 & Calculus 2 at most colleges and universities.)

1.

Preliminary Review Topics

1. Pythagorean Trigonometric Identity (1)
2. Trig ID Movie (II)
3. Trig ID Movie (III)
4. Exponential Functions: Graphs
5. Logarithmic Functions: Graphs
6. Derivatives of Inverse Functions
7. Oil Pipeline Optimization Problem

1.1.

Pythagorean Trigonometric Identity (1)

This applet shows the derivation of one of the most frequently used trigonometric identities. [br][br]How, specifically, does it relate to the Pythagorean Theorem?

1.2.

1.3.

1.4.

1.5.

1.6.

1.7.

2.

Limits & Continuity

1. Limits: Introductory Questions
2. One CRAZY Limit !!!
3. Cont in uity
4. IVT Dynamic Illustrator
5. Continuity Questions
6. Vertical Asymptote (Defined as a Limit)
7. Horizontal Asymptote (Defined as a Limit)
8. One Special Limit

2.1.

Limits: Introductory Questions

The questions you need to answer are contained in the applet below.

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

3.

Derivatives of Special Functions

1. Derivatives (Intro)-2
2. When does f'(x) = f(x)?
3. Important Trig Limits
4. Mean Value Theorem (Without Words)

3.1.

Derivatives (Intro)-2

[color=#0000ff][b]Students:[/b][/color][br][br]Use the limit definition of a derivative to find the derivatives of the functions given to you at the beginning of class. Use this applet ONLY for the sole purpose of checking each result afterwards.[br][br][color=#9900ff][b]Be sure to DRAG THE WHITE POINT around as well! [/b][/color] (This should help you better understand the relationship between the graph of the original function and the graph of its derivative function.)

3.2.

3.3.

3.4.

4.

Derivatives & Differentation Techniques

1. Average Velocity
2. Average to Instantaneous (Intro)
3. Constructing Graphs of Derivative Functions (Intuitively)
4. Derivatives (Intro)-1
5. Derivatives (Intro)-2
6. 3-Way Color-Changing Derivative Grapher
7. Sketch the Derivative (Warm Up)
8. Graphing The Derivative of a Function
9. Derivative of a Product
10. Derivative of Sine & Cosine Functions (Quick Investigation)
11. Derivative of Tangent Function (Investigation)
12. Implicit Differentiation: Intro
13. Folium of Descartes
14. Implicit Differentiation (Quick Check)
15. Derivative of Arcsine Function
16. Derivative of Arccosine Function
17. Derivative of Arctangent Function
18. Derivatives of Inverse Functions
19. Derivative of Natural Log Function

4.1.

Average Velocity

[b][color=#1e84cc]Recall that VELOCITY is a vector quantity (quantity having both MAGNITUDE and DIRECTION). [br][/color][color=#980000]SPEED, on the other hand = | VELOCITY |, thus always making it a non-negative quantity.[/color][/b][br][br]Thus, there are times when velocity can be [color=#cc0000]negative.[/color] [br][br]The following graph and table provides information with respect to a person driving away from home.[br]Let [i]t [/i]= the number of hours that have passed. [br]Let [i]d[/i] = this person's displacement from home. [br][br]Study the graph and table carefully. (They display the same information.) [br]Then, answer the questions that appear below the applet.

1.

What was the average velocity for the first half hour of your trip?

2.

What was your average velocity from t = 0.5 hr to t = 1 hr?

3.

What was your average velocity between t = 1.5 hrs to t = 2 hrs?[br]Explain what your answer physically means with respect to the context of this story.

4.

What was your average velocity for the entire trip? (Assume your trip took 2.5 hours.)

5.

Between what two listed 1/2-hr increments was your [b]average speed[/b] the greatest?

4.2.

4.3.

4.4.

4.5.

4.6.

4.7.

4.8.

4.9.

4.10.

4.11.

4.12.

4.13.

4.14.

4.15.

4.16.

4.17.

4.18.

4.19.

5.

Related Rates

1. Related Rates (1)
2. Related Rates (2)
3. Falling Ladder !!!
4. Related Rates (3)
5. Related Rates: Adjustable Cone with dh/dt Constant
6. Related Rates: Adjustable Cone with dV/dt Constant
7. Related Rates (Rolling Carts Problem)
8. Linearization Checker (Calculus)

5.1.

Related Rates (1)

Suppose [color=#0000ff][b]particle A [/b][/color]and [color=#cc0000][b]particle B[/b][/color] start at the same spot. [br][br]Suppose [color=#0000ff][b]particle A[/b][/color] [color=#0000ff]starts to head east at a speed of 10 m/s.[/color] [br]At the same time [b][color=#0000ff]A[/color][/b] leaves its initial position, [color=#cc0000][b]particle B[/b][/color] [color=#cc0000]starts heading north at a speed of 5 m/s[/color]. [br][br]Play to animate, then answer the questions that follow. [br]([color=#ff00ff][b]Please don't click the checkbox yet. Do so when prompted to.) [/b][/color]

1.

At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 3 sec?

2.

At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 5 sec?

3.

At what rate is the distance, [i]w[/i], between [i]A[/i] and [i]B[/i] changing after 6 sec?

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

6.

Linearization of Functions

1. Linearization Illustrator (Calculus)
2. Linearization Checker (Calculus)

6.1.

Linearization Illustrator (Calculus)

6.2.

7.

Closed Interval Method

1. Critical Numbers & Critical Points of Functions (Intro)
2. The Closed Interval Method (Application Problem)

7.1.

Critical Numbers & Critical Points of Functions (Intro)

Interact with the following applet for a few minutes. [br]Then, answer the question that follows. [br][br]

KEY QUESTION:

What does it mean for a number [i]c[/i] to be defined as a [color=#38761d][b]CRITICAL NUMBER[/b][/color] of a function [i]f[/i]?

7.2.

8.

Optimization Problems

1. Optimization Challenge 1
2. Optimization: Maximizing Area of a Rectangle Inside a Right Triangle
3. Critical Numbers & Critical Points of Functions (Intro)
4. Honeybee Population (Growth Rate Question)
5. Oil Pipeline Optimization Problem
6. Distance From Point to Graph

8.1.

Optimization Challenge 1

[color=#9900ff][b]In the applet below, [i]A[/i] and [i]B [/i]are 5 units apart.[/b][/color][br](This distance will change if you move point [i]B[/i], so keep it where it is for now.) [br][br][color=#38761d][b]Slide the green unlabeled slider. [/b][/color][br]What does this imply about the two lines?[br][br]Nonetheless, the goal of this problem is to[b] [color=#9900ff]determine how far (to the left) [i]D[/i] needs to be placed from [i]C[/i][/color] in order to minimize total area enclosed by both triangles.[/b] Even though you can use this applet to obtain an approximate value of this distance, use calculus to [color=#9900ff]determine an [i]exact value[/i] of this approximate distance[/color]. [br][br][color=#9900ff]Retry this problem for [i]AB[/i] = some other distance. [/color] (You can move point [i]B[/i] to make this happen.) [br][br][br][b][color=#9900ff]What if [i]AB[/i] = [i]x[/i] units? [/color][color=#9900ff]Can you find an expression (in terms of [i]x[/i]) for the distance [i]DC[/i] [/color][color=#0000ff]that minimizes the sum of the areas of both triangles? [/color][/b][br][br][br] [br][br][br][br]

Quick (Silent) Demo

8.2.

8.3.

8.4.

8.5.

8.6.

9.

Riemann Sums; Indefinite and Definite Integrals

1. Definite Integral Illustrator (I)

9.1.

Definite Integral Illustrator (I)

[b]Note: [br][/b][color=#0000ff]This applet only generates nonnegative areas. [br]Thus, incorrect values will be displayed for any areas underneath the [i]x[/i]-axis.[br][br][/color]For an applet that also includes [b][color=#ff7700]trapezoidal approximations[/color][/b], go to [url=https://www.geogebra.org/m/yUpHvBYc]Definite Integral Illustrator (II)[/url].

10.

Areas and Volumes

1. Definite Integrals: Sum
2. What Makes a Function Even?
3. What Makes a Function Odd?
4. Odd Functions: Another Look (Take 2)!
5. U-Substitution (Definite Integrals)
6. Areas Between Curves (Intro)
7. Area Between Curves
8. Area Between 2 Curves
9. Solids of Revolution (Discs)
10. Volume of Solid of Revolution about x-axis
11. Volume by Cylindrical Shells: Illustrated
12. V. of Solid NOT of Revolution
13. Areas & Volumes: Culminating Activity

10.1.

Definite Integrals: Sum

[color=#000000][b]Note: [br][/b]This geometric property of definite integrals only holds true if function [i]h[/i] is integrable over the interval [a,b]. [br]Feel free to adjust the function, sliders, and the locations of the [/color][color=#1e84cc][b]BIG BLUE POINTS[/b][/color][color=#000000] on the graph. [br][br]Can you think of a function [i]h [/i](and values for [i]a, c, [/i]and [i]b[/i]) for which this illustrated property does [b]not [/b]apply?[/color]

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