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Slope Fields

Resources from OpenStax Calc II
College Board YouTube
& Khan Academy Calc AB

Slope Fields are a unit 7 topic that show the expected slope for any point in the x-y plane. You will not be asked to create these graphs but you may have to interpret them.

direction field (slope field) is a mathematical object used to graphically represent solutions to a first-order differential equation. At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point.


An applied example of this type of differential equation appears in Newton’s law of cooling, which we will solve explicitly later in this chapter. First, though, let us create a direction field for the differential equation.

T′(𝑡) = −0.4(𝑇−72)

Here 𝑇(𝑡)T(t) represents the temperature (in degrees Fahrenheit) of an object at time 𝑡,t, and the ambient temperature is 72°F.72°F. Figure 4.6 shows the direction field for this equation.

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Please feel free to email and text me questions. I made this page to initially document our schedules/meet ups. Soon, I will add more information to address any questions you have about working with them in these interesting times.

We will continue to use Skype to connect (michael.briscoe_2). If we lose connection, please call me (571.641.7611) and we will do some reading of Call of the Wild to wrap up the time. I know Charles read some of this book, I want to make sure he is comprehending the paragraphs so we will go through it a little bit slowly.

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Types of Inheritance



Incomplete Dominance

Mendel’s results, that traits are inherited as dominant and recessive pairs, contradicted the view at that time that offspring exhibited a blend of their parents’ traits. However, the heterozygote phenotype occasionally does appear to be intermediate between the two parents. For example, in the snapdragon, Antirrhinum majus (Figure 12.7), a cross between a homozygous parent with white flowers (CWCW) and a homozygous parent with red flowers (CRCR) will produce offspring with pink flowers (CRCW). (Note that different genotypic abbreviations are used for Mendelian extensions to distinguish these patterns from simple dominance and recessiveness.) This pattern of inheritance is described as incomplete dominance, denoting the expression of two contrasting alleles such that the individual displays an intermediate phenotype. The allele for red flowers is incompletely dominant over the allele for white flowers. However, the results of a heterozygote self-cross can still be predicted, just as with Mendelian dominant and recessive crosses. In this case, the genotypic ratio would be 1 CRCR:2 CRCW:1 CWCW, and the phenotypic ratio would be 1:2:1 for red:pink:white.

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This week is a bit of a holding pattern for the girls as we wait for specific College Board direction. That said, we know there are some foundational ideas about mathematical limits that we can strengthen.

For Omar, the big focus has been to lock down the highest grades possible in each course by Thursday.



This Week’s Schedule

  • Monday: 2:30-4pm
  • Wednesday: 2:30-4pm
  • Friday: 2:30-4pm


I am sending you an invoice that brings up to date through March 29th, then I will send you a weekly invoice Friday nights for the hours we worked each week. If this causes you any issues, just let me know.

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Sum & Difference Properties for sin and cos


Section 1. For the following exercises, find the exact value.

1. cos(7𝜋/12)

2. cos(𝜋/12)

3. sin(5𝜋/12)

4. sin(11𝜋/12)

5. tan(−𝜋/12)

6. tan(19𝜋/12)

Section 2. For the following exercises, rewrite in terms of sin𝑥 and cos 𝑥.

7. sin(𝑥 + 11𝜋/6)

8. sin(𝑥 − 3𝜋/4)

9.  cos(𝑥 − 5𝜋/6)

10.  cos(𝑥 + 2𝜋/3)

Section 3. Use the unit circle and/or the properties for finding sin and cos values to solve these questions:

11. 225°

12. 300°

13. 320°

14. 135°

15. 210°

16. 120°

17. 250°

18. 150°

19. 5𝜋/4

20. 7𝜋/6

Section 4. For the following exercises, simplify the given expression.

21. csc(𝜋/2−𝑡)

21. sec(𝜋/2−𝜃)

22. cot(𝜋2−𝑥)

23. tan(𝜋/2−𝑥)

24. sin(2𝑥)cos(5𝑥)−sin(5𝑥)cos(2𝑥)


Section 5. For the following exercises, find the requested value.

1.  If cos (𝑡)= 1/7 and 𝑡 t is in the fourth quadrant, find sin (𝑡).

2. If cos (𝑡)= 2/9 and 𝑡 t is in the first quadrant, find sin (𝑡).

3.  If sin (𝑡) = 3/8 and 𝑡 t is in the second quadrant, find cos (𝑡).

4.  If sin (𝑡) = −1/4 and  and 𝑡 t is in the third quadrant, find cos (𝑡).

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Double Angle Properties for Trigonometry

Use the rules above to solve the questions listed below. For the following exercises, find the exact values of a) sin(2𝑥), b) cos(2x), and c) tan(2x) without solving for𝑥.

5. If sin 𝑥 = 1/8, and 𝑥 is in quadrant I.

6. If cos 𝑥 = 2/3, and 𝑥 is in quadrant I.

7. If cos 𝑥 = −1/2, and 𝑥 is in quadrant III.

8. If tan 𝑥 = −8, and 𝑥 is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

9. cos (2𝜃) = 3/5 and 90° ≤ 𝜃 ≤ 180°

10. cos (2𝜃) = 1/2 and 180° ≤ 𝜃 ≤ 270°

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Half-Angle Properties

There are three rules to use in these questions:


Warm-up #1. Given that sin 𝛼 = −4/5  and 𝛼 lies in quadrant IV, find the exact value of cos(𝛼/2).

For the following exercises, find the exact value using half-angle formulas.

2.  sin (𝜋/8) 

3. cos (−11𝜋/12)

4. sin (11𝜋/12)

5. cos (7𝜋/8)

6. tan (5𝜋/12)

7. tan (−3𝜋/12)

8. tan (−3𝜋/8)

For the following exercises, find the exact values of a) sin(𝑥/2), b) cos (𝑥/2), and c) tan (𝑥/2) without solving for 𝑥,  when 0° ≤ 𝑥 ≤ 360°. 

9. If tan 𝑥 = −4/3, and x is in quadrant IV.

10. If sin 𝑥 = −12/13, and 𝑥 is in quadrant III.

11.  If csc 𝑥 = 7, and 𝑥 is in quadrant II.

12. If sec 𝑥 = −4, and 𝑥 is in quadrant II.

Video For Practice if Necessary (Let me know what you think about the usefulness).

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This initial video is focused on Geolocation. The following videos connect to YouTube channels for a more complete experience. Find a channel you like then follow along step by step.

Flutter Training Channels: Free Code Camp

Flutter Training Channel: Google Developers

(You may want to Skip to video 6 after listening to this quick video.)

Flutter Training Channel: Net Ninja

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You can always see what I am posting for Kushaan and Medhnaa:

Schedule from Khan Academy

April 1 Update

It seems like they both have time given Pinnacle’s week, but need to catch up in math.

Kushaan seems to be about a chapter behind the class. His page currently have links to three types of rules we reviewed Sunday. He will need to work problems in his notebook to get adjusted to each type of formula.

Medhnaa’s test is going to be challenging. She needs to go back through all the course material Monday and the first half of Tuesday to get a list of notes and organize a study guide. There is a practice test in ALEKS she can use to see the 25 question types that should be on the midterm. That said, she will need to truly memorize the unit circle by Tuesday night and work a variety of functions requiring “inverse trigonometric functions.”

“Inverse Trig Functions”: Sine and Cosine take an angle in a triangle and give the ratio of two sides. The value of sine and cosine is always between 1 and -1. Inverse functions say, “You give me the number between 1 and -1, and I will give you the angle.” The most difficult part for students is that there are two angles for each value.

Medhnaa also has readings from arguments by the Founding Fathers that we are using to prepare for the informational text portion of the SAT. It also aligns with her history class.

This Week’s Schedule

I think three days this week makes sense but could easily go to 2 or 4 if the need arises. Right now my plan is:

  • Wednesday 5:15-8pm

This Week

  • Monday: 4:30-7pm
  • Tuesday: 5-8:45pm

Previous Invoice

In getting everyone moved to mostly virtual, I feel behind on invoices. Here is the invoice I sent since March 3.