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Chain Rule

There are two big parts to mastering the Chain Rule:

  1. Identify the chain rule is necessary.
  2. Use the chain rule by taking the derivative of the outside function, then the inside function.

We use the chain rule to take the derivative of a function with two components.

f'(g(x)) = f'(g(x)) * g'(x)

Common functions requiring Chain Rule:

sin^2 x \frac{1}{cos x} \frac{1}{e^x}

How to master:

  1. Watch two of the three videos below that explain the Chain Rule with examples.
  2. Complete the worksheet of quick practice to get down the steps.
  3. Use quick practice to state whether Chain Rule is necessary.

Maintenance: Be sure you can quickly identify whether Chain Rule applies to functions.

NancyPi Describes Chain Rule

Organic Chemistry Tutor provides examples of the Chain Rule

Khan Academy has a great overview of the Chain Rule

Distrust Authority?

This hands-on example shows how the chain rule works using wheels representing each function:

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Chapter 2. Solar Energy Powers the Water Cycle

Solar energy is an important part of the water cycle. The water cycle describes how water moves from the Earth, into the air, and back to the Earth’s surface. It is powered by heat from the sun. Without the sun, there would be no water cycle. The sun heats the water up and makes some of the water evaporate. When water evaporates it turns into a gas that goes up in the air. Eventually it gets so high that it can form clouds. When water is a gas it is called steam or water vapor. This water can condense back into water or even freeze into ice when it is high in the sky. It weighs so little that is able to float (for a while) because the sun is heating the Earth which continues to push air upward.

Water in clouds can eventually fall back to Earth as rain or snow. Water sticks to other water very easily. In clouds, tiny drops of water connect with other drops of water and get bigger and bigger. When they get too big, they will start to fall back to earth. If this water immediately runs into a river, lake or ocean, it is still in the water cycle. If this water is immediately absorbed by a plant’s roots, then it helps the plant make energy (more on that next chapter!). If that water falls in your mouth, it is delicious!

Think about the energy that this process takes! The sun is warming water on earth, which makes it move. As water drops get higher, they gain the potential energy to fall back to earth. When the drops get big enough, they fall back to earth. The energy from falling rain can move pebbles and dirt as water runs into streams. And, the energy stored in water can be used by plants to grow tall, make seeds, and sink roots deep into the earth. All this energy starts with the sun and is transformed over and over again. It’s a great example of how energy changes forms but never is destroyed!

Humans use water’s energy. Smart!

Humans have figured out that energy from the water cycle can be used to produce electricity. Hydropower is creating energy (such as electricity) from moving water. When water is released from a reservoir, which is a large lake that stores water, it enters a turbine. The turbine spins the water. The turbine is connected to a generator, so when the turbine spins the generator produces electricity. Hydropower uses energy to create work. That work is making electricity. Hydropower uses moving water, which we consider to be renewable (remember “renewable” from chapter 1?). Why is moving water renewed? Because the water cycle allows water to circulate across the earth all the time. The water cycle is an endless, recharging cycle, which makes hydropower a renewable energy source. 

The Environment & You

If you want to show some love to your planet, then try cutting down your shower time. According to USGS, old showers used up to 5 gallons of water per minute. Water-saving shower heads produce about 2 gallons per minute. Cutting back even a few minutes, can save many gallons of water. Which adds up over the years and decades you will be showering!

Topic for Another Day

Some readers start asking, where does the Sun get its energy? If you want to explore this idea, search for “Sun” and “fusion” to learn how the sun transforms energy from hydrogen into solar energy and helium.

Writing Topic

Do you agree solar energy powers the water cycle? Write a paragraph explaining your answer.

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You can see the word “person” in personification. Authors personify when they describe things that aren’t people using people words. You may have written about “angry clouds” or “happy flowers”, but you know that clouds don’t get angry and flowers do not have emotions. As you read more good books, you will see lots of examples of personification. Be sure to list them below.

If it’s helpful here is a video describing personification with a cartoon.

Personification Video

So glad you are enjoying reading and exploring some new poems. Here is the video explaining “personification”.

Personification is Thrilling

Here are lyrics from Michael Jackson’s song Thriller:

“You try to scream but terror takes the sound before you make it
You start to freeze as horror looks you right between the eyes
You’re paralyzed
‘Cause this is thriller, thriller night
And no one’s gonna save you from the beast about to strike.”

“Thriller” by Michael Jackson

Here terror and horror are not people but they are treated like people. Terror does not literally “take” and horror does not literally “look”, but together they paint a great picture about the emotion of the situation. Note that people can be “paralyzed” and “saved” so those lines do not include personification.

Examples of Personification

  • The comet raced across the sky.
  • The sun happily spread light across the land.

What’s your favorite example of personification?

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Lillian Gilbreth

Before we begin: Write down what you know about Lillian Gilbreth in a notebook. (If you are just learning about her, write her name on the top of the page.)

Purdue University is proud of Lillian’s work! This short video provides an energetic overview.

This 2020 video from PBS highlights Lillian’s role as a STEM pioneer.

Vox explored the Gilbreth’s home life to highlight how her engineering leadership and mindset impacted her family.

Interested in Industrial Organization Psychology? Here is a 90-second video that explains the Gilbreths’ roles and the field they pioneered.

Discussion Questions:

  1. Do you consider Lillian Gilbreth a pioneer? List three pieces of evidence from the videos to support your point.
  2. Define IO Psychology and provide an example from one of the videos.
  3. If there are any words or topics you do not understand, list them on your page.
  4. What is the key takeaway you want to remember about Lillian Gilbreth?
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AP Calculus: Unit 2 Concepts

The slope of this line is given by an equation in the form of a difference quotient:

𝑚= \frac{𝑓(𝑥)−𝑓(𝑎)}{𝑥−𝑎}

We can also calculate the slope of a secant line to a function at a value a by using this equation and replacing 𝑥 with 𝑎+ℎ, where ℎ is a value close to 0. We can then calculate the slope of the line through the points (𝑎,𝑓(𝑎)) and (𝑎+ℎ,𝑓(𝑎+ℎ)). In this case, we find the secant line has a slope given by the following difference quotient with increment ℎ:

𝑚= \frac{𝑓(a+h)−𝑓(𝑎)}{a+h−𝑎} 𝑚= \frac{𝑓(a+h)−𝑓(𝑎)}{h}


Let 𝑓 be a function defined on an interval containing 𝑎. If 𝑥≠𝑎 is on the interval, then

𝑄= \frac{𝑓(𝑥)−𝑓(𝑎)}{𝑥−𝑎}

is a difference quotient. Also, if ℎ ≠ 0 is chosen so that 𝑎+ℎ is in the interval, then

𝑄= \frac{𝑓(𝑎+ℎ)−𝑓(𝑎)}{ℎ}

is a difference quotient with increment ℎ.

Defining the Derivative

Let 𝑓(𝑥) be a function defined in an open interval containing 𝑎. The derivative of the function 𝑓(𝑥) at 𝑎, denoted by 𝑓′(𝑎), is defined by

𝑓′(𝑎)= \lim\limits_{𝑥→𝑎} \frac{𝑓(𝑥)−𝑓(𝑎)}{𝑥−𝑎}

provided this limit exists. Alternatively, we may also define the derivative of 𝑓(𝑥) at 𝑎 as

𝑓′(𝑎)= \lim\limits_{ℎ→0} \frac{𝑓(𝑎+ℎ)−𝑓(𝑎)}{ℎ}.

Video Introduction

Want to see these numbers in action? This tool from Wolfram uses a “snowball” to show the rate of change for different functions.

Notes Check

Which definitions match these images?

This figure consists of two graphs labeled a and b. Figure a shows the Cartesian coordinate plane with 0, a, and x marked on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)) and (x, f(x)). There is also a straight line that crosses these two points (a, f(a)) and (x, f(x)). At the bottom of the graph, the equation msec = (f(x) - f(a))/(x - a) is given. Figure b shows a similar graph, but this time a + h is marked on the x-axis instead of x. Consequently, the curve labeled y = f(x) passes through (a, f(a)) and (a + h, f(a + h)) as does the straight line. At the bottom of the graph, the equation msec = (f(a + h) - f(a))/h is given.

Video Deep Dives


Organic Chemistry Tutor

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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.


Let me know what works best for you. I suggested changes merely to help Charles stay in a weekday/weekend flow. Thanks!

  • Monday 4-5pm
  • Wednesday 4-5pm
  • Thursday 4-5pm
  • Friday 12-1pm

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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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AP Updates: We now know that the AP Calculus AB exam will be May 12 (complete AP schedule here). The test will focus on the first 7/8 of the course. We have reviewed the first of 7 units as of today. This week, April 6-13, we will focus on Unit 2 which defines the derivative and the basic rules for finding derivatives. Since the test will be open notes, they do not need to memorize rules. But, the test will allow 15-20 minutes per page of questions, so it will be important for them to be able to use these rules fast and accurately.

  • Yasmine will complete all Unit 2 in Khan Academy this week. Yasmine has re-test and project information in the next 7 days.
  • Sofia is wrapping up the Unit 1 test in Khan Academy today, and we will also complete all Unit 2 in Khan Academy this week.

Unit 2 direct link:

The AP tests will be open notes. So, by the end of the week you should be able to ask to see their notebooks that they have started. These notes should be clear, organized and easy to use to answer any question.

For Omar, the big focus has been to lock down the highest grades possible in each course by April 24. We just accurately completed the most recent quiz (Section 33). There is one quiz that we may need to do a retake. He is emailing homework and the quiz that will wrap up the work affecting his grade. In that email, I encouraged him to offer to meet her during the office hours before the end of the quarter.

  • Section 34: Even and odd functions (whether functions are symmetrical like a butterfly; the key strategy is use -x as the input and see if the output changes)
  • Section 35: Parent graphs (the foundational graph types like y = x, y = 1/x, y = sin x, etc.
  • Section 36: Piecewise functions (functions that have two or more rules)

When we wrap up all the grades for the course, Omar and I will finish going through the Math content in the remainder of the Pre-Calc course so he can choose which Calculus course to take next year.



This Week’s Schedule

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


  • Friday, April 3: 2:30-4pm
  • Sunday, April 5: 2:30-4pm