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Overview: There are five practice worksheets ready now: Limits, Power Rule, Product Rule, Quotient Rule, and Chain Rule. After you finish each sheet, you can click these links after you work through the worksheets for Limits Worksheet Answers, Power Rule Answers, Product Rule Answers, Quotient Rule Answers, and the Chain Rule Answers.

Tuesday we will talk though this limit definition of the derivative one more time. We are going to highlight examples of f(x) and how the structure shows that we are finding f‘(x) at a certain point. I want to talk about the concept so the fluency makes more sense.

\lim_{h \to 0} \frac{f(x + c) - f(c)}{h}

You are seeing this definition where f(c) is already solved, so f(c) equals a number like, 1000. Then, we see the structure of the function because f(x + c) is never solved. It just shows how f(x) works. Here are the solution steps we will dive deeply into.

The remainder of this page has definitions and tools that we will review up until the test on Thursday.

Tools/Definitions from the Last Quiz

Definition of a continuous function.

  • Functions have limits if the left and right limits go to the same place
\lim_{x \to 1-} f(x) = \lim_{x \to 1+} f(x)

limit of a piece-wise function

When you have time for a 10-minute video. Here is another voice (Sal Khan!) to help make the piece-wise/continuity/approaching ideas make more sense.

These were problems; they seemed better in the group test. But, look through those questions while covering up the answers. Do you feel comfortable with these questions now?

Power Rule: The rule we use all the time when we have polynomials (but not rational functions). Here is the simplest example:

f(x) = x^2, then f'(x) = 2x

Product rule: This may require some memorization, but what is the derivative of (f(x))(g(x))?

h(x) = (3x-7)x^2, then h'(x) = (3)(x^2)+(3x-7)(2x)

We say something like “Derivative of the first times the second plus the derivative of second times the first.”

Because we now have the quotient rule, I am going to encourage us to keep this idea of derivative of one times the other. The product rule has the sum of these two types. The quotient rule has the difference of these two types (and a denominator).

Quotient Rule: For rationale functions we can’t use the Power Rule and the Chain Rule is too complicated. We use this rule for functions like:

\frac{(5x-4)}{x^2} or \frac{sin(x)}{cos(x)}

These functions have derivatives using one rule:

\frac{(5x-4)}{x^2} -> \frac{5(x^2)-(5x-4)(2x)}{(x^4)}

Chain rule to calculate a derivative: This may require some memorization, but what is the derivative of (f(g(x))?

h(x) = (2x-9)^2, then h'(x) = 2(2x-9)*(2)

Because the function on the inside is 2x-9 and the function on the outside is ( )2. The derivative of the outside function is 2( ). Power Rule!! The derivative of the inside function is 2. That means altogether we get 2(2x-9)x2.

Derivatives of sinusodial function: What is the derivative of sin? (If you memorize one, then you know the derivative of cos is similar but has a different sign.)

Integration is Antidifferentiation

Indefinite integral (antidifferentiation): Using the power rule, product rule, and chain rule backwards requires the persistence to check your work over and over again. (Also, remember the last 2 problems we did emphasizing adding in the constant term “+c”)

Bigger Problems

Find the critical points of a function: This is why we take derivatives. The process is to take the derivative and set it equal to 0.

Write the intervals over which the function is increasing or decreasing: Another reason to take the derivative. If the derivative is positive, then the rate of change is positive. If the derivative is negative, then the rate of change is?

Horizontal asymptote of a graph: You can solve these with limits if you are interested in extremely large or extremely negative values. In the middle of a function, you use critical points. Question: Given a function, how will you find the local minimum and local maximum values?

Topics you memorized for the quiz

  • Power Rule
  • Product Rule
  • Chain Rule
  • derivative of sin
  • derivative of cos
  • derivative of e^x
  • antiderivative using power rule (If f'(x)=6x, what is f(x)?)

Today let’s go over any of the topics/explanations that you think are troubling. Then, let’s discuss the homework problems that do not seem possible. Note that the calculator is not available but you can quickly sketch graphs by plotting points and calculating critical points.