Lesson 2
Continuing with our study of derivatives, you will explore the "shortcuts" for computing derivatives. You will be looking at proofs and using digital tools to assist you in your visual understanding of derivatives. I have provided the objective goals for this lesson and the big idea addressed by this Webercise.
California Content Standards
Big Idea
Objective goals for lesson 2:
California Content Standards
- 4.1 Students demonstrate of an understanding of the derivative of a function as the slope of the tangent line to the graph
- 4.3 Students understand the relation between differentiability and continuity
- 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential and logarithmic functions
- 5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
- 7.0 Students compute derivatives of higher orders.
Big Idea
- How is a derivative computed and how can we recognize what derivative rules to use?
Objective goals for lesson 2:
- Students will be able to recognize and use the power, product, quotient, and chain rule.
- Students will be exposed to the idea of the derivative as the slope of a tangent line to a graph.
- Students will know if a function is differentiable based on its graph.
- Students will memorize and be able to use derivatives of trigonometric functions.
- Students will continue their introduction on proof writing, particularly in the rules of derivatives.
- Students will compute higher ordered derivatives
Lesson 2 Webercise
1) “Why do I need to learn this?” You tell me! There are hundreds of opinions online, some educated…some not so much. I have provided a few links I would like you to take a look at, then let me know why you think YOU should learn this (and please make it more than “to pass your class”).
Ask Dr. Math
Physics Forum
CHS AP Calculus
Ask Dr. Math
Physics Forum
CHS AP Calculus
2) Watch this video to review the derivative of a function as the slope of the tangent line to a graph. You won't be asked to do anything with it at the moment. However, do pay close attention as, not only will you be quizzed on tested on this concept, but you this is also fundamental to your understanding of derivatives.
Caculus: Derivatives 1
Caculus: Derivatives 1
|
3) Proofs are going to be an essential part of your higher lever math courses. As such, you are to be able to derive the product and the quotient rule and be able to explain the proofs to your peers. Here are the general methods for deriving these.
Proof of the Product Rule From Calculus
Proof of Quotient Rule
Proof of the Product Rule From Calculus
Proof of Quotient Rule
4) The chain rule is so important and students tend to find it as one of the more difficult rules of derivatives. While there are many proofs of this online, most of those I found require the use of tools we don’t have knowledge of just yet. So, I have provided an alternative proof. Remember that the important thing is not to write down the computations, but to be able to EXPLAIN why those steps were taken and why we are allowed to do as such
Chain Rule Webercise Proof
Chain Rule Webercise Proof
5) To gain a graphical understanding of the derivative, take some time looking at this interactive website where you can change the function and move around the tangent line:
GeoGebraTube
GeoGebraTube
6) Now that you have seen a graphical representation of the above given functions and their derivatives, compute the derivative of the functions at the given points. Also, find their second higher order derivative by computation.
7) For the above given functions, give the intervals for which they are differentiable and which they are not. You can look at the graph of these to answer this question.
8) For the above given functions, analyze and explain the relationship between their continuity and their differentiability. That is, are the above functions continuous but not differentiable? If so, state the intervals for which this is the case.
Here is a quick review on these concepts:
Differentiability vs Continuity
7) For the above given functions, give the intervals for which they are differentiable and which they are not. You can look at the graph of these to answer this question.
8) For the above given functions, analyze and explain the relationship between their continuity and their differentiability. That is, are the above functions continuous but not differentiable? If so, state the intervals for which this is the case.
Here is a quick review on these concepts:
Differentiability vs Continuity
9) As will always be the case, this webercise will end with a timed test (don’t worry, it won’t be graded! It is merely so that YOU have an idea of where you’re at in understanding the topic).
Differential Calculus Problems
Differential Calculus Problems