Derivatives

Unit 2: Derivatives


 * __ Summary of Unit: __**

__Prerequisite Skills:__
posted by Kendalyn Moulder In order to understand and do well with derivatives, you must first understand and be able to apply limits with basic and complex functions. When you look at a function, it can either be continuous or discontinuous and this is determined by looking at the limit. Limits are used mainly with rational functions because polynomials are known to be well behaved. They are continuous everywhere. Rational functions; however, are not continuous everywhere because they can be seen as fractions and because of this have numbers that will make them undefined.

Now dealing with limits specifically here are some basic steps on how to ues them:
 * 1) You will be asked to look at a graph at a specific point
 * 2) First look at what the graph does coming from the negative direction and moving toward the positive direction
 * 3) Then look at what the graph does coming from the positive and heading toward the negative
 * 4) Then you need to look at the **Double-sided limit** which is where you look at the graph coming from both directions
 * 5) Then look at what the graph does AT THAT SPECIFIC POINT!!!

**Limits in relation to derivatives:**
Now that you know a little bit about limits here is how they are applied to derivatives:
 * First of all a derivative deals with finding the instantaneous velocity of a graph at a certain point; but this isn't always easy because most graphs are not comprised of straight lines, in this case you need to find the slope of a tangent line
 * The **definition of derivative** is the most basic way to find this slope
 * The definition of derivative deals with looking at the limit of a function as you approach your given point on the graph(this will be in the form of a double-sided limit)
 * You will plug your function into the definition of derivative and that will simplify to an equation that you plug your point into in order to find the slope
 * Helpful resources:**

If you need more help understanding limits you can look at the limits portion of this site Or you could look at this video [|Limits and derivatives video(no math just explanation)]

Here is the definition of derivative formula



All of us were complaining about factoring cubes so I figured this would be a good prerequisite skills video -Bailey Mann [|Cubic Factoring (Sum of Cubes and Difference of Cubes)]
 * __ Practice Problems: __**

By: Michael White 1. Differentiate the following 3x^2 -1 a. 9x^2 -1 b. 6x -1 c. 6x d. 4x +2
 * Beginner:**

2. Differentiate the following 7x^3 +4 a. 21x^2 +4 b. 28x^3 c. 3x +11 d. 21x^2

**On The Way:**
1. Find the tangent line of F(x)= (3x -5)^10 a. F'(x)= 30(3x -5)^9 b. F'(x)= 90x^9 -50 c. F'(x)= 10(3x -5)^9 d. F'(x)= 30x^9

2. Find the derivative of F(x)= (2x+4)^2 a. F'(x)= 2(2x +8) b. F'(x)= 4(2x +4) c. F'(x)= 4x^2 +16x +16 d. F'(x)= 4x

3. Find the derivative of F(x)= (1/2x +3/4)^3 a. F'(x)= 4/5(x -1/2)^3 b. F'(x)= -1/2(3/4x)^2 c. F'(x)= 3/2(1/2x +3/4)^2 d. F'(x)= 2/3x^1/2


 * Got it:**

1. Find the derivative of F(x)= (x^2 -x +1)^2 a. F'(x)= (4x -2)(x^2 -x +1) b. F'(x)= 2x^2 -2x +1 c. F'(x)= 4x^3 -6x^2 +6x -2 d. F'(x)= 4x -3

2. Find the Tangent line of F(x)= (5x +2)/(x^2 -1) a. F'(x)= 5x^2 -3x -2 b. F'(x)= (-5x^2 -4x -5)/(x^2 -1)^2 c. F'(x)= -5x^2 -4x -5 d. F'(x)= (5x^2 -4x -5)/(x^2 -1)^2

3. Find the derivative of F(x)= fifth root of (3x^4 -1) a. F'(x)= (x +2)(4x -7) b. F'(x)= 9x^16 -6x^4 +1 b. F'(x)= 1/5(3x^4 -1)^-4/5 d. F'(x)= 12x^3/5(3x^4 -1)^4/5

4. Find the derivative of F(x)= 2sec(x) a. F'(x)= 2/cos(x) b. F'(x)= 2cot(x) c. F'(x)= 2sec(x) 2tan(x) d. F'(x)= 2/sin(x)


 * Rockstar:**

1. Consider the curve y^2 = 4+x and cord AB Joining points A(-4,0) and B(0,2) on the curve. a. Find the x and y-coordinate of the point on the curve where the tangent line is parallel to chord AB.

KEY to PRACTICE PROBLEMS:

Just an introductory video. ~Caylin McCubbin [|Differential Calculus & Instantaneous Velocity] More detailed explanation of what a derivative is-Triston Payne [] Recap of what we covered today 9/5/13. ~Caylin McCubbin Slope of a Line Secant to a Curve
 * __ Videos: __**

Recap of what is covered in 2.1 of our Calculus books. ~Caylin McCubbin Slope of Secant Line Example

Derivatives of sin x, cos x, tan x, e^x and ln x ~Caylin McCubbin

examples using multiple rules ~Caylin McCubbin


 * __ Sites: __**
 * Here is an introduction to derivatives with the definition of a derivative: [|Intro to Derivatives] -Emily Sullivan
 * This a review of how to find a max and min using first derivative [|Max and Min Review] - Bailey Mann
 * Here is a cheat sheet of quick derivative facts, rules, and commonly used derivatives [|Derivative Cheat Sheet] - Bailey Mann
 * Overview of product rule with practice problems [|Product Rule] -Shannon Murphy
 * Overview and examples of the quotient rule Quotient Rule -Shannon Murphy
 * Overview of chain rule with practice problems [|Chain Rule] -Shannon Murphy
 * Review and help with how to find the equation of a tangent line Equation of tangent line -Shannon Murphy


 * __ Careers/Uses: __** -Emily Sullivan
 * 1) Differentiation can be used to find the maximum or minimum area or volume of something -This site explains step-by-step how to find the maximum volume of a box [|Volume using Differentiation]
 * 2) A career in financial engineering: Financial engineers use computer models and calculus to make trading and investment decisions; they also use derivatives to evaluate and measure the risk of financial decisions.

The AP Calculus test-prep book began the differentiation section with the introduction of derivatives using the definition. The book stated that a derivative is the instantaneous rate of change of a point on a graph. The book also included all of the rules and formulas needed to evaluate derivatives. Here is the list of the rules/formulas: This AP Calculus book was helpful because it gave a specific example for each rule of derivatives, and it had all the rules in one list which is easier to use. It was a beneficial source and I would recommend checking it out from the library, especially when it gets closer to the AP test.
 * __ Annotated Bibliography: __** -Emily Sullivan
 * Hockett, Shirley, and David Bock. AP Calculus. 10th ed. Barron's Educational Series Inc., 2010. 111-157. Print.