Wednesday's Test

Here’s a list of topics that will be covered on this Wednesday’s 3.1-5 Test

Test 3.1-5 Topics
Higher order derivatives – fractional exponents, chain rule
Derivatives – General Power Rule
Derivatives – Single-term denominator
Derivatives – Product/Quotient Rule
Derivatives – Chain Rule
Higher Order Derivatives – Polynomials
Higher Order Derivatives – Polynomial/Product Rule
Derivatives – Chain Rule (table)
Derivatives – Chain Rule, Numerical Evaluation
Determine Polynomial given derivative information
Marginal Profit/Revenue/Cost Equations
Derivatives – Product and Quotient (Graphically)
Elasticity
Position/Velocity/Acceleration/Speed

That’s it! I’ll be in early on Wednesday, and available after school today…

Serious-minded people have few ideas. People with ideas are never serious.
- Paul Valery

Higher Order Derivative

Higher Order Derivatives
Higher Order Derivatives are a derivative of a derivative.

Example :

The second derivative describes how the first derivative is changing.


Derivatives and Speed
If you are given an equation for speed, the derivative of that equation is velocity and the derivative of the velocity equation is acceleration.

When you have a positive velocity that goes to a more positive velocity you have an acceleration.
When you have a negative velocity that goes to a more negative velocity you have an acceleration.

If you graph the velocity equation on your calculator you can find the acceleration. As the velocity is the slope of the line tangent to the speed equation, the acceleration is the maximum or minimum of the velocity equation.

Finding the Equation when given the Derivative Values

If you are given several of the derivative values you can find the equation that started it. If they give you dervatives up to a third derivative then you know the equation is a third degree polynomial, and so on.

EXAMPLE PROBLEM
Find the equation if you are given these derivatives.

1. You are given the values of the derivatives.



2. Depending on the degree you right the several polynomials with a, b, and c.


3. You start with the last equation and you make it equal to the last derivative value.


4. Just work your way up! (plug in answers of previous derivatives)

5. The equation will be equal to this : (ANSWER)

LINKS:

http://tutorial.math.lamar.edu/AllBrowsers/2413/HigherOrderDerivatives.asp

http://www.pinkmonkey.com/studyguides/subjects/calc/chap4/c0404f01.asp

PERSONALIZATION:

my favorite inspiration quote.

good better best, never let it rest, until the good is better, and the better is the best.

KYONG YOUR UP NEXT!!!! :)

Thursday's Quiz Topics

Here’s a list of topics that will be covered on this Thursday’s 3.3-4 Quiz

Quiz 3.3-4 Topics
Average Cost Function
Marginal average cost function
Revenue, Cost, Profit functions
Marginal Revenue, Marginal Cost, Marginal Profit functions
Actual Cost vs. Marginal Cost
Elasticity
Find the polynomial given information about its derivatives
Higher-order derivatives (third derivative, fractional exponents) – expressions
Higher-order derivatives (third derivative, fractional exponents) – evaluate
Higher-order derivatives (second derivative, fractional exponents)
Higher-order derivatives (rational functions, chain rule)
Position/Velocity/Acceleration/Speed

That’s it! I’ll be in early on Thursday…

"I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I --
I took the one less traveled by,
And that has made all the difference."

- Robert Frost
The Road Not Taken

3.4 Marginal Functions in Economics

Marginal Analysis

Marginal analysis is the study of the rate of change of economic quantities

So in this lesson MARGINAL is the same as DERIVITIVE OF

Marginal Cost: the derivative of the cost function
Marginal Revenue: the derivative of the revenue equation
Marginal Profit: the derivative of the profit equation

Cost Functions

To find the actual cost of the last individual product produced (x):

To find the marginal cost of the number of products produced (x):
Find the derivative of the cost function

To find the average cost per unit produced:

Use the notation for the average cost per unit produced:

Revenue Functions

To find the revenue:

To find the marginal revenue:

Find the derivative of the revenue function

Profit Functions

To find the profit:

To find the marginal profit:

Find the derivative of the profit function

Elasticity of Demand

To find the elasticity of demand:

Elastic: if (E>1) then an increase in price will cause a decrease in revenue (revenue will react in an opposite manner from the price)

Inelastic: if (E<1)> then an increase in price will cause an increase in revenue (revenue will react in a similar manner as the price)

Unitary: if (E=1) then an increase in price will cause the revenue to stay about the same

Example Problem

Find the elasticity of demand from the demand equation if Sally is selling Prepstock tickets for $12:

Step 1: Resolve the equation in terms of x. Step 2: Plug in to the elasticity equation

Step 3: Find the elasticity of any price (p), in this case plug in 12

Here is a site with some practice problems:

http://www.math.purdue.edu/~rzhao/Courses/MA223/Lesson17.pdf

KJ, you are up next with lesson 3.5 High Order Derivatives.

"Cherish your vision and your dreams as they are the children of your soul; the blueprints of your ultimate achievements."

Thursday's Quiz Topics

Here’s a list of topics that will be covered on this Thursday’s 3.2-3 Quiz

Quiz 3.2-3 Topics
Product Rule
Quotient Rule
Chain Rule

Not a lot of surprises… I’ll be mixing and matching them, of course. And give you both formulas and data to calculate answers. But after those problems you tackled in class today, I’m certainly not worried…

That’s it! I’ll be in early on Thursday…

An Old Cherokee describes an experience going on
inside himself....
"It is a terrible fight and it is between two wolves.
One is evil - he is anger, envy, sorrow, regret,
greed, arrogance, self-pity, guilt, resentment,
inferiority, lies, false pride, superiority, and ego.

The other is good - he is joy, peace, love, hope,
serenity, humility, kindness, benevolence, empathy,
generosity, truth, compassion, and faith. This same
fight is going on inside you - and inside every
other person, too."

The grandson thought about it for a minute and then
asked his grandfather, "Which wolf will win?"

The old Cherokee simply replied, "The one you feed."

If I were on the Cartesian Plane...

Ok, Barbara Walters once asked an interview subject, "If you were a tree, what type of tree would you be?" So we'll take the same random idea and apply to math! Here's a college application essay question from the University of Chicago:

The Cartesian coordinate system is a popular method of representing real numbers and is the bane of eighth graders everywhere. Since its introduction by Descartes in 1637, this means of visually characterizing mathematical values has swept the globe, earning a significant role in branches of mathematics such as algebra, geometry, and calculus. Describe yourself as a point or series of points on this axial arrangement. If you are a function, what are you? In which quadrants do you lie? Are x and y enough for you, or do you warrant some love from the z-axis? Be sure to include your domain, range, derivative, and asymptotes, should any apply. Your possibilities are positively and negatively unbounded.
Inspired by Joshua Nalven, a graduate of West Orange High School, West Orange, NJ

For some extra credit, send me your response at mathmaverick@flintridgeprep.org by Sunday, 5pm. Have a great weekend!

Section 3.3 The Chain Rule

The Chain Rule
The Chain Rule is used to find the derivative of a function with more than one layer.
Let's take the function



With the tools we have learned over the last couple of days, we would have to expand this function and then use the power rule to find the derivative. Having to expand a polynomial to the 15th power, however, would not be much fun. Instead, why don't we think of the function f(x) as two different functions, g(x) and h(x). Lets set f(x) equal to g(h(x)). How does this help? Well, lets look at how we would seperate f(x) into its components. Lets use the following definitions of g(x) and h(x).






Now we can use the Chain Rule for layered operations. It looks like this:

So, if

and

then



Further Information:
http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html

Next Up: Nicole


"Draw a line and live above it."