Thursday's Test Topics

Here’s a list of topics that will be covered on this Thursday’s 2.5-6, 3.1-2 Test

Test Topics 2.5-6, 3.1-2
Determining one-sided limits from a graph
Intermediate Value Theorem
Definition of the Derivative
Product Rule
Quotient Rule

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

"Sometimes when I get up in the morning, I feel very peculiar. I feel like I've just got to bite a cat! I feel like if I don't bite a cat before sundown, I'll go crazy! But then I just take a deep breath and forget about it. That's what is known as real maturity."
- Snoopy

Chapter 3.2
Product and Quotient Rules

This section covers two new rules to make finding the derivative easier. The product rule will help us find the derivative of a long function, and the quotient rule will help us find the derivative of the quotient of two functions.

The Product Rule:

d/d(x) [ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

The Derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

The Product Rule is very useful for extremely long functions when using 4-step process to find the function will take too much time. So for shorter equations such as x^2, use the 4-step process or the four basic rules for the derivative.

Example of Product Rule:

Find the derivative of f(x) = (7x^3+3x^2+4x-10) (8x^3-4x^2+3x).

f'(x) = (7x^3 + 3x^2+4x-10) (24x^2-8x+3) + (8x^3-4x^2+3x) (21x^2+6x +4)*

*Note: You are able to leave the function unfactored, so you don't have to simplify.

The Quotient Rule

d/d(x) [f(x)/g(x)] = [g(x) f'(x) - f(x) g'(x)] / [g(x)]^2

The derivative of the quotient of two function is equal to the denominator multiplied by the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the square of the denominator.

h(x) = Hi / Lo
"The derivative is equal to LoDHi minus HiDlo divided by what is squared down below."

Example of Quotient Rule:

f(x) = X / (x^2 +1)
f'(x) = {(x^2+1)(1)-[(x) (2x)]} / (x^2+1)^2 which reduces to (1-x^2) / (x^2+1)^2*

* Note: You are able to leave the function unfactored, so you don't have to simplify.

Links for More Assistance:
http://www.mathwords.com/p/product_rule.htm
http://www.mathwords.com/q/quotient_rule.htm
http://en.wikipedia.org/wiki/Product_rule
http://en.wikipedia.org/wiki/Quotient_rule
http://mathworld.wolfram.com/ProductRule.html
http://mathworld.wolfram.com/QuotientRule.html

Next for Wednesday November 11 2006 is Danica on Chapter 3.3, The Chain Rule.

Personalization:

Theorem: 1=2

Proof:

let A=1
let B=A
multiply both sides of (2.) by A, you get AB=A2
subtract B2 from both sides, you get AB-B2=A2-B2
factor left and right hand sides, you get B(A-B)=(A-B)(A+B)
divide both sides by (A-B), you get B=A+B
plug A=1 and B=A into (6.), you get 1=2

3.1 Basic Rules of Differentiation

Four Basic Rules of Differentiation:

1. Derivative of a Constant:
The derivative of a constant function is always equal to 0.





Proof:












2. The Power Rule
The derivative of a power function is equal to the exponent times the quantity of the variable raised to the power of the exponent minus 1.






Proof:




















3. Derivative of a Constant Multiple of a Function
The derivative of a constant time a function is equal to the constant times the derivative of the function.






Proof:














4. The Sum Rule
The derivative of the sum or difference of 2 functions is equal to the sum or difference, respectively, of their derivatives.







Proof:


















Example Problem:
Find the derivative of the function f by using the rules of differentiation.










Links to Derivatives:
http://archives.math.utk.edu/visual.calculus/2/index.html
http://mathworld.wolfram.com/Derivative.html

Brian, you're up next with 3.2 The Product and Quotient Rules

Derivative Joke:
Q: What is the first derivative of a cow?
A: Prime Rib!

Section 2.6

Derivative:

Slope of the tangent line

Instantaneous rate of change

Step 1: f(x) = x² + 5x

f(x + h) = (x + h)² + 5(x + h)

= x² + 2xh + h² + 5x + 5h

The Slope Formula:
f(x + h) – f(x)
x + h – x

Step 2: [(x² + 2xh + h² + 5x) – (x² + 5x)] / h

Step 3:
(2xh + h² +5h) / h
lim h à 0
Step 4:
2x +h + 5 = 2x + 5 make the h = 0 because the limit of h is 0
lim h --> 0

Tip: You must put the equation of the line tangent to the chosen point in point-slope formula or the slope intercept form.

i.e. if according to our step 4, f(3) = 11 you will put it in:

Point Slope: y – 24 = 11(x – 3)

Slope Intercept: y = 11x – 33 + 24 = 11x – 9

Since at the time of the development of Calculus many different mathematicians from different countries found it at the same time, there are many different notations that mean essentially the same thing.

i.e. f¹(x), y¹, dy/dx, Dx(y), Dxf(x)

Differentiability

Many functions have derivatives, but some functions have areas where you

cannot draw a tangent line.

i.e.

When the graph of a function has a derivative at a point or interval, it is said to be differentiable

Hint: Differentiability implies continuity

Examples
To find the average rate of change:

Essentially step two above with numbers in place of the variables

For f(t) = 2t² what is the average velocity of [22,23] seconds?

f(23sec) = 1058ft f(22sec) = 968ft

(1058 – 968) / (23 – 22) = 90 / 1 = 90 ft/sec

To find the instantaneous rate of change:

Essentially the entire 4 step process with numbers in place of the variables

For the same f(t) what is the instantaneous velocity at [22] seconds?

f(t + h) = 2(t + h)²

= 2(t² + 2th + h²)

= 2t² + 2th + 2h²

(2t² + 4th + 2h² – 2t²) / h = 4t +2h = 4t

lim h --> 0

[22sec] --> 4t = 88 ft/sec

for Extra assistance visit http://tutorial.math.lamar.edu/AllBrowsers/2413/Differentials.asp

Victim for next Lesson : Drew Titus

Quiz 2.3-5 Topics

Here’s a list of topics that will be covered on this Wednesday’s 2.3-5 Quiz

Quiz 2.3-5 Topics
Determining limits from a graph
Determining one-sided limits from a graph
Determining limits from an equation
Determining one-sided limits from an equation
Determining discontinuity from an equation
Cost/Revenue/Profit/Breakeven formulas and equations
Intermediate Value Theorem

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

"There is nothing noble in being superior to someone else. The true nobility is in being superior to your previous self."
-Hindu Proverb

And here's a funny telemarketing phone call...

Thursday's Test Topics

Here’s a list of topics that will be covered on this Thursday’s 2.3-4, 5.1-3, 12.1-2 Test.

Compound Interest problems – I will give you the formulas, you decide which one applies.
Function Modeling – Cost, Revenue, Profit functions, Determine profit(loss), Breakeven point
Evaluate special angle trig functions.
Determine a limit from a graph.
Determine limits of rational expressions (as x approaches a value).
Determine limits of rational expressions (as x increases without bound – the Great Battle!).
Simplify an exponential expression.
Solve an exponential equation.
Solve a trigonometric equation (special angles).

That’s it! The format is as expected – ½ calculator, ½ Non-Calculator. I’ll be in my classroom until 3 on Wednesday and before school on Thursday. I’ll also be available online after Back to School Night on Wednesday.

See you in class!

2.5 One-Sided Limits and Continuity

One-Sided Limits

The function f has the right-hand limit L as x approaches a from the right, written

The function f has the left-hand limit L as x approaches a from the right, written

A limit can exist if and only if the one-sided limits agree.

Find the limit, the right-hand limit, and the left-hand limit from this graph.

Limit =

Right-hand limit =

Left-hand limit =

A Function is Continuous if it satisfies all of the following
1) f(a) exists.

2) a limit, , must exist

3)

*Remember that a function is continuous if you can trace the entire graph with your finger

Continuous Functions
1) Polynomials
2) Constants
3) Sin and Cos
4) Exponential
5) Logarithmic (on their domain)

Is the function on the graph continuous?

No, because the function is not continuous at x = -2

Intermediate Value Theorem

If f(x) is a continuous function on a closed interval [a, b] and M is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) =M

Existence of Zeros of a Continuous Function

If f is a continuous function on a closed interval [a, b], and if f(a) and f(b) have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval [a, b]

Example Problem

Calculate the left and right hand limits of f (x) at 2.

As x approaches 2 from the left, f (x) = 3 - x and f(x) = 1 at x=2 so the left-hand limit is 1

As x approaches 2 from the right f(x) = x/2 and f(x) = 1 at x=2 so the right hand limit is 1.

Here are some helpful links:

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

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

Kyong you are up next!

"If you see a friend without a smile, give him one of yours"

2.4 Limits

Expressing Limits
The function F has the limit L as x approaches a, written

if the value f(x) can be made as close to the # L as we please by taking x sufficiently close to (but not equal to) a. This means that the limit is the number you are approaching as x gets closer to a.

Evaluating Limits
By Graphing

(The Limit as X approaches a of F(x) is L)

If the equation had a hole where L is the limit would still be the same.

.

(Infinite Limit is true by both sides, increasing without bounds.)

DNE = Does not exist

Your fingers do not meet when tracing.
BUT if you are looking for or it is possible.

Positive and negative signs indicate direction. Positive = traveling from right to left and Negative = traveling from left to right.

In this case the limits would be:

and

Evaluating Limits

Direct Substitution

To find the limit without looking at a graph. I plugged in 2 for the equation. 2 squared plus 6 equals 10. 10 is the limit as x approaches 2.

If you have a hole in your graph than you want to try and see the removable discontinuity.

transforms to

transforms to


now you can plug in 2 for the equation 4(x+2).
The limit is 16. The limit of 4(x+2) as x approaches 2 is 16.

Limits at Infinity

Rules:



1. If the degree of P(x) is greater than the degree of Q(x) , then the limit will be + or - infinity.
2. If the degree of P(x) is less than the degree of Q(x), then the limit will be a horizontal asympotote at 0.
3. If the degree's of the function are a tie than there will be a horizontal asymptote at the ratio of the leading coefficients.


Example Problem




What is the limit of the function as x approaches infinity?

The function has the same degrees on both top and bottum. You can use the infinity rule that the degree's of the function are a tie. Because they are a tie, this means that the limit is a horizontal asymptote at the ratio of the leading coefficients. In this case the ratio is 17/7.

ANSWER: 17/7


some extra sites on limits:

http://www.coolmath.com/limit1.htm
http://www.calculus-help.com/funstuff/phobe.html
http://curvebank.calstatela.edu/limit/limit.htm

Reminder:

hey nicole....you are up next!


Since it is almost halloween here is a cute joke:

Q: What do you get if you divide the cirucmference of a jack-o-lantern by its diameter?
A: Pumpkin Pi!

2.3: Functions and Mathematical Models

The Four Steps of Solving Word Problems:
1: FORMULATE - you must convert the problem into math language. eliminate extraneous information and pull out the information that will help you solve the problem
2: SOLVE - find an answer by plugging in information in the problem to equations you made in step one
3: INTERPRET - figure out what that number you solved for means
4: TEST - determine whether or not your number make sense in the circumstances of the problem

Rational Functions:
Just as rational numbers can also be expressed as fractions, so too can rational functions.

rational function f(x):






Special Cases (Economics):
Profit is Revenue minus Cost.
Revenue is Price(Retail) times Quantity.
Cost is Price(Wholesale) times Quantity plus any Fixed Costs.

demand equation f(x) = price








supply equation f(x) = price










The intersection of these graphs is economic equilibrium, when supply equals demand.











Sample Problem:

The Rigerbird portable light company has determined their weekly supply and demand equations to be




and




respectively, where p is measured in dollars and x is measured in units of 100. What is the equilibrium quantity and price?

First, we recognize that the equilibrium is the intersection of the graphs. So, we solve for that intersection. We can graph both equations on the same axes so we can see it.



















Then, we can use the intersect function on our calculators to determine that the two points of intersection are (-8, 80) and (8, 80). These answers would mean -800 portable lights per week at $80 and 800 portabel lights per week at $80, respectively. Only one of these answers would make sense in the problem, the second answer, because you cannot phsyically sell -800 portable lights in a week, as the first answer suggests. So, the answer to this problem is 800 lights per week, sold at $80 per light.


Links:
http://en.wikipedia.org/wiki/Supply_and_demand
http://www.netmba.com/econ/micro/supply-demand/

Nicole, you're up next!


"Giving up is the easiest thing you could ever do, but holding it all together when everyone expects you to crumble, now that is true strength."