AP Calculus

Went over homework

Quotient Rule:

identities:

The Chain Rule.

A composite function uses the output of one function for the input of the other function: f(g(x)).

Trigonometric applications of the Chain Rule

Simple and Composite uses of the chain rule

Trig stuff, and note the use of the SQUEEZE THEOREM at the end

Derivatives- Displacement, Velocity, Acceleration

Anti- Derivative= integral= undoing the derivative

Parametric Mode
-Can be changed under MODE menu. Click PAR
-Can be used for particle problems to analyze a one or two dimensional motion
-Easier to see if you use the bouncing ball when you graph it
-You can plot displacement, velocity (derivative of displacement), and acceleration (2nd derivative of displacement and derivative of velocity) in this mode just like in function mode
-It’s useful because you can see the motion, unlike in function mode

Derivative of Acceleration= a jerk

Questions of Worksheet
16

30

38

Chain Rule

Homework- ON ASSIGNMENT CALENDAR

CLASS OCTOBER 14 2009

WE GOT OUR TESTS BACK
THEY WERE SCALED

QUESTIONS THAT WE WENT OVER:
PART 1: 2,4,6,7
QUESTION 2: REMEMBER TO FACTOR OUT AS MUCH AS POSSIBLE-SYNTHETIC DIVISION ON CUBIC. ANSWER B

QUESTION 4, WHAT DOES THIS FUNCTION LOOK LIKE?
ANSWER: C
THE LIMIT OF THE FUNCTION APPROACHING INFINITY IS ZERO, SO THE X AXIS IS THE VERTICAL ASYMPTOTE.
THE DENOMINATOR IS THE DIFFERENCE OF SQUARES OF ONE, MEANING THERE ARE TWO VERTICAL ASYMPTOTES, AT 1 AND -1.

QUESTION  6: WHAT MUST BE TRUE ABOUT THE FUNCTION?
ANSWER: E
ALL OF THESE OPTIONS CAN BE TRUE, BUT NONE OF THEM HAVE TO BE.

QUESTION 7: SOLVE FOR K
ANSWER: -1/2
MAKE SURE TO WRITE
OR ELSE IT MAKES NO SENSE

PART 2

QUESTION 3: FIND f ‘(2)
ANSWER: f ‘(2)=42
THERE WERE MULTIPLE WAYS TO DO THIS, ALL USING SOME FORM OF THE DIFFERENCE QUOTIENT.

QUESTION 5: FIND DISCONTINOUS POINTS AND LABEL
THIS CAME DOWN TO FACTORING CORRECTLY AND KNOWING YOUR DIFFERENT TYPES OF DISCONTINUITY

THEN WE PLAYED A GAME PLOTTING DERIVATIVE FUNCTIONS
FRONT TEAM WON!!!!!

DISPLACEMENT, VELOCITY, AND ACCELERATION
s(t)= DISPLACEMENT
v(t)=VELOCITY
a(t)=ACCLERATION

THIS LED TO THE DISCOVERY OF SECOND DERIVATIVES:

s’(t)=v(t)
s”(t)=v’(t)=a(t)

AKA: THE DERIVATIVE EQUATION OF A DISPLACEMENT EQUATION IS VELOCITY, AND THE DERIVATIVE EQUATION OF A VELOCITY EQUATION IS ACCELERATION, MAKING THE SECOND DERIVATIVE OF DISPLACEMENT ACCELERATION.

Binomial Expansion

Example:

Definition of Derivative: f ‘ (x)

Forward Difference (quotient difference)=

Backward Difference=

Symmetric Difference (often used with table)=

At a Point=

Derivative Worksheet:

1.

2.

f’(x)=(-2)

POWER FUNCTION RULE (also on page 92)

Power function continued…

Example 1

Example 2

Calculator:

On the AP exam, a calculator can be used for finding x intercepts, numerical derivatives and numerical integration.

To find the derivative at x=2 when f(x)=x^5:

Plug in x^5 as Y1 –> Math –> Option #8 –>and then plug in nDeriv(Y1,x,2)–> 2nd calc gets a value.

This will get you an answer of 80.00004.  Don’t forget that .00004 is a small margin of error.

PROPERTIES OF DIFFERENTIATION (pg 93)

NOTATION OF DERIVATIVE

Derivative with respect to x of function y:

REMINDER:

HW: pg 95- 10 Qs, #1-25 odd

-Challenge problem

-Major Quiz Friday

finds the slope of the tangent line at f=c

more specific; at a point

use when you have a function and a point

any function at any point

can be used for every problem

as x and x+h get closer, it approaches the slope of the tangent line.

Derivative: Slope of the tangent line.

Find when taking limit of the slope of the secant line

Derivative Worksheet:

2. The answer is -3 because it is the slope of the secant line. Since the equation 9-3x is linear, it has the same slope.

4.

7.

10.

review of limits terminology:

HOMEWORK: limits graded worksheet

Challenge problem due 10/14

work on derivative packet

Major quiz Friday 10/9

-All polynomial functions continues
-Trig functions continues
-Piecewise may be continues
-Rational functions may not be continues
-Continues if:
f(c) – exist
lim(x) – exist
f(c)= lim f(x)

Quiz Clarifications

Question #1

Continuity (at point c)

Cusp

a cusp is a singular point of the curve.

(http://img.photobucket.com/albums/v48/punkdbaby/limits5.jpg)

NO DERIVATIVE AT A CUSP

Step Discontinuity

(http://www.mathwords.com/s/s_assets/s157.gif)

Removable Discontinuity

Asymptote

NO LIMIT

(http://jwilson.coe.uga.edu/EMT668/EMAT6680.Folders/Barron/Write-ups/Assignment%201/Figure4.gif)

Homework for next class

pg 58-59

23, 27, 31, 37, 41, 49, 55, 57, 65, 67, 70