2.1 Bar Graphs, Circle Graphs, and Time Plots
Features of a Bar Graph
1. Bars can be _ or _
2. Bars are of uniform _ and evenly _
3. The Lengths of the barts represent values of the variable being displayed, t
5.4 Geometric Distribution
Suppose we have an experiment where we will repeat binomial trials until we get
the FIRST success. We use n to denote the first trial that is successful.
Say we were trying to
3.1 Measures of Central Tendency: Mean, Median, and Mode
Mode: the value that occurs the most frequently
Find the mode
5
3
7
2
4
4
2
4
8
3
4
3
A data set can have two modes. If there ar
12.1 The Fundamental Counting Principle and Permutations
FACTORIALS:
Definition: A factorial of n objects is denoted n! and represents :
n! = n (n 1) (n 2). 2 1
4! =
Question:
7! =
15!=
How many different ways can we seat 4 people in only 3 chairs?
Method
2.3 Ogives and Stem and Leaf Displays
Ogive -is a graph that displays cumulative frequencies. It is useful for determining
the number of data values above or below a specified value
Using the data from when we
3.2 Measures of Variation
Range:
Standard Deviation: a measure that lets us know by how much a group of data
differs
Sample standard deviation s =
(x x )2
n 1
Sample Variance: same as sample standar
5.3 More Binomial Distribution
Graphs of Binomial Distributions
A waiter at a restaurant has learned over time that the probability that a lone diner
leaves a tip is .7. During one lunch hour he serves six lo
12.2 COMBINATIONS
Question 1: How many arrangements of 4 people in 3 chairs can we make, where order doesnt
matter?
Method 1: List all Possiblilities
ALL THE PERMUTATIONS ARE:
A
A
A
A
A
A
B
B
C
C
D
D
C
D
B
D
C
B
C
C
C
C
C
C
A
A
B
B
D
D
B
D
A
D
A
B
B
B
B
B
4.2 Probability of Compound Events
We want to find the probability of two events occurring TOGETHER. For this we
will use compound probability rules
Independent Events: the occurrence or nonoccurrence of one
10.2 BEST Fit Lines
We have all drawn best fit lines, and we have also used LinReg ax+b on the
calculator. This is how the calculator actually finds the BEST fit line.
* y=a + bx *
a is
INTRO TO CALCULUS
REVIEW FINAL EXAM
NAME:
DATE:
A. Equations of Lines (Review Chapter)
y = mx + b (Slope-Intercept Form)
Ax + By = C (Standard Form)
y y1 = m(x x1) (Point-Slope Form)
Problems:
1.
Find the equation of a line passing through point (5, -2) w
Evaluating Limits Analytically
Dividing out/ Cancellation Technique
x3 1
lim
x1 x 1
Remember this limit from yesterday. We examined it yesterday and determined
x3 1
lim
=3
that x1 x 1
Now lets see why.
Complete
1.1 Introduction to Calculus
PreCalculus
Calculus
Today we are going to explore both of these situations.
Consider the following graph of y = x 2 + 1 .
Suppose we want to find the sl
5.5 Derivatives of Bases other than e
Recall that
d x
e = e x
dx
Now if base is a which is anything other than e
d x
[a ] =
dx
Now recall that
d u
e = eu du
dx
So for a base other than
1.2 Finding Limits Graphically and Numerically
Limit: As a function approaches a given x value what y value does it look like the
graph is getting closer and closer to
Example:
x3 1
lim
x1 x 1
We read thi
Continuity and One Sided Limits
Definition of Continuity
Continuity at a Point: A function F is continuous at c if the following three conditions
are met.
1. f(c) is defined
2. lim f (x) exists
xc
3. lim f (x)
5.2 Integrating with natural logs
First to review take the derivative of f (x) = ln(3x 2 + 4x)
Notice how we have the entire original inside function on the bottom and the
derivative of that in
2.1 Definition of Derivative
Definition of Tangent Line with Slope m
y
f (c + x) f (c)
lim
= lim
= m
x0 x
x0
x
The line passing through (c,f(c) with slope m is the tangent line to f at the point
(c
5.4 Exponential Functions: Derivatives and Integration
Definition of the Natural Exponential Function
e is the inverse of ln, that is ln(e x ) = x and eln x = x
They undo each other.
Using this property
Name
Relative Extrema Fizz
1. Use the 1st derivative test to find all relative extrema of
f (x) = 2x3 + 9x2 + 24 You must produce a chart to receive full credit.
, Per
rm: MM.th (369mg gemomw a x
, 2'4 x:( {:(f
t 98 +49} K I I
O ~<» (x 3) +"(~n>~¢<~u*
Name_
Relative Extrema Fizz
1. Use the 1st derivative test to find all relative extrema of
f (x) = 2x 3 + 9x 2 + 24 You must produce a chart to receive full credit.
2. Use the se
2.4 Chain Rule
Composite Function: A composite function is a function that is made up of one
function inside another function
Example
y = (3x 2 + 2)3
f (x) = x 2 + 1
y = sin(2x + 1)
Rewrite the above
2.2 Basic Differentiation Rules and Rates of Change
Differentiability of a Function
A function is differentiable at any point in its domain except in three cases
1.
2.
3.
The Power Rule
Kuta Software ~ Infinite Calculus Name
Differentiation Product Rule Date Period
Differentiate each function with respect to x.
1) y = x3(3x4 2) 2) f(x) = x2(-3x2 2)
"33 ~ 5 ix} 7 3" ~ "-432 m 3; .1
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{.512 x ~§ 3.3; «3.
Product and Quotient Rules
The Product Rule
We use the product rule to find the derivative of two functions that are being
multiplied together so that we can find the derivative faster than having to multiply
the