6.4 Exponential Growth and Decay
Goals
- know where the exponential growth model comes from
- solve exp. growth and decay problems
-apply to half-life problems
Exponential Growth and Decay (continuous)
.where does it come from?
If some amount of something
6.3: Goals: To use tabular integration
Tabular Method:
f ( x) g ( x)dx in which f can be differentiated repeatedly to become zero and g can be integrated
repeatedly without difficulty.
Use the tabular method:
1) x 2 e x dx
2) x 3 sin xdx
3.6 The Chain Rule
Goals
Understand the chain rule and how to use it.
Find the derivative:
a ) y (2 x 3) 2
Power integer chain rule:
Chain Rule: If
b) (2 x 3) 20
d n
u nu n 1u '
dx
F ( x) f ( g ( x) then F '( x) f '( g ( x) g '( x)
Find the derivative:
1)
6.2: Integration by Substitution
Goals
- practice the process of integrating by substitution
Examples: evaluate the indefinite integral using the indicated expression to be substituted
1) cos(5 x)dx, u 5 x
2) x sin(4 x 2 )dx, u 4 x 2
Examples: evaluate th
3.1A - Derivative of a function
Goals
- To use the definition of the derivative to find
(1) the derivative, and
(2) the slope of a curve at a point
Ex. Find the slope (rate of change) of the curve at the given point:
f ( x ) 4 x 2 , a 1
Derivative of a fu
3.3A Rules for Differentiation
Goals
- be able to use shortcuts to find the
derivatives of various functions
- be able to use the derivative to find tangent lines
Notations for derivative:
Function
y
Derivative
y
dy
dx
y
f(x)
f (x)
d
f ( x)
dx
df
dx
f(x)
Implicit Differentiation
Goals
- be able to derive implicitly
- be able to find the slope at a point given a problem that requires implicit differentiation
-find the second deriv implicitly
What is implicit differentiation?
- it is a process by which you
3.2A Differentiability
Goals
- Understand what differentiable means
- be able to determine if a function is differentiable
Left and Right Hand Derivatives
rule - For a function to be differentiable at a point, we
say that the left and right hand derivativ
Sect. 6.5: Goals: To use partial fractions to take the antiderivative
If f ( x)
a
P ( x)
where P and Q are polynomials with degree of P less than the degree of Q, and if
Q( x)
Q(x) can be written as a product of distinct linear factors, then f(x) can be
3.5 Derivatives of Trig Functions
Goals
- learn the derivatives of trig functions
- be able to use these derivatives in various derivative applications
d
sin x cos x
dx
d
tan x sec 2 x
dx
d
cot x csc2 x
dx
d
cos x sin x
dx
d
sec x sec x tan x
dx
d
csc x c
6.1 Differential Equations and Initial Value Problems
Goals
- understand definite vs. indefinite integrals
- solve simple differential equations
- use the initial value to find the original function
-graph a slope field given a differential equation
Evalu
Section 2.1: limits
2.1A Rates of Change and Limits
Goals
- understand what a limit is
- find limits by substitution
- find limits when substitution is not possible (with and
without calc)
Limit: Intended height of a function ( y-value) at a given value o
3.9 Derivatives of Exponential and
Logarithmic Functions
Goals
- learn the new deriv rules
- be able to apply the new deriv rules to given problems
d x
e ex
dx
d u
e eu u '
dx
d u
a a u ln a u ', a 0, a 1
dx
d
1
ln u u '
dx
u
ln u
remember : log a u
ln a
Section 2.3: continuity
Objective: Understand continuity and how limits are involved
Be able to determine if a function is continuous and identify location and types of
discontinuity
Determining continuity:
A function f(x) is only continuous at an interio
3.4A Velocity and Other Rates of Change
Goals
- Calculate the rate of change within familiar formulas
- Use derivatives and rate of change within particle motion and other applications
-Understand how a particle moves with respect to its velocity, acceler
3.8 Derivatives of Inverse Trig Functions
Goals
- Learn the inverse trig derivatives
- Be able to use inverse trig derivatives in different derivative application problems
Inverse Trig Function Derivatives:
d
1
sin 1 u
u'
dx
1 u2
d
1
cos 1 u
u'
dx
1 u2
Section 2.4: Rates of change and tangent lines
Objective: Be able to find the average rate of change
Be able to find the equations of tangent lines and normal lines given a curve and a point on
curve
Be able to find the slope of a curve at a point
Be able
Section 2.2: limits involving infinity
Objective: find limits involving infinity
Memorize:
1
0
x x
1) lim
any real #
0
x x any int eger 0
2) lim
3) lim polynomial
x
sin x
0
x
x
4) lim
Properties of limits as x same as section 2.1
Examples:
a ) lim(2
x
AP Statistics Chapter 4 Practice Test Name: : lQ/H\Q/
Surveys and experiments Per _ Date
0\ Part 1: Multiple Choice. Circle the letter corresponding to the best answer.
1. You wish to survey people who have brought in their cars for service drini the past
AP Statistics Chapter 10 Notes: Comparing Two Population Parameters
10.1: Comparing Two Proportions
Conditions for Comparing Two Proportions
Random We have two random samples, from two distinct populations
Independence Each sample must be selected indep
The Practice of Statistics for AP*, 4th Edition Glossary
Chapter 4
Anonymity When the names of individuals participating in a study are not known even
to the director of the study.
Bias The design of a statistical study shows bias if it systematically fav
AP Statistics Chapter 11: Inference for Distributions of Categorical Data
11.1 Chi-Square ( 2 ) Goodness of Fit Test
Goodness of Fit
A goodness of fit test is used to help determine whether a population has a certain hypothesized
distribution, expressed a
AP Statistics Chapter 9 Notes: Testing a Claim
9.1: Significance Test Basics
Null and Alternate Hypotheses
The statement that is being tested is called the null hypothesis (H0). The significance test is
designed to assess the strength of the evidence agai
AP Statistics Chapter 12: More about Regression
12.1 Inference for Linear Regression
Sample Computer Output for a Linear Data Analysis
For the above, the linear equation is y = 7.0647 + 0.36583x
The Standard Error of the slope (SEb) = 0.01048
S = the Stan
AP Statistics Chapter 4 Designing Studies
4.1: Surveys and Samples
Population, Census and Sample
The population in a statistical study is the entire group of individuals we want
information about. For example, all registered voters in a given county.
A
AP Statistics Chapter 6 Discrete, Binomial and Geometric Random Vars.
6.1: Discrete Random Variables
Random Variable
A random variable is a variable whose value is a numerical outcome of a random phenomenon.
Discrete Random Variable
A discrete random vari
AP Statistics Chapter 8 Notes: Estimating with Confidence
8.1 Confidence Interval Basics
Point Estimate
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic
from a sample is called a point estim
AP Statistics Chapter 1 Notes - Exploring Data
1.1/2: Categorical Variables and Displaying Distributions with Graphs
Individuals and Variables
Individuals are objects described by a set of data. Individuals may be people, but they
may also be animals or