Calculus
1.3 Part IIEvaluating
Limits
Analytically
Goal: Students will develop and use
strategies for finding limits (non wellbehaved)
Property of Limits:
Example 1:
A Strategy for Finding Limits
Cancellation Techniques
When direct substitution fails, the
Calculus
5.2 The Natural
Logarithmic
Function:
Integration
Goal: Students will learn how to develop
and use properties of natural log and to
integrate functions involving natural
logs.
Exploration:
What do these functions have in
common?
The degree in the
Calculus
1.4 Part IIContinuity and
OneSided Limits
Goal: Students will determine continuity at a point and
on an open interval and determine onesided limits and
continuity on a closed interval
Properties of Continuity
Example 7:
Discuss the continuity o
Calculus
1.3Evaluating
Limits
Analytically
Property of Limits:
Many times to find the limit you can use direct
substitution.
These functions are continuous at x = c.
Example 1: Evaluating Basic Limits
Properties of Limits
Example 2: Limit of a Polynomial
Calculus
4.5
Integration
by
Substitution
Goal: Students will learn how to
use pattern recognition to find an
indefinite integral.
Integration by Substitution:
Back in derivatives
Use chain rule!
Now to integrate
Use integration by substitution!
Example 1:
Calculus
4.2 Area
Goal: Students will learn how to
approximate the area of a plane region.
Sigma Notation:
Use Sigma Notation to write and evaluate a sum.
Sigma Notation:
Use Sigma Notation to write and evaluate a sum.
Properties of Summation:
Example 1:
Increasing and
Calculus 3.3
Decreasing Functions and the
First Derivative Test
Goal: Students will determine the intervals on
which the function is increasing or decreasing and
use the First Derivative Test to find relative
Increasing & Decreasing:
Test I
Calculus AB
P.3: Functions and Their Graphs
Objective: Students will learn how to use
function notation and identify their graphs
Vocabulary
Implicit form:
Explicit Notation:
x2+2y=
1
y=(1x2)
Function Notation:
f(x) = (1x2)
Function:
Each domain element
Calculus
2.1 The
Derivative and
the Tangent
Line Problem
Goal: Students will find the slope of the tangent line to a
curve at a point and use the limit definition to find the
derivative of a function.
Definition of a Derivative
The derivative of a functi
Calculus AB
P.3: Functions and Their Graphs
Objective: Students will learn learn how to use
function notation and identify their graphs
Rigid TransformationsInside
Parent Graph:
When you add/subtract a number in the inside of
the parenthesis, you shift t
Calculus
1.4 Part IContinuity and
OneSided
Limits
Goal: Students will determine continuity at a
point and on an open interval and determine
onesided limits and continuity on a closed
interval
What does it mean to be continuous?:
In math, the term contin
Calculus
5.6 Inverse
Trigonometric
Functions
Goal: Students will learn how to
differentiate trigonometric inverse
functions.
Bases Other than e
Why do none of the six trig functions have an inverse function?
Trig functions are periodic.
Therefore they are
Calculus
4.6
Numerical
Integration
Goal: Students will learn how to
approximate definite integrals using the
Trapezoidal Rule and Simpsons Rule.
Approximation Techniques:
Some functions do not have antiderivatives
If the function does not have an antideri
Calculus 3.6 A Summary of
Curve Sketching
Goal: Students will learn how to analyze
and sketch the graph of a function.
Checklist:
x and yintercepts
symmetry
domain and range
continuity
asymptotes
Checklist, contd:
differentiability
critical numbers
Calculus
1.5 Infinite
Limits
Goal: Students will determine infinite limits from
the left and the right and sketch vertical
asymptotes of the graph of a function
Example 1:
Consider the function
Analytically:
Whats happening at x = 2?
Graphically:
Example
Calculus 3.7
Goal:
Optimimization
Students will learn how to solve
Problems
applied minimum and maximum
problems.
Key Phrases that you want to Optimize:
Greatest profit
Least cost
Least time
Greatest voltage
Optimization size
Greatest strength
Least dista
Calculus
2.5 Implicit
Differentiatio
n
Goal: Students will distinguish between functions
written in implicit form and explicit form and use implicit
differentiation to find the derivative of a function.
Explicit vs Implicit
Example 1
Rewrite in explicit f
Calculus
4.1
Antiderivatives &
Indefinite
Integration
Goal: Students will learn how to
understand antiderivatives and use of
basic integral notation.
Inverse Functions:
Function:
Inverse function:
Inverse Functions:
If
.
then
Example 1:
Find the general s
Calculus AB
P.2: Linear Models and Rates of Change
Objective: Students will learn how to identify
linear models and use rates of change.
Finding the slope of a line
The formula for slope of a line passing
through points (x1, y1) and (x2, y2) is:
Slope
A l
Calculus
2.3 Product and
Quotient Rules and
HigherOrder
Derivatives
Goal: Students will find the derivatives of
higherorder derivatives using the Product and
Quotient Rules
Product Rule
The first times the derivative of the second plus the
derivative of
Calculus 3.5 Limits at Infinity
Goal: Students will determine finite and
infinite limits at infinity and determine
the horizontal asymptotes of a funciton.
Limits at Infinity (End Behavior):
Back in Chapter 1:
Now:
Example 1:
Use the graphing utility to e
Calculus 3.9 Differentials
Goal: Students will learn how find the differential of
a function using differentiation formulas and
understand the concept of tangent line
approximation.
Tangent Line Approximation:
Find the tangent line approximation of
All ta
Calculus
5.5 Bases Other
Than e
Goal: Students will learn how to
differentiate and integrate functions
with bases other than e.
Bases Other than e
Need base e to differentiate & integrate but other bases
are still useful.
HalfLife Problems
where n is the
Calculus
1.1 A
Preview of
Calculus
What is Calculus?
Calculus is the mathematics of change
Precalculus:
Analyze constant
velocity
Analyze slope of a
line
Analyze tangent line
to a circle
Analyze the area of a
rectangle
Calculus:
Analyze velocity of an
ac
Calculus AB
P.1: Graphs and Models
Objective: Students will learn how to
identify graphs and their characteristics.
Example 1:
Complete the table. Then use the resulting points
to sketch the graph of the equation:
x
4
y
5
2
4
0
3
2
2
4
1





Inter
Concavity and the 2nd
Calculus 3.4
Derivative Test
Goal: Students will determine the intervals on
which the function is concave up and concave down
and use the 2nd Derivative Test to find relative
Definition of Concavity:
Test for Concavity:
Example 1:
De
Calculus
4.3 Reimanns
Sums &
Definite
Integrals
Goal: Students will learn how to understand
the definition of a Reimanns Sum and to
evaluate definite integrals.
Reimann Sums:
f is defined on [a, b]
is a partition of [a, b] given by
is called a Reimann Sum
Calculus 3.1 Extrema on
Interval
Goal: Students will understand the definition of
extrema and relative extrema and find the extrema
on a closed interval.
Overview
Absolute Maximum:
Relative (Local) Maximum:
Absolute Minimum:
Relative (Local) Minimum:
Defi
Calculus
6.1 Slope Fields
and Eulers
Method
Goal: Students will learn how use
initial conditions and slope fields
to solve differential equations.
Example 1: Verifying Solutions
Determine whether the function is a solution of the differential
equation y 
Calculus
5.7 Inverse
Trigonometric
FunctionsIntegrals
Goal: Students will learn how to
integrate trigonometric inverse
functions.
Integrals involving Inverse Trig Functions
Let u be a differentiable function of x, and let a > 0.
Example 1
Find the integra