Trigonometric Functions
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Trigonometric Functions
1 / 26
Topics for the 4th Exam
1
Trigonometric Functions - Derivatives and Antiderivatives
2
Logarithmic and Exponential Functions
Rules for differentiation
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Rules for differentiation
1 / 16
Recall
The derivative of a function f (x ) at a x = x0 is given by
f (x + x0 ) f (x0 )
.
x
x 0
f (x ) = lim
Math 100 (Inst.
The Real Number System and Inequalities
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
The Real Number System and Inequalities
1 / 25
Outline
1
The Real Number System
2
Inequalities and the Real Number Line
The Completeness A
Rational Inequalities and the Absolute Value
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Rational Inequalities and the Absolute Value
1 / 34
Outline
1
Rational Inequalities
2
The Absolute Value of a Number
Equations and In
Optimization Problems
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Optimization Problems
1 / 15
Example
Suppose a company has determined that its total revenue R for a
particular product is given by R = x 3 + 450x 2 + 52500x wh
Optimization Problems
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Optimization Problems
1 / 16
Examples
Suppose a company has determined that its total revenue R for a
particular product is given by R = x 3 + 450x 2 + 52500x w
One-Sided and Innite Limits
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
One-Sided and Innite Limits
1 / 18
Outline
1
One-Sided Limits
2
Lecture Exercises
Math 100 (Institute of Mathematics)
One-Sided and Innite Limits
2 /
Lines and the Derivative
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Lines and the Derivative
1 / 24
Outline
1
Lines
Equations of a Line
Parallel and Perpendicular Lines
2
The Derivative
Limit Denition of the Derivative
Findin
Limit of a Function and Limit Theorems
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Limit and Limit Theorems
1 / 25
Outline
1
Operations on Functions
2
Limit of a Function
3
Limit Theorems
Math 100 (Institute of Mathematics
Integrals Yielding the Natural Logarithmic Function
and Logarithmic Differentiation
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
1 / 17
Outline
1
Integrals Yielding the Natural Logarithmic Function
2
Logarithmic Differentiation
Innite Limits and Limits at Innity
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Innite Limits and Limits at Innity
1 / 29
Outline
1
Innite Limits
2
Limits at Innity
3
Lecture Exercises
Math 100 (Institute of Mathematics)
In
Implicit Differentiation and Equations of Tangent and
Normal Lines
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Implicit Differentiation and Equations of Tangent and Normal Lines
1 / 21
Example
Given xy = 1. Find
Math 100 (Inst
The Two-Dimensional Coordinate System,
Functions and their Graphs
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
2D System and Functions
1 / 47
Outline
1
Functions and their Graphs
Graphs of Relations and Functions
Polynomial
Exponential Growth and Decay
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Exponential Growth and Decay
1 / 16
Exponential Growth and Decay
There are many natural phenomena in which the rate of increase or
decrease of a certain
Derivative of Exponential Functions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Derivative of Exponential Functions
1 / 24
The Exponential Function
We recall the exponential function and some of its properties.
Denition (Expon
Differential Equations and Applications
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Differential Equations and Applications
1 / 22
Outline
1
Differential Equations
Separable Differential Equations
Particular Solution of a Diff
Derivative Tests
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Derivative Tests
1 / 22
Outline
1
Increasing and Decreasing functions
2
Relative and Absolute Extrema
Math 100 (Inst. of Mathematics)
Derivative Tests
2 / 22
Intuiti
Derivative of Logarithmic Function
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Logarithmic Functions
1 / 17
Outline
1
The Logarithmic Function
2
Derivative of Logarithmic Functions
3
Examples
Math 100 (Institute of Mathema
Denite Integrals
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Denite Integrals
1 / 20
Outline
1
Fundamental Theorem of Calculus
2
Properties of Denite Integral
Math 100 (Institute of Mathematics)
Denite Integrals
2 / 20
Mat
Continuity of Functions
Mathematics 100
Institute of Mathematics
Math 100 (Institute of Mathematics)
Continuity
1 / 21
Outline
1
Continuity of Functions
2
Lecture Exercises
Math 100 (Institute of Mathematics)
Continuity
2 / 21
Continuity
Math 100 (Institu
Areas of Plane Regions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Areas of Plane Regions
1 / 17
Outline
1
Finding the Area of a Bounded Region
2
Area Between Two Curves
Math 100 (Inst. of Mathematics)
Areas of Plane Regions
2
Antiderivatives
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Antiderivatives
1 / 25
Outline
1
Antiderivatives
2
Basic Integration Rules
3
The General Power Rule/Substitution Rule
Math 100 (Inst. of Mathematics)
Antiderivatives