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OUTLINE
1.
2.
3.
4.
5.
6.
CHAPTER 1:
Function and graph
Compositions of functions
Limit
Continuity
The intermediate value theorem
Others types of exercises
6. 1 Odd and even function
6. 2 Vertical line test
6. 3 Inverse function
1. Function and graph
2. Compositions of functions
6. Find f(g(x)):
2. Compositions of functions
3. Limit
3. Limit
3. Limit
Example (Mid Sem 1 2014-2015)
Example (Mid Sem 1 2016-2017)
4.
Continuity
Example (Mid Sem 2 2015-2016)
5. The intermediate value theorem
Example (Mid Sem 2 2015-2016)
6. Other types
6.1 Odd and even function
6. Other types
6.1 Odd and even function
6. Other types
6.2 Vertical line test
6. Other types
6.3 Inverse function
CHAP 2:
2. Differentiability
3. Rules
Chain rule of differentiation
4. Implicit differentiation
Use
4. Tangent line of y=f(x) at (x0,y0)
5. Average and Instantaneous
6. Logarithmic differentiation
Method:
1. Take natural logarithms of both sides of an equation y = f(x) and use the
Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y′.
7. Linear Approximation and differentials
1. The side of the cube are found to be 10cm in length with a possible error of
no more than 2cm. What is the maximum possible error in the volume of the
cube, and find the percentage error
Content
• Chapter 3: Application of differentiation.
• Chapter 4: Integration.
• Chapter 5: Application of integration.
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Chapter 3: Application of differentation
1. Related rates.
2. Maxima, Minima and Optimization Problems.
3. Mean Value Theorem, First Derivative Test, Concavity and Shape of
Curves.
4. Curves Sketching.
5. Indefinite Form and L’ Hospital’s Rule.
6. Newton’s Method.
7. Anti-derivatives and Indefinite Integrals.
3.1 Related rates
• Goals: calculating the unknown rate of change by relating it to other rates
of change which are known.
• Strategy:
1. draw a picture (if possible) of the situation, label all relevant variables.
2. identify all the variables and rates of change given.
3. identify which factor you are going to determine.
4. write an equation involving the changing variables, including the function
whose rate you are going to find.
5. apply implicit differentiation to the equation, substitute all the known
numbers to find the desired factor.
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Example 1.
Answer: 25600pi. mm^3/s
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Example 2.
Answer: 140 cm^2/s.
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Example 3.
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Example 4.
Answer: 21,39 km/h.
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Example 5.
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Example 6.
Answer: 2mi/h.
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Homework
Answer: 48 cm^2/s
Answer:
a) – 0,56
b) 1,79
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Homework
Answer: 55.4 km/h
Answer: 65 mi/h
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3.2 Maxima, Minima and Optimization
Problems
• Optimization problems require us to find the optimal (best) way of
doing something.
• These problems can be reduced to finding the maximum or
minimum vakue of a function.
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3.2.1. Absolute max/min
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3.2.2. Local maximum/minumum
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3.3 The Extreme Value Theorem
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3.4. Fermat's Theorem
There may be an extreme value at c even when f'(c) does not
exist
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3.5. Critical number
NOTE. critical numbers lies in the DOMAIN of the given function.
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3.5. Critical number
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3.5. Critical number
x = 0 is a critical
number of the
function
Example. Find the
critical number of the
function
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Homework.
Find the critical numbers of the following functions.
Answer:
33. 0
34. 4/3
35. 0;2
36. 1- căn 5; 1+ căn 5
42. +- 1; 0.
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3.6. The Closed Interval Method
EXAMPLE.
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Answer: x=600 y=1200
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Homework
Answer: T=3.96
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3.3 Mean Value Theorem, First Derivative Test
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Exercises. (Rolle's Theorem )
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The Mean Value Theorem
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Example.
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Solution.
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Exercises. (Mean Value Theorem )
Answer:
23) 16
24)
25) No.
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Exercises. (Mean Value Theorem )
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Increasing/Decreasing Test.
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Exercises. (Increasing/Decreasing Test )
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The First Derivative Test.
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Exercises. (The First Derivative Test )
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Exercises. (The First Derivative Test )
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(The First Derivative Test )
Find the local maximum , local minimum of the function using the first derivative test
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upward
downward
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Concavity Test.
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Inflection Point.
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The Second Derivative Test
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Example.
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Exercises.
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Exercises.
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Exercises.
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Exercises.
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Horizontal asymptote.
Determine whether the following function has a horizontal asymptote or NOT
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Infinite Limits at Infinity
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Indeterminate forms.
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L'Hopital's Rule.
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Example.
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Indeterminate products.
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Indeterminate differences.
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3.6 newton’s method
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Exercises.
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3.7 Anti-derivatives and Indefinite Integrals
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Exercises.
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REVIEW Chapter 3
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Exercise 1.
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Exercise 2.
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Exercise 3.
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Exercise 3.
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Chapter 4: Content
1. Areas under Curves
2. The definite integral and its properties
3. The fundamental theorem of calculus
4. Techniques of integration
5. Approximate integration
6. Improper integrals
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4.1 Areas under curves
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Definite Integrals.
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Riemann Sum
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Example.
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Area under a curve.
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Exercise.
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Properties of the Integral.
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Properties of the Integral.
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Homework.
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Comparison Properties.
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Comparison Properties.
Given the following definite integral
Prove that.
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The Fundamental Theorem of Calculus - Part 1.
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Example. (basic)
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Example. (advanced)
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The Fundamental Theorem of Calculus - Part
2.
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Examples.
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Indefinite Integrals
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Indefinite Integrals
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Exercises.
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The net change theorem
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Applications.
Examples.
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Applications.
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Applications.
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Applications.
Examples.
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Applications.
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Example.
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Homework
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Homework
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Substitution Rule
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Homework
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Substitution Rule for Definite Integral
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Homework
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Symmetric Functions
Example. Evaluate the following integrals
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note-taking
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=
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Trigonometric substitution
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Trigonometric substitution
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Trigonometric substitution
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Trigonometric substitution
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Trigonometric substitution
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Trigonometric substitution
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note-taking
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Integration of Rational Functions
Case 1.
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Integration of Rational Functions
Case 1.
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Integration of Rational Functions
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Integration of Rational Functions
Find the antiderivative of the following function
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Integration of Rational Functions
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Integration of Rational Functions
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Intergration by Parts
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Intergration by Parts
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Midpoint Rule
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Error of Midpoint Rule
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Trapezoidal Rule
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Error of Trapezoidal Rule
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Simson Rule
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Improper Integrals - Type 1
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Improper Integrals - Type 1
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Improper Integrals - Type 1
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Improper Integrals - Type 1
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Improper Integrals - Type 2
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Improper Integrals - Type 2
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Improper Integrals - Type 2
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Improper Integrals - Type 2
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Mean value theorem
or
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Chapter 5.
Application of Integration
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Area between curves.
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Area between curves.
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Area between curves.
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Area between curves.
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Area between curves.
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Volume.
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Volume by Cylindrical Shells
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Volume by Cylindrical Shells
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Average value of a function.
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Average value of a function.
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Arc Length.
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Arc Length.
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4.5 Approximation of integration
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4.6 Improper integrals
Type 1 ( Infinite Intervals)
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Type 2
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Chapter 5: Application of Integration
1. Area between Curves
2. Areas Enclosed by Parametric Curves
3. Volume of a solid
4. Net change theorem
5. Lengths of curves
6. The Average Value of a Function
7. Application to Engineering, Economics and Science
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5.1 Areas between curves
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Steps to slove
• Draw graphs
• Find intersections between f(x) and g(x)
• Find out f(x) > g(x) or g(x) > f(x)
• Write and evaluate the value of the integral
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5.2 Areas enclosed by parametric curves
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5.3 Volume of a solid
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5.4 Net change theorem
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5.5 Lengths of curves
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5.6 Average value of a fumction
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5.7 Applications to engineering and
economics
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• Consumer surplus
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