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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. ISE Academic Club 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. ISE Academic Club Example 1. Answer: 25600pi. mm^3/s ISE Academic Club Example 2. Answer: 140 cm^2/s. ISE Academic Club Example 3. ISE Academic Club ISE Academic Club Example 4. Answer: 21,39 km/h. ISE Academic Club Example 5. ISE Academic Club ISE Academic Club Example 6. Answer: 2mi/h. ISE Academic Club Homework Answer: 48 cm^2/s Answer: a) – 0,56 b) 1,79 ISE Academic Club Homework Answer: 55.4 km/h Answer: 65 mi/h ISE Academic Club 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. ISE Academic Club 3.2.1. Absolute max/min ISE Academic Club 3.2.2. Local maximum/minumum ISE Academic Club 3.3 The Extreme Value Theorem ISE Academic Club 3.4. Fermat's Theorem There may be an extreme value at c even when f'(c) does not exist ISE Academic Club 3.5. Critical number NOTE. critical numbers lies in the DOMAIN of the given function. ISE Academic Club 3.5. Critical number ISE Academic Club 3.5. Critical number x = 0 is a critical number of the function Example. Find the critical number of the function ISE Academic Club 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. ISE Academic Club 3.6. The Closed Interval Method EXAMPLE. ISE Academic Club Answer: x=600 y=1200 ISE Academic Club Homework Answer: T=3.96 ISE Academic Club 3.3 Mean Value Theorem, First Derivative Test ISE Academic Club Exercises. (Rolle's Theorem ) ISE Academic Club The Mean Value Theorem ISE Academic Club Example. ISE Academic Club Solution. ISE Academic Club Exercises. (Mean Value Theorem ) Answer: 23) 16 24) 25) No. ISE Academic Club Exercises. (Mean Value Theorem ) ISE Academic Club Increasing/Decreasing Test. ISE Academic Club Exercises. (Increasing/Decreasing Test ) ISE Academic Club The First Derivative Test. ISE Academic Club Exercises. (The First Derivative Test ) ISE Academic Club Exercises. (The First Derivative Test ) ISE Academic Club (The First Derivative Test ) Find the local maximum , local minimum of the function using the first derivative test ISE Academic Club upward downward ISE Academic Club Concavity Test. ISE Academic Club Inflection Point. ISE Academic Club The Second Derivative Test ISE Academic Club Example. ISE Academic Club Exercises. ISE Academic Club Exercises. ISE Academic Club Exercises. ISE Academic Club Exercises. ISE Academic Club Horizontal asymptote. Determine whether the following function has a horizontal asymptote or NOT ISE Academic Club Infinite Limits at Infinity ISE Academic Club Indeterminate forms. ISE Academic Club L'Hopital's Rule. ISE Academic Club Example. ISE Academic Club Indeterminate products. ISE Academic Club Indeterminate differences. ISE Academic Club 3.6 newton’s method ISE Academic Club Exercises. ISE Academic Club 3.7 Anti-derivatives and Indefinite Integrals ISE Academic Club ISE Academic Club Exercises. ISE Academic Club REVIEW Chapter 3 ISE Academic Club Exercise 1. ISE Academic Club Exercise 2. ISE Academic Club Exercise 3. ISE Academic Club Exercise 3. ISE Academic Club 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 ISE Academic Club 4.1 Areas under curves ISE Academic Club Definite Integrals. ISE Academic Club Riemann Sum ISE Academic Club Example. ISE Academic Club Area under a curve. ISE Academic Club Exercise. ISE Academic Club Properties of the Integral. ISE Academic Club Properties of the Integral. ISE Academic Club Homework. ISE Academic Club Comparison Properties. ISE Academic Club Comparison Properties. Given the following definite integral Prove that. ISE Academic Club The Fundamental Theorem of Calculus - Part 1. ISE Academic Club Example. (basic) ISE Academic Club Example. (advanced) ISE Academic Club The Fundamental Theorem of Calculus - Part 2. ISE Academic Club Examples. ISE Academic Club Indefinite Integrals ISE Academic Club Indefinite Integrals ISE Academic Club Exercises. ISE Academic Club The net change theorem ISE Academic Club Applications. Examples. ISE Academic Club Applications. ISE Academic Club Applications. ISE Academic Club Applications. Examples. ISE Academic Club Applications. ISE Academic Club Example. ISE Academic Club Homework ISE Academic Club Homework ISE Academic Club Substitution Rule ISE Academic Club Homework ISE Academic Club Substitution Rule for Definite Integral ISE Academic Club Homework ISE Academic Club Symmetric Functions Example. Evaluate the following integrals ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club note-taking ISE Academic Club ISE Academic Club = ISE Academic Club Trigonometric substitution ISE Academic Club Trigonometric substitution ISE Academic Club Trigonometric substitution ISE Academic Club Trigonometric substitution ISE Academic Club Trigonometric substitution ISE Academic Club Trigonometric substitution ISE Academic Club note-taking ISE Academic Club Integration of Rational Functions Case 1. ISE Academic Club Integration of Rational Functions Case 1. ISE Academic Club Integration of Rational Functions ISE Academic Club Integration of Rational Functions Find the antiderivative of the following function ISE Academic Club Integration of Rational Functions ISE Academic Club Integration of Rational Functions ISE Academic Club Intergration by Parts ISE Academic Club Intergration by Parts ISE Academic Club ISE Academic Club ISE Academic Club Midpoint Rule ISE Academic Club Error of Midpoint Rule ISE Academic Club Trapezoidal Rule ISE Academic Club Error of Trapezoidal Rule ISE Academic Club Simson Rule ISE Academic Club Improper Integrals - Type 1 ISE Academic Club Improper Integrals - Type 1 ISE Academic Club Improper Integrals - Type 1 ISE Academic Club Improper Integrals - Type 1 ISE Academic Club Improper Integrals - Type 2 ISE Academic Club Improper Integrals - Type 2 ISE Academic Club Improper Integrals - Type 2 ISE Academic Club Improper Integrals - Type 2 ISE Academic Club ISE Academic Club Mean value theorem or ISE Academic Club Chapter 5. Application of Integration ISE Academic Club Area between curves. ISE Academic Club Area between curves. ISE Academic Club Area between curves. ISE Academic Club Area between curves. ISE Academic Club Area between curves. ISE Academic Club Volume. ISE Academic Club Volume by Cylindrical Shells ISE Academic Club Volume by Cylindrical Shells ISE Academic Club Average value of a function. ISE Academic Club Average value of a function. ISE Academic Club Arc Length. ISE Academic Club Arc Length. ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club 4.5 Approximation of integration ISE Academic Club ISE Academic Club ISE Academic Club 4.6 Improper integrals Type 1 ( Infinite Intervals) ISE Academic Club Type 2 ISE Academic Club ISE Academic Club 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 ISE Academic Club 5.1 Areas between curves ISE Academic Club 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 ISE Academic Club ISE Academic Club 5.2 Areas enclosed by parametric curves ISE Academic Club 5.3 Volume of a solid ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club ISE Academic Club 5.4 Net change theorem ISE Academic Club ISE Academic Club 5.5 Lengths of curves ISE Academic Club ISE Academic Club 5.6 Average value of a fumction ISE Academic Club 5.7 Applications to engineering and economics ISE Academic Club ISE Academic Club • Consumer surplus ISE Academic Club ISE Academic Club