MTH101 Final Term Mega File by GumNaaM Helpers

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TABLE OF CONTENTS :

Lesson 1 :Coordinates, Graphs, Lines Lesson 2 :Absolute Value Lesson 3 :Coordinate Planes and Graphs Lesson 4 :Lines Lesson 5 :Distance; Circles, Quadratic Equations Lesson 6 :Functions and Limits Lesson 7 :Operations on Functions Lesson 8 :Graphing Functions Lesson 9 :Limits (Intuitive Introduction) Lesson 10:Limits (Computational Techniques) Lesson 11: Limits (Rigorous Approach) Lesson 12 :Continuity Lesson 13 :Limits and Continuity of Trigonometric Functions Lesson 14 :Tangent Lines, Rates of Change Lesson 15 :The Derivative Lesson 16 :Techniques of Differentiation Lesson 17 :Derivatives of Trigonometric Function Lesson 18 :The chain Rule Lesson 19 :Implicit Differentiation Lesson 20 :Derivative of Logarithmic and Exponential Functions Lesson 21 :Applications of Differentiation Lesson 22 :Relative Extrema Lesson 23 :Maximum and Minimum Values of Functions Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem Lesson 25 :Integrations Lesson 26 :Integration by Substitution Lesson 27 :Sigma Notation Lesson 28 :Area as Limit Lesson 29 :Definite Integral Lesson 30 :First Fundamental Theorem of Calculus Lesson 31 :Evaluating Definite Integral by Subsitution Lesson 32 :Second Fundamental Theorem of Calculus Lesson 33 :Application of Definite Integral Lesson 34 :Volume by slicing; Disks and Washers Lesson 35 :Volume by Cylindrical Shells Lesson 36 :Length of Plane Curves Lesson 37 :Area of Surface of Revolution Lesson 38:Work and Definite Integral Lesson 39 :Improper Integral Lesson 40 :L’Hopital’s Rule Lesson 41 :Sequence Lesson 42 :Infinite Series Lesson 43 :Additional Convergence tests Lesson 44 :Alternating Series; Conditional Convergence Lesson 45 :Taylor and Maclaurin Series

Lecture # 23

Maximum and Minimum Values of Functions

Absolute Extrema Finding Absolute Extrema for a continuous function Summary of extreme behaviors of functions over

Problems involving finite closed intervals Problems involving intervals that are non-finite and closed

Absolute Extrema

Previously we talked about relative maxima, relative minima of functions. These were like the highest mountain and the deepest valley in a given vicinity or neighborhood. Now we will talk about absolute maximum and minimum values of functions. These are like the highest peak in a mountain range, and the deepest valley.

Absolute maximum means that the value is the maximum one over the entire domain of the function. Absolute minimum means that the value is the minimum one over the entire domain of the function. If we think of the Earth’s surface as defining some function, then its absolute maximum will be Mt. Everest, and absolute minimum will be the Marianna Trench in the Pacific Ocean near Hawaii.

Example

Consider the following picture of the graph of f (x) = 2x+1 on the interval [0,3). The minimum value is 1 at x = 0. But is there a maximum?