Lecture 2.1 Derivative - Chapter 2 Derivatives and...

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Chapter 2: Derivatives and Differentials Lecture 1: Lecture 1: Derivatives By Prof. Phan Quoc Khanh 11/09/16 Prof. Phan Quoc Khanh
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Outline 1. Tangent Lines and Rates of Changes 2. The Derivative, Differentials 3. Higher-Order Derivatives 4. Velocity and Acceleration 11/09/16 Prof. Phan Quoc Khanh
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1. Tangent Lines… Tangent points – the line just touches the curve Notice that a tangent line may also intersect the graph 11/09/16 Prof. Phan Quoc Khanh
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Rates of Change Average rate of change of f over the interval [ a , a + h ] ( ) ( ) f a h f a h Instantaneous rate of change of f at a point a Slope of the Tangent Slope of Secant 0 ( ) ( ) ( ) lim h f a h f a f a h 11/09/16 Prof. Phan Quoc Khanh
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Tangent Line and Secant Line , ( ) a f a , ( ) a h f a h secant line tangent line 11/09/16 Prof. Phan Quoc Khanh
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Secant Lines and Tangent The secant line PQ passing through two point P(a, f(a)) and Q(x, f(x)) has the slope 11/09/16 Prof. Phan Quoc Khanh ( ) ( ) PQ f x f a m x a P Q by letting Q P x a
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Tangent Line The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope 11/09/16 Prof. Phan Quoc Khanh ( ) ( ) lim x a f x f a m x a P Equation of tangent line: - If m : y m ( x a ) f ( a ) - If m   : x a (vertical line)
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Rates of Changes Let y be a quantity that depends on another quantity x. So, y=f(x) 11/09/16 Prof. Phan Quoc Khanh 1 2 2 1 2 1 2 1 2 1 If changes from to , then the change in (increment of ) is The corresponding change in is ( ) ( ) The difference quotient is called the average rate of change x x x x x x x x y y y y f x f x y x x x 1 2 of y with respect to x over the interval [ , ] x x
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Average Rate of Change Ex. Compute the average rate of change of 2 ( ) 3 2 f x x x over [0,2].
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