Application od derivatives

Application of Derivatives : Application of Derivatives By Malay

What will we learn? : What will we learn? Introduction Application in physical problems Application in functions Increasing, decreasing, strictly increasing and strictly decreasing functions 2

Introduction : Introduction dy/dx means rate of change of y with respect to x. Differentiation gives instantaneous rate of change. dy/dx is positive if y increases as x increases and is negative if y decreases as x increases. 3

Application in physical problems : Application in physical problems Q: Find the rate of change of the area of a circle with respect to its radius r when r = 3. Q: The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2cm/minute. When x =10cm and y = 6cm, find the rates of change of the perimeter. Q: A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate. 4

Application in functions : Application in functions 5

Different types of function : Different types of function Let I be an open interval contained in the domain of a real valued function f. Then f is said to be Increasing on I if x1 < x2 in I ? f (x1) = f (x2) for all x1, x2 ? I. Strictly increasing on I if x1 < x2 in I ? f (x1) < f (x2) for all x1, x2 ? I. Decreasing on I if x1 < x2 in I ? f (x1) = f (x2) for all x1, x2 ? I. Strictly decreasing on I if x1 < x2 in I ? f (x1) > f (x2) for all x1, x2 ? I. (a,b) – Open interval a and b means all the points between a and b excluding a and b. [a,b] – Close interval a and b means all the points between a and b including a and b. 6

First derivative method : First derivative method Let f be continuous on [a, b] and differentiable on the open interval (a,b). Then f is increasing in [a,b] if f '(x) > 0 for each x ? (a, b) Proof (a) Let x1, x2 ? [a, b] be such that x1 < x2. Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c between x1 and x2 such that f (x2) – f (x1) = f '(c) (x2 – x1) i.e. f (x2) – f (x1) > 0 (as f '(c) > 0 (given)) i.e. f (x2) > f (x1) Thus, we have x1

Example : Example Q: Show that the function f given by f (x) = x3 – 3x2 + 4x, x ? R is strictly increasing on R. Q: Find the intervals in which the function f given by f (x) = x2 – 4x + 6 is (a) strictly increasing (b) strictly decreasing 8

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