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DIFFERENTIAL CALCULUS, , Monday, October 18, 2021 9:22 PM, , Differential Calculus, , Using the concept of ‘differential coefficient’ or ‘derivative’, we can easily define velocity and, acceleration. Though you will learn in detail in mathematics about derivatives, we shall introduce, this concept in brief in this Appendix so as to facilitate its use in describing physical quantities, involved in motion., , Suppose we have a quantity y whose value depends upon a single variable x, and is expressed, by an equation defining y as some specific function of x. This is represented as:, , y=f9 (1), , This relationship can be visualised by drawing a graph of function y = f(x) regarding y and xas, Cartesian coordinates, as shown in Fig. 3.30 (a)., , , , , , , , , , , , x =f ,, y y = f(x) y, Y| PRIN 5 x2 0:00 gl, ro:, 0 x XtAX x” 0 5d xe, (a) (b), , Consider the point P on the curve y = f(x) whose coordinates are (x, y) and another point Q, where coordinates are (x + Ax, y + Ay). The slope of the line joining P and Q is given by:

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——, Q, , Fig. 1.7, , Then from the graph we can see that at maximum or minimum valu of slope o tothe graph is, , zero. Thus,, , lS, , #. 0 at maximum or minimum value of y., dx, , j PT, fy is maximum if ——, By putting * = Owe will get different values of x. At these values of x, value of Y ae, . os wh F', (double differentiation of y with respect to x) is negative at this value of x. Similarly y is minimum —s is, , 2, Le = ~ve for maximum value of y, , de, , 2, and ‘ =+ve for minimum value of y, dx, , Note That at constant value ofy also = 0 but in this case 12 is zero., , dx?, , Sample Example 1.4 Find maximum or minimum values of the functions, (@) y=25x? + 5-10 () y=9-(x-3), , Salution fa\ Rew marion ond mintension satan sian nan ans Y, , a

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Serre ey mame cone rE YarUS, WE CaN pul — =U, , Y Lsoe-10=0 :. x=!, or 1080 # ems, , 2, Further, 4 ¥ 659, oe, @ ws 1 oh, or cat Pov value at Tees y as minimum vale t= +. Subsiing x= in gv, equation, we get, 1, Yin =25(2) +5-10{ 24, ) y=9-(2-3) 29-3? 94 6, or y=Gr~x?, 9-6-2, , For minimum or maximum value of y we will substitute 2” 9, dy, , or 6-2r=0, or v=?, , 2, To check whether value of y is maximum or minimum at x =3we will have to check whether “2 is, dx, , positive or negative., 2, i, de, 2, , d*y. ‘, or <r is eeative at x=3 Hence, value of y is maximum. This maximum value of yi,, , Ymax =9-(3-3) =9, , Let us Find a Derivative! ( ay,, , To find the derivative of a function y = f(x) we use the slope formula:, , Ax —0, A—41 0, , X, 44), , A, , My nh