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UMD MATH 220 - The Derivative

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Calculus 220 section 1 3 The Derivative notes by Tim Pilachowski We now take the idea of section 1 2 the slope of a curve at a point P and treat it more formally Given a function f x the first derivative of f is a formula which provides the slope of the curve at any point on f Note that the derivative of a function is itself a function The process of finding a derivative is called differentiation dy d The first derivative of f has several notations f f x and f x dx dx 2 2 i e f 2 Example A Given f x x 3 8 x 2 estimate the derivative evaluated at x 2 3 3 2 2 to other points on f The pictures below show a series of secant lines connecting the point 2 f 2 3 3 2 2 the secant lines approach the tangent line If we As the second point approaches 2 f 2 3 3 continued to zoom in the secant lines would come closer and closer to horizontal So our estimate for 2 is 0 f 2 3 2 Example A extended Given f x x 3 8 x 2 estimate f 2 3 While the visual sequence above seems to point to a particular result it is far from accurate A point misplaced by even a little bit will change the picture greatly We can use decimal approximations to gain a little more 2 to place the second point on the precision In the table below h will indicate the amount added to x 2 3 curve of f The slope of the secant line thus formed is calculated using the familiar linear slope formula 2 2 f 2 2 h f 2 2 f 2 h f 2 3 3 3 3 y m x h 2 2 2 2 h 3 3 f 2 2 3 10 709296863 10 709296863 10 709296863 10 709296863 10 709296863 10 709296863 10 709296863 10 709296863 10 709296863 h 1 0 1 0 01 0 001 0 0001 0 00001 0 000001 0 0000001 0 00000001 2 2 3 h f 2 2 3 h secant slope 0 632993162 1 532993162 1 622993162 1 631993162 1 632893162 1 632983162 1 632992162 1 632993062 1 632993152 6 810317378 10 661307068 10 708807965 10 709291965 10 709296814 10 709296863 10 709296863 10 709296863 10 709296863 3 898979486 0 479897949 0 048889795 0 004897979 0 000489888 0 000048990 0 000004899 0 000000497 0 000000000 2 2 the slope of So with an accuracy to nine decimal places as the second point approaches 2 f 2 3 3 2 is 0 the secant lines approaches 0 So our estimate for f 2 3 Neither of the above methods is sufficient mathematically because even with the added precision of decimals we cannot be sure that our answer is exact as opposed to being extremely close Also neither method is practical to reproduce for every point on the curve However notice first of all that the slope of each sequential f x h f x secant line is the difference quotient As h approaches 0 sequence of secant lines approaches h the tangent line and the sequence of slopes approaches the slope of the tangent 2 Example A one last time Given f x x 3 8 x 2 derive a formula for f x then calculate f 2 3 Answers In mathematical notation Given f x x 3 8 x 2 lim h 0 Example B Given f x x 2 find f x h f x 3x 2 8 0 h d 2 x Answer 2x dx We can go through a similar process for any function of the form f x x r where r could be any number Specifically we d find that f x r x r 1 Such a function is called a power function and this property of derivatives is called the power rule Example C Given y x find dy Answer 1 dx Note that the answer to Example C fits with what we already know about lines in slope intercept form Example D Given y 5 find f x Answer 0 Note that the answer to Example D fits with what we already know about horizontal lines Example E Given f x x find f x Answer Example F Given f x 1 x 3 3 Example G Given f x x find f x Answer x 2 1 2 x 3 x4 5 find f x Answer 3x 8 3 Some final notions about derivatives and differentiability In line with the concept of slope of the curve at a point being equal to the slope of the tangent line at that point f x 0 implies that f is increasing Likewise f x 0 implies that f is decreasing However the implication does not go the other way The fact that a function is increasing does not imply that f x 0 The simplest example is y x 3 Although x 3 is increasing everywhere on its domain its derivative is not always positive d 3 The curve momentarily levels out precisely when x 0 then continues upward At this point x 0 A dx similar limitation exists for decreasing functions Another important concept for differentiability of functions is the idea of being continuous Specifically the existence of a derivative at a point implies that the function is continuous at that point i e its graph is not disconnected pieces However the implication does not go the other way A function may be continuous but may not be differentiable at all points Two examples are shown below On the left f x 25 x 2 is defined and has values at x 5 and x 5 but the derivative does not exist since the tangent is a vertical line whose slope is undefined On the right f x x 5 is defined and continuous for all real numbers but the derivative does not exist at x 0 because the tangent line on the left has a positive slope while the tangent line on the right has a negative slope


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