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1 Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point Pa,fa can now be defined as te line passing troug te point P wit slope f a. 1. Consider te grap of te function g sown below. - - Place te following quantities in increasing order: g, g 1, g 0, g 1, g. If fx x x, find f and use it to find te equation of te line tangent to te grap of f at x. First find f using te first definition given above f f f Tus te slope of te tangent line is. Since f, te line passes troug te point,. Tus te equation of te tangent line is y x y x 9 y x 7 Below is a grapical confirmation. Te function f is in red and te tangent line at x isingreen. y tangent line x. We can interpret f a as te rate of cange of te function f at a. 1
2 For example, suppose N ft represents te number of elk living in te Smoky Mountain National Park t years after 1999? a. Wat is te meaning of f t? b. Wat are te units of f t? c. Wat does te statement f 5 mean? d. Wat is appening if f t 0?. Te function D gv represents te force of drag (in pounds) on an aircraft traveling at a speed of v miles per our. a. Wat is te meaning of g v? b. Wat are te units of g v? c. Wat does te statement g mean? d. Wat does te statement g 600 mean? Te derivative as a function: Te derivative of f is te function f x fx fx for all x for wic te it exists. Te notation f x is read "f prime of x". Notice te difference quotient in te it expression. Oter notation & terminology: Note tat te derivative "outputs" te slope of f at te point x,fx Te process of finding te derivative is called differentiation A function is differentiable at x if its derivative exists at x. A function is differentiable on an open interval a,b if it is differentiable at every point in a,b Notations used for te derivative: f x, dx, D xy, y, d fx, D dx x fx Te symbols dx d and D are called differential operators and sould be tougt of as functions wose domain and range are also functions. Te notation is read as "te derivative of y wit respect to dx x" (we say "dee-y dee-x"). Tis is te notation used by Leibnitz wen e presented te calculus in 168. Because of tis it is often referred to as te Leibnitz notation. For te function y fx, te idea of te Leibnitz notation is dx Δy Δ Δx fx Δx fx f Δ Δx x Wen we want to state te value of a derivative using te Leibnitz notation at a specific number a we will write dx xa Examples: 1. Find te derivative of fx x x
3 f x fx fx x x x x x x x x x x x x 0 x x Recall tat f x is te slope of f at x,fx. Compare te graps of f and f and discuss tis relationsip.. Find te derivative of y t dt ft ft t t t t t t t t t t t t t t 1 t t 1 t t 1 t Find te slope of te tangent line at te points,1 and 6, 1 dt t dt t Is tis function differentiable at te point,0? Interpret tis grapically in terms of te tangent line at tis point.. Find te derivative of gx x g x gx gx x x x x xx xx x x xx xx xx xx 0 x x x xx Grap g and g and discuss teir relationsip.. A grapical example. Consider te grap of te function f sown below. Use it to sketc a grap of f ontesameaxes.
4 - - We first note were tere appear to be orizontal tangent lines. At tese points te derivative must be zero (wy?). Below te locations of orizontal tangent lines are sown - - Tese orizontal tangent lines appear to occur at about x andx. Tus f f 0. So tat te grap of f as x-intercepts of,0 and,0. Note furter tat between x and x te slope of te tangent line is always negative (reacing its maximum negative value at about x 1 ). Tus f x 0 over te interval,. Alsoforx and for x, te slope of te tangent line is always positive so tat f x 0 over te intervals, and,. Putting all tis togeter, a sketc of te grap of f is sown below in green. f' - - f 5. Anoter grapical example. Below is te grap of y gx. Use it to estimate te following values: g, g, g 0, g
5 Derivatives from te left and from te rigt. Te one-sided it fx fa x a x a is called te derivative from te left of f. and te one-sided it x a fx fa x a is called te derivative from te rigt of f. Note tat f is differentiable at a if and only if te derivative from te rigt and from te left bot exist and are equal. Recall te greatest integer function fx x. Remember tat tis function is not continuous at any integer. Let s look at te derivative from te left and from te rigt at x 0 for tis function. Te derivative from te left is fx f0 x 0 x x 0 x x Since x 1 for 1 x 0, so tat as x gets close to zero from te left we ave 1 being divided by a smaller & smaller negative number. Te derivative from te rigt is fx f0 x 0 x 0 x x x 0 Since x 0 for 0 x 1, so tat as x gets close to zero from te rigt we ave 0 being divided by a smaller & smaller positive number. Tus, fx x is NOT differentiable at x 0(i.e.f 0 DNE). In fact, tis same argument can sow tat fx x is not differentiable at all integers. 5
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