Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

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1 Mat Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit equation y= f ( x) at te point a, f( a ) to be. We also saw tat te velocity of an object wit position function s f ( t) = at time t = a is In fact, limits of te form ( + ) ( ) 0 f lim a f a arise wenever we calculate a rate of cange in any of te sciences or engineering, suc as a rate of reaction in cemistry or a marginal cost in economics. Since tis type of limit occurs so widely, it is given a special name and notation. Definition 1 Te derivative of a function f at x a, is =, denoted by f ( a) f if tis limit exists. ( + ) ( ) ( a) = lim (1) or f ( a) f a f a 0 ( ) ( ) f x f a = lim x a x a () Notes: ( ) f ( a) is te slope of te tangent to te curve y= f ( x) at te point a, f( a ) f ( a) is te instantaneous rate of cange of y f ( x) If we sketc te curve y f ( x) = wit respect to x wen x= a =, wen te derivative is large ( and terefore te curve is steep), te y- values cange rapidly. Wen te derivative is small, te curve is relatively flat, and te y-values cange slowly. In particular, if s= f ( t) is te position of a particle tat moves along a straigt line, ten f ( a) of cange of te displacement s wit respect to te time t. In oter words, f ( a) is te rate is te velocity of te particle at time t = a. Te speed of te particle is te absolute value of te velocity. 1

2 Exercise 1 Find te derivative of te function ( ) 3 f t t t = + at te number a. Te Derivative as a Function Now we cange our point of view and let te number a vary. If we replace a in te equation (1) above we obtain ( x) ( + ) ( ) f = (3) 0 f lim x f x. So we can regard f as a new For any number x for wic tis limits exists, we assign to x te number f ( x) function, called te derivative of f and defined by equation (3). Te value of f at x, f ( x) geometrically as te slope of te tangent to te grap of f at te point x, f( x ). ( ), can be interpreted Notes: Te function f is called te derivative of f because it as been derived from f by te limiting operation in equation (3). { } Te domain of f is te set x f ( x) If f ( x) R and it may be smaller tan te domain of f. exists, we say tat f as a derivative at x or tat f is differentiable at x. If f exists at any x in te domain of f, we say tat f is differentiable. Te process of calculating a derivative is called differentiation. Exercise Differentiate f ( x) x =. x 1

3 . 3 Exercise 3 a) If f ( x) = x x, find a formula for f ( x) b) Illustrate by comparing te graps of f and f. Exercise 4 If f ( x) = x+ 5, find te derivative of f. State te domain of f. Differentiable on an Interval; One -sided Derivatives Definition 1) A function f is differentiable on an open interval ( ab, ) [ or ( a, ) or (,a) or (, ) if and only if f ( x) exists for any x in te interval. ] ) A function f is differentiable on a closed interval [ ab, ] if and only if it is differentiable on ( ab, ) and if te rigt-and derivative at a and te left-and derivative at b exist. f ( ) ( ) ( ) lim f a+ f a a = f ( b) = lim _ + 0 ( + ) ( ) f b f b Exercise 5 Were is te function f ( x) = x differentiable? Give a formula for f. Illustrate wit te graps of f and f. 3

4 Exercise 6 Matc te functions graped in #7 30 wit te derivatives graped in te accompanying figures a) d). 4

5 Exercise 7 a) Te grap of te accompanying figure is made of line segments joined end to end. At wic points of te interval ' [ 4,6] is f not defined? Give reasons for your answer. b) Grap te derivative of f. Exercise 8 Compare te rigt-and and left-and derivatives to sow tat te functions are not differentiable at te point P. 5

6 Oter notations If we use te traditional notation y= f ( x) to indicate tat te independent variable is x and te dependent variable is y, ten some common alternative notations for te derivative are as follows: dy df d f x y f x Df x Df x x dx dx dx ( ) = = = = ( ) = ( ) = ( ) Te symbols D and d/ dx are called differentiation operators because tey indicate te operation of differentiation, wic is te process of calculating a derivative. Te symbol dy/ dx, wic was introduced by Leibniz, sould not be regarded as a ratio ( for te time being); it is a. synonym for f ( x) If we want to indicate te value of a derivative dy/ dx in Leibniz notation at a specific number a, we use te notation dy wic is a synonym for f ( a) dx = x a. Teorem If a function is differentiable at x= a, ten f is continuous at x= a. Proof Notes 1) Te converse is false: a function could be continuous, but not differentiable )Te contrapositive is always true: if a function is not continuous at x = a,ten it is not differentiable at x= a 6

7 In conclusion, wen does a function not ave a derivative at a point? 1) a corner, were te one-sided derivatives differ ) a cusp, were te slope of te tangent approaces from one side and from te oter 3) a vertical tangent, were te slope of te tangent approaces from bot sides or approaces from bot sides 4) a discontinuity Higer Derivatives If f is a differentiable function, ten its derivative f is also a function, so f may ave a derivative of its own, denoted by ( f ) = f. Tis new function f is called te second derivative of f. d dy d y Using Leibniz notation, we write dx = dx. dx Exercise 9 Let g( x) 1 4 ( 3, ). = + x. Differentiate te function. Ten find an equation of te tangent line at Exercise 10 Te figure sows te grap of a function over a closed interval. At wat points does te function appear to be: a) differentiable? b) Continuous but not differentiable? c) Neiter continuous nor differentiable. Exercise 11 Does any tangent to te curve y = x cross te x-axis at x = 1? If so, find an equation for te line and point of tangency. If not, wy not? Exercise 1 Sow tat f ( x) x sin, x 0 = x 0, x = 0 1 is differentiable at 0 x = and find ( 0) f. Exercise 13 a) If g( x) = x 3, sow tat g '0 ( ) does not exist. b) If a 0, find g' ( a ). c) Sow tat te grap as a vertical tangent line at ( 0,0 ) 7

8 Answers:1) a + ; ) ( ) x 1 ; 3) 3x 1; 4) 1 x + 5 ; 5) all x except 0; 6) 7 b, 8 a, 9 d, 30 c; + = ; 11) y 1= ( x 1) ; 1) ( ) 9b) x y 7 1 f 0 = 0. Solutions 8