Section 3.2 The Derivative as a Function Graphing the Derivative

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1 Math 80 Derivatives Section 3. The Derivative as a Function Graphing the Derivative ( ) In the previous section we defined the slope of the tangent to a curve with equation y= f ( ) at the point a, f( a ) to be. We also saw that the velocity of an object with position function s f ( t) = at time t = a is In fact, limits of the form ( + ) ( ) h 0 f lim a h f a arise whenever we calculate a rate of change in any of the sciences or h engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. Definition The derivative of a function f at a, is =, denoted by f ( a) f if this limit eists. ( + ) ( ) ( a) = lim () or f ( a) h f a h f a h 0 ( ) ( ) f f a = lim a a () Notes: ( ) f ( a) is the slope of the tangent to the curve y= f ( ) at the point a, f( a ) f ( a) is the instantaneous rate of change of y f ( ) If we sketch the curve y f ( ) = with respect to when = a =, when the derivative is large ( and therefore the curve is steep), the y- values change rapidly. When the derivative is small, the curve is relatively flat, and the y-values change slowly. s f t f a is the rate In particular, if = ( ) is the position of a particle that moves along a straight line, then ( ) of change of the displacement s with respect to the time t. In other words, f ( a) is the velocity of the particle at time t = a. The speed of the particle is the absolute value of the velocity. Eercise Find the derivative of the function ( ) f = at the number a.

2 The Derivative as a Function Now we change our point of view and let the number a vary. If we replace a in the equation () above we obtain ( ) ( + ) ( ) f = (3) h h 0 f lim h f. So we can regard f as a new For any number for which this limits eists, we assign to the number f ( ) function, called the derivative of f and defined by equation (3). The value of f at, f ( ) geometrically as the slope of the tangent to the graph of f at the point, f( ). ( ), can be interpreted Notes: The function f is called the derivative of f because it has been derived from f by the limiting operation in equation (3). { } The domain of f is the set f ( ) If f ( ) R and it may be smaller than the domain of f. eists, we say that f has a derivative at or that f is differentiable at. If f eists at any in the domain of f, we say that f is differentiable. The process of calculating a derivative is called differentiation.. 3 Eercise a) If f ( ) =, fin a formula for f ( ) b) Illustrate by comparing the graphs of f and f. Eercise 3 If f ( ) =, find the derivative of f. State the domain of f. Eercise 4 Find f if f ( ) =. + Eercise 5 (#4) Differentiate f ( ) = and find the slope of the tangent line at =. + Differentiable on an Interval; One-sided Derivatives Definition ) A function f is differentiable on an open interval ( ab, ) [ or ( a, ) or (,a) or (, ) if and only if f ( ) eists for any in the interval. ] ) A function f is differentiable on a closed interval [ ab, ] if and only if it is differentiable on ( ab, ) and if the right-hand derivative at a and the left-hand derivative at b eist.

3 Eercise 6 ( #7 30) Match the functions graphed in #7 30 with the derivatives graphed in the accompanying figures a) d). 3

4 Eercise 7 (#3) a) The graph of the accompanying figure is made of line segments joined end to end. At which points of the interval ' [ 4,6] is f not defined? Give reasons for your answer. b) Graph the derivative of f. Eercise 8 (#37, 40) Compare the right-hand and left-hand derivatives to show that the functions are not differentiable at the point P. Eercise 9 Where is the function f ( ) = differentiable? Give a formula for f. Illustrate with the graphs of f and f. 4

5 Other notations If we use the traditional notation y= f ( ) to indicate that the independent variable is and the dependent variable is y, then some common alternative notations for the derivative are as follows: dy df d f y f Df Df d d d ( ) = = = = ( ) = ( ) = ( ) The symbols D and d/ d are called differentiation operators because they indicate the operation of differentiation, which is the process of calculating a derivative. The symbol dy/ d, which was introduced by Leibniz, should not be regarded as a ratio ( for the time being); it is a. synonym for f ( ) If we want to indicate the value of a derivative dy/ d in Leibniz notation at a specific number a, we use the notation dy which is a synonym for f ( a) d = a. Theorem If a function is differentiable at = c, then f is continuous at = c. In conclusion, when does a function not have a derivative at a point? ) a corner, where the one-sided derivatives differ ) a cusp, where the slope of the tangent approaches from one side and from the other 3) a vertical tangent, where the slope of the tangent approaches from both sides or approaches from both sides 4) a discontinuity 5

6 Eercise 0 (#8) Let g( ) 4 ( 3, ). = +. Differentiate the function. Then find an equation of the tangent line at Eercise (#0) Find dy d = 3 if y =. Eercise (#46) The figure shows the graph of a function over a closed interval. At what points does the function appear to be: a) differentiable? b) Continuous but not differentiable? c) Neither continuous nor differentiable. Eercise 3 (#54) Does any tangent to the curve y = cross the -ais at =? If so, find an equation for the line and point of tangency. If not, why not? Eercise 4 (#58) a) Let f ( ) be a function satisfying ( ) = 0 and find f ( 0). b) Show that f ( ) f for sin, 0 = is differentiable at 0 0, = 0. Show that f is differentiable at = and find ( 0) f. Answers:0b) y 7 + = ; ) /3; 3) y = ( ) ; 4) a) f ( 0) = 0; b) ( ) f 0 = 0. 6