f(a + h) f(a) f (a) = lim

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1 Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + ) f(a) wen tis limit exists. Tis gives te slope of te tangent to te curve y = f(x) wen x = a Example Last day we saw tat if f(x) = x 2 + 5x, ten f (a) = 2a + 5 for any value of a. Terefore f (1) = 7, f (2) = 9, f (2.5) = 10 etc... Te value of f (a) varies as te number a varies, ence f is a function of a. We can cange te variable from a to x to get a new function, called Te derivative of f f (x) 0 f(x + ) f(x) (f (x) is defined wen f is defined in an open interval containing x and te above limit exists). Note tat wen calculating tis limit for a particular value of x, 0 and te value of x remains constant. Note also tat if x is in te domain of f, it must satisfy te following 3 conditions: 1. x must be in te domain of f. 2. lim 0 f(x+) f(x) must exist at x. 3. f must be defined in an open interval containing x. Te domain of te function f may be smaller tan te domain of te function f since 2 or 3 may fail for some values of x in te domain of f. Example Wat is f (x) wen f(x) = x 2 + 2x + 4?. Wat is te domain of f (x)? 1

2 17 16 Example Consider te function in te example above f(x) = x 2 + 2x + 4. Te grap, y = f(x) is sown below along wit te grap of te new function f (x) = 2x + 2. We can see ow te grap of f 15 (x) is related to te slope of te tangents to te grap of f f( x) = x 2 + 2! x f' ( x) = 2! x + 2 g( x) = 7 4! ( x ( x) = 10 q( x) = (-3, 7) (0,4) (-1, 3) (-1, 3) (0,4) Fill in <, > or = as appropriate: Wen f(x) is decreasing te function f (x) 0 Wen f(x) is increasing te function f (x) 0 At te turning point x = 1, f (x)

3 Example Consider te function f(x) = x. Does lim 0 f(x+) f(x) exist wen x > 0? Does lim 0 f(x+) f(x) exist wen x < 0? Does lim 0 f(x+) f(x) exist wen x = 0? Wat is te domain of f (x)? Alternative Notation Using y = f(x), to denote tat te independent variable is y, tere are a number of notations used to denote te derivative of f(x) : f (x) = y = dy dx = df dx = d dx f(x) = Df(x) = D xf(x). Te symbols D and d are called differential operators, because wen tey are applied to a function, dx tey transform te function to its derivative. Te symbol dy sould not be interpreted as a quotient dx rater it is a limit originating from te notation dy dx y x 0 x. Wen we evaluate te derivative at a number a, we use te following notation f (a) = dy. dx x=a Differentiability Definition Wen a function f is defined in an open interval containing a, we say a function f is differentiable at a if f (a) exists. [ Tat is, conditions 1, 2 and 3 from page 1 must be satisfied for x = a.] It is differentiable on an open interval, (a, b) (or (a, ) or (, a)) if it is differentiable at every number in te interval. Example Let f(x) = x. Is f(x) differentiable at 0? If f(x) differentiable on te intervals (, 0) and (0, ).. Is f(x) continuous at 0? 3

4 M1 O Examples O Te following teorem sows tat if a function as a discontinuity at a point a, ten it cannot be differentiable at a. (Note by te previous example, te converse is not true; a function can be continuous at a, but not differentiable at a). Teorem If f is differentiable at a, ten f is continuous at a. Proof Lets assume tat f is a function wic is differentiable at a. Ten we know tat f(x) f(a) lim x a x a = f (a) exists. To sow tat f is continuous at a, we must sow tat lim f(x) = f(a) x a or equivalently tat lim x a (f(x) f(a)) = 0. We ave lim x a (x a) = 0. So by our rules of limits we ave ( ) f(x) f(a)) ( f(x) f(a)) ) lim(f(x) f(a)) (x a) lim(x a) x a x a x a x a x a x a = f (a) 0 = 0 Points were functions are Not Differentiable A function f can fail to be differentiable at a point a in a number of ways. Te function migt be continuous at a, but ave a sarp point or kink in te grap, like te grap 20 of f(x) = x at 0. wit(plots,implicitplot); implicitplot Te function migt not be continuous or migt be undefined at a. Plot a circle. O T:=implicitplot(x^2=1, x=-1..1, y=-1..1); T := PLOT... Te function migt be continuous but te tangent line may be vertical, i.e. lim x a f(x) f(a) O implicitplot y 3 = x 2, x =K10..10, y =K5..5, gridrefine = 5 ; Example (1.1) 18 (1.2) 16 Identify te points in te graps below were te functions are not differentiable x a = ± f( x) = x 3 8 y O K10 K x

5 Higer Derivatives We ave seen tat given a function f(x), we can define a new function f (x). We can continue tis process by defining a new function, f (x) = d dx f (x). Tis is te second derivative of te function f(x). Tis function gives te slope of te tangent to te curve y = f (x) at eac value of x. We can ten define te tird derivative of f(x) as te derivative of te second derivative, etc... Example Let f(x) = x 2 + 2x + 4. We saw above tat te derivative of f(x) is f (x) = 2x + 2. Find and interpret te second derivative of f(x); f (x) = Te second derivative gives us te rate of cange of te rate of cange. In te case of a position function s = s(t) of an object moving in a straigt line, te derivative v(t) = s (t) gives us te velocity of te moving object at time t and te second derivative a(t) = v (t) = s (t) gives us te acceleration of te moving object at time t. Tis is te rate of cange of te velocity at time t. Example Te position of an object moving in a straigt line at time t is given by s(t) = t 2 + 2t + 4. Wat is te velocity and acceleration of te object after t = 5 seconds? Te second derivative is also denoted by f (x) = Notation d ( dy ) dx dx = d2 y dx 2 = y. Te tird derivative of f is te derivative of te second derivative, denoted Higer derivative are denoted d dx f (x) = f (x) = y = y (3) = d ( d 2 y ) dx dx 2 = d3 y dx 3 f (4) (x) = y (4) = d4 y dx 4, f (5) (x) = y (5) = d5 y dx 5, etc... 5

6 Example If f(x) = x 2 + 2x + 4, find f (4) (x) and f (5) (x). 6

7 Old Exam Questions 1. Find te derivative of te function using te limit definition of te derivative. f(x) = x x 5 2. Wic of te statements given below is false? (a) If f is differentiable at x = a, ten lim x a f(x) f(a) x a must equal f(a). (b) If f is differentiable at x = a, ten a must be in te domain of f. (c) If f is differentiable at x = a, ten lim x a f(x) f(a) x a must exist. (d) If f is differentiable at x = a, ten f must be continuous at x = a. (e) If f is differentiable at x = a, ten lim 0 f(a+) f(a) 0 + f(a+) f(a) 7

8 1. Te grap of te function f(x) is sown below: Wic of te following gives te grap of f (x)? (a) (b) (e) (c) None of te above (d) 8

9 1. Find te derivative of te function using te limit definition of te derivative. Old Exam Question, Sample Solution f(x) = x x 5 Note te format of te solution below. It is important to carry te limits and sow all calculations in order to recieve full credit f (x) 0 f(x + ) f(x) 0 x+ x+ 5 x x 5 (x + )(x 5) x(x + 5) 1 0 (x + 5)(x 5) x 2 + x 5x 5 x 2 x + 5x 1 0 (x + 5)(x 5) x 2 + x 5x 5 x 2 x+ 5x 1 0 (x + 5)(x 5) 5 0 (x + 5)(x 5) (x + 5)(x 5) = 5 (x 5)(x 5) = 5 (x 5) 2 9

10 2. Wic of te statements given below is false? If f is differentiable at a, 1. a must be in te domain of f. 2. lim 0 f(a+) f(x) must exist at a. 3. f must be defined in an open interval containing a. f(x) f(a) (a) If f is differentiable at x = a, ten lim x a must equal f(a). false, it is not required x a tat tis limit is f(a). For example consider f(x) = x 2 + 2x + 4 from te notes. f (x) = 2x + 2. f f(x) f(1) (1) x 1 = 4 f(1) = 7. x 1 (b) (c) If f is differentiable at x = a, ten a must be in te domain of f. True see 1 above. If f is differentiable at x = a, ten lim x a f(x) f(a) x a must exist. True see 2 above. (d) If f is differentiable at x = a, ten f must be continuous at x = a. True by te teorem given in notes. (e) If f is differentiable at x = a, ten lim f(a+) f(a) 0 f(a+) f(a) 0 + True since te limit exists only if te laft and rigt and limits exist and are equal. 3. Te derivative must be positive wen f(x) is increasing and negative wen it is decreasing. In particular f (x) > 0 for all values of x bigger tan 4 in tis instance. Terefore te answer is (a). 10