1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) =

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1 Theorem 13 (i) If p(x) is a polynomial, then p(x) = p(c) 1 Limits Fining its graphically Examples 1 f(x) = x3 1, x 1 x 1 The behavior of f(x) as x approximates 1 x 1 f(x) = 3 x 2 f(x) = x+1 1 f(x) = 2 x 0 The behavior of f(x) as x = 0 3 Let f(x) = { 1 x 2 0 x = 2 Examples (Limits that fail to exist) x 1 x 0 x x 0 x ( 2) sin1 x 0 x ( 2) f(x) = 1, but f(x) = 0 x 2 Formal efinition of it f(x) = L f becomes arbitrarily close to L as x approaches c ɛ-δ efinition f(x) L < ɛ, 0 < x c < δ for each ɛ, δ st 13 Fining its analytically Theorem 11 b,c real numbers, n: positive integer (i) b = b, (ii) x = c, (iii) xn = c n Theorem 12 f,g functions with its f(x) = L, g(x) = K 1 bf(x) = bl 2 f(x) ± g(x) = L ± K 3 f(x)g(x) = LK 1 4 if K 0 5 f(x) g(x) = L K (f(x))n = L n Examples x 2 (4x2 + 2) = x 2 4x2 + 2 x 2 = = 18

2 Examples x 3 1 x 1 x+1 1 x x 1, x 2 +x 6, x 3 x+3 Theorem 18 (The squeeze theorem) If h(x) f(x) g(x) in an open interval containung c, except possibly at c an if h(x)=l= g(x) f(x) = L then Example 1 sin x = 1, x 0 x A = tanθ θ sinθ θ 1 cosθ sinθ cosθ sinθ 1 θ Now apply squeeze thm 2 x 0 3 x 0 tan x = x x 0 sin 4x=4 x x 0 sin x x 1 = cos x x 0 sin 4x 4x =4 x 0 1 cos x = 0 x 0 x sin x x x 0 sin y y = 4 1 = 1 cos x 14 Continuity an one-sie its Definition: A function f is continuous at c if f(c) is efine, f(x) exists an f(x) = f(c) The function f is continuous on an open interval (a, b) if it is continuous at each point in (a, b) Discontinuity: jump, infinity, not efine One-sie its it from the right f(x) = L, + it from the left f(x) = L Examples greatest integer function f(x) = [x] [x] = 1, [x] = 0 x 0 x 0 + Theorem 110 f:function, c, L: real numbers Then f(x) = L iff f(x) = L an f(x) = L + 2

3 Definition: A function f is continuous on the close interval [a, b] if f is continuous on (a, b) an f(x) = f(a), f(x) = f(b) x a + x b Examples f(x) = 1 x 2 x x2 = 0 = f( 1), x 1 1 x2 = 0 = f(1) f(x) is continuous on [ 1, 1] Theorem 111 b: real number f, g are continuous at x = c, then the following functions are continuous: (a) bf (b) f ± g (c) fg () f if g(c) 0 g Theorem 112 If g is continuous at x = c, f is continuous at g(c), then the composite (f g)(x) = f(g(x)) is continuous at x = c Examples Test for continuity { sin 1 (a) f(x)=tan x(not efine at x = π+nπ,n N) (b) f(x) = x 0 x 2 0 x = 2 (c) f(x) = xsin 1, x xsin 1 x x x Theorem 113 (Intermeiate value theorem) If f is continuous on the close interval [a, b] an k is any number between f(a) an f(b), then there is at least one number c in [a, b] st f(c) = k Examples f(x) = x 3 + 2x 1 f(x) has a root in [a, b] 15 Infinite its Definition: f: function efine in an open interval containing c(except possibly at c) f(x) = means that for each M > 0, δ > 0 st f(x) > M whenever 0 < x c < δ f(x) = 3

4 means that for each N > 0, δ > 0 st f(x) < M whenever 0 < x c < δ f(x) = means that f(x) > M for c δ < x < c In each of the above cases, we say x = c is a vertical asymptate Theorem 114 If h(x) = f(x) where f(x), g(x) are continuous at x = c, g(x) with f(c) 0 an g(c) 0, then h(c) has a vertical asymptate at x = c Examples (a) f(x) = 1 2(x+1) (b) f(x) = x2 +1 x 2 1) (c) f(x) = cot x () f(x) = x2 +2x 8 x 2 4 Theorem 115 If f(x) =, g(x) = L 1 (f(x) ± g(x)) = { if L > 0, 2 f(x)g(x) = if L < 0 3 ( g(x) ) = 0 f(x) 2 Differentiation 21 The erivative an the tangent line problem Definition: If f is efine on an open interval containing c, an if the it y = f(c+ x) f(c) = m x x exists, then the line through (c, f(c)) with shape m is the tangent line to the graph of f at (c, f(c)) The erivative of f at x is efine by f (x) = f(c+ x) f(c) x provie the it exists Forr all x for which the it exists, f is a function of x Remark: The process of fining the erivative of a finction is calle ifferentiation A function f is ifferentiable at x if f (x) = exists f is calle ifferentiable on (a, b), if it is ifferentiable at each point x in (a, b) 4

5 Notation: f (x), y, y, f Examples (a) f(x) = x 2 + 1, f(x) = x (f (c) = f(x) f(c) ) x c Differentiability: Not ifferentiable if (1) f (c) = f(x) f(c) x c f(x) f(c) x c or (2) f (c) = = ± Examples (a) f(x) = x 2, (b) f(x) = x 1 3 f(x) f(c) + x c Theorem 21 If f is ifferentatiable at x = c, then f is continuous at x = c proof = f(x) f(c) f(x) f(c) = (x c) x c = f(x) f(c) (x c) x c (x c) f (c) = 0 This means f(x) = f(c),ie f(x) is continuous at c 22 Bsic ifferentiation rule Theorem 22 If c is a real number, then c = 0 Theorem 23 If n is any rational number, then f(x) = x n is ifferentiable an f (x) = nx n 1 In particular, we have x = 1 Examples Ex2 p109 Theorem 24 If f is ifferentiable an c is any real number, then (cf(x)) = cf (x) Theorem 25 If f(x) an g(x) are ifferentiable, then f ± g are ifferentiable an (f ± g)(x) = f (x) ± g (x) Examples (a) f(x) = 1 x 2, (b) f(x) = x 3 4x + 5 5

6 Theorem 26 (sin x) = cos x, (cos x) = sin x sin(x + y) = sin xcos y + cos xsin y 23 prouct an quotient rules Theorem 27 (prouct rule) If f(x) an g(x) are ifferentiable, then (f(x)g(x)) = f (x)g(x) + f(x)g(x) Theorem 28 (quotient rule) If f(x) an g(x) are ifferentiable with g(x) 0, then ( f(x) ) = f (x)g(x) f(x)g(x) g(x) g(x) 2 Examples Fing the erivatives 1 h(x) = (3x 2x 2 )(5 + 4x) 2 y = 5x 2 x y = 3x 2 sin x 4 The tangent line of f(x) = 3 1 x x+5 5 y = 1 cos x = csc x cot x sin x at ( 1, 1) High orer erivatives 1 st orer y f (x) 2 n orer y f (x) n th orer y n f n (x) y f 2 y 2 f 2 2 n y n n n f Theorem 28 (tan x) = sec2 x, (cot x) = csc2 x (sec x) = sec xtan x, (csc x) = csc xcot x 24 Chain rule Theorem 210 If y = f(x) is a ifferentiable function of u an u = g(x) is a ifferentiable function of x then y = f(g(x)) is a ifferentiable function of x an 6

7 y = y u u or f(g(x)) = f (g(x)) g (x) Examples Fing y if (1) y = (x + 1) 3 (2) general power rule y = (u(x)) u, n rational, u(x) is ifferentiable u = nu(x)n 1 (3) f(x) = 3 (x 2 1) 2 then y (4) g(t) = 7 (5) f(x) = x (6) y = ( 3x 1 (7) (2t 3) 2 3 x 2 +4 x 2 +3 )2 (sin u) = cos u u, (tan u) = sec2 u u, (8) y = cos 2 x (9) f(t) = sin 3 4t (cos u) = sin u u (cot u) = csc2 u u 25 Implicit ifferentiation If the function y of x is given implicitly by f(x, y) = 0, we can still fin y in terms of f(x, y) an the chain rule Examples 2, 4, 5, 6, 7 3 Application of ifferentiation 31 Extreme on an interval Definition:(extreme) f: function efine on an interval I containing c 1 f(c) is the minimum of f on I if f(c) f(x), x I 2 f(c) is the maximum of f on I if f(c) f(x), x I They are also calle the extreme values, absolute min or absolute max 7

8 Theorem 210 (extreme value theorem) If y = f(x) is continuous on a close interval [a, b], then f has both a minimum an a maximum on the interval Definition:(relative extrema) 1 If open interval containing c on which f(c) is a maximum, then f(c) is calle a relative maximum of f, or f has a relative maximum at (c, f(c)) 2 If open interval containing c on which f(c) is a minimum, then f(c) is calle a relative minimum of f, or f has a relative minimum at (c, f(c)) Definition:(critical numbers) Let f be efine on c If f (c) = 0 or f is Not ifferentiable at c, then c is a critical number of f Theorem 210 If f has a relative minimum or relative maximum at c, then c is a critical number of f proof (i) If f is Not ifferentiable, then by efinition c is a critical number (ii) If f is ifferentiable at x = c, f (x) is > 0, = 0, < 0 If f (c) > 0, ie f (c) = f(x) f(c) > 0 This implies that f(x) f(c) > 0 in a neighborhoo of x c x c c, x c left of c, x < c, an f(x) < f(c) f(c) is Not a relative minimum, right of c, x > c, an f(x) > f(c) f(c) is Not a relative maximum So, f (c) > 0 contraicts the hypothesis that f(c) is a relative extremum Fining extreme on a close interval [a, b] 1 Fin the critical numbers of f in (a, b) 2 Evaluate f at each critical number 3 Evaluate f at the enpoints a an b 4 The least of these is the minimum an the largest is the maximum Examples Fin the extrema of f(x) = 2x 3x 2 3 on [ 1, 3] p Rolle s theorem an the MVT Theorem 33 (Rolle) f: continuous on [a, b] an ifferentiable on (a, b) If f(a) = f(b), then c (a, b) st f (c) = 0 8

9 proof (a) If f(x) = = f(a) = f(b), f is const on the interval [a, b], f (x) = 0 for any x (a, b) (b) If f(x) > for some x (a, b) By extreme value theorem, f has a maximum at some point c in the interval Since f(c) >, the maximum is NOT at the enpoints This implies that f(c) is relative maximum, by theorem f c = 0, since f is ifferentiable at c (c) If f(x) < for some x (a, b), we get a relative minimum by the argument above Theorem 34 (MVT) If f is continuous on [a, b] an ifferentiable on (a, b), then c (a, b) st f (c) = f(b) f(c) b a proof Define the function g(x) by g(x) = f(x) f(b) f(c) (x a) f(a) b a Then g(a) = 0 = g(b), g is continuous on [a, b] an ifferentiable on (a, b) By Rolle s theorem c (a, b) st g (c) = 0 Now 0 = g (c) = f (c) = f(b) f(c) b a ie f (c) = f(b) f(c) b a EX 3 p Increasing an ecreasing functions an the 1 st erivative test Definition: A function f is increasing on an interval I if for any x 1, x 2 I, x 1 < x 2 implies that f(x 1 ) < f(x 2 ) ecreasing on an interval I if for any x 1, x 2 I, x 1 < x 2 implies that f(x 1 ) > f(x 2 ) Theorem 35 Test for increasing an ecreasing f: continuous on [a, b], ifferentiable on (a, b) 1 If f (x) > 0 x I, then f is increasing on [a, b] 2 If f (x) < 0 x I, then f is ecreasing on [a, b] 1 If f (x) = 0 x I, then f is constant on [a, b] 9

10 proof If f (x) > 0 x I For any x 1, x 2 (a, b) with x 1 < x 2 By MVT x 1 < c < x 2 st f (c) = f(x 2) f(x 1 ) x 2 x 1 > 0 f(x 2 ) > f(x 1 ) ie f is increasing on (a, b) Examples 1 p180 Fining functions on which a function is increasing or ecreasing 1 Locate the critical numbers of f in (a, b), an use these numbers to etermine the test intervals 2 In each of the intervals, etermine the sign of f (x) at one testnumber 3 Use the above Theorem 35 Theorem 36 c is a critical number of f that is continuous on an open interval I containing c If f is ifferentiable on the interval, except possibly at c, then 1 If f (x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)) 2 If f (x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)) 3 If f (x) is positive on both sies of c or is negative on sies of c, then f neither a relative minimum nor a relative minimum Examples 3, 4 p Concavity an 2 n erivative test Definition: f: ifferentiable on I, The graph of f is concave upwar on I if f (x) is increasing on I, concave ownwar on I if f (x) is ecreasing on I The above are equivalent to the following: f is concave upwar on I the graph of f lies above of its tangent lines on I f is concave ownwar on I the graph of f lies below of its tangent lines on I 10

11 Theorem 37(Test for concavity) f function on I whose 2 n erivative exists on I 1If f (x) > 0 for all x I, then f is concave upwar on I 2If f (x) < 0 for all x I, then f is concave ownwar on I Definition: f: continuous on open interval I, c I If the graph of f has a tangent line at the point (c, f(c)), then the point (c, f(c)) is a point of inflection of f is the concavity of f changes from upwar to ownwar(or ownwar to upwar) Theorem 38 If (c, f(c)) is a point of inflection, then either f (x) = 0 or f oes Not exist at x = c Examples f(x) = x 4 4x 3 Fin the points of inflection f (x) > 0 = 4x 3 12x 2 f (x) = 12x 2 24x = 12x(x 2) f (x) = 0 x = 0, 2 - < x < 0 0 < x < 2 2 < x < test value x = 1 x = 1 x = 3 f ( 1) > 0 f (1) < 0 f (3) > 0 -upwar ownwar upwar Theorem 38 2 n erivative test f: a function st f (c) = 0 an f (x) exists on an open interval containing c 1 If f (c) > 0, then f has a relative minimum at (c, f(c)) 2 If f (c) < 0, then f has a relative maximum at (c, f(c)) If f (c) = 0, the test fails That is, f may have a relative maximum, relative minimum or neither In this case, we shoul use 1 st erivative test 35 Limits an infinity Definition: L: a real number 1 f(x) = L means that for every ɛ > 0, M > 0 st f(x) L < ɛ whenever x > M 2 f(x) = L means that for every ɛ > 0, N < 0 st f(x) L < ɛ x whenever x < N 11

12 In both cases, we say that y = L is a horizontal asymptote for the function f(x) as x Theorem 38 Limits at infinity IF t > 0 rational number, c is any real number, then c = 0 x r If x r is efine for x < 0, then c = 0 x x r If Ex 3, 4, 5 f(x), (f(x) + g(x))= (f(x)g(x))= g(x) exists, then f(x)+ g(x), f(x) g(x) Definition: 1 f(x) = means f(x) > M whenever x > N 36 A summary an sketching Guie lines: 1 Determine the omain an range of the function 2 Deter the intercepts, asymptotes an symmetry of the graph 3 Locate the x-value for which f (x) an f (x) either are zero or o Not exist Use these results to etermine relative extreme an points of inflection Examples 1, 2, 4, Differentials The tangent line at (c, f(c)) is y = f(c) + f (c)(x = c) when x c x is vert small, the change y = f(c + x) f(c)can be approximate by y = f(c + x) f(c) f (c) x (1) 12

13 When x 0, enote it by, we then efine the ifferential to y = f (x) Examples 2, before 4, 5 (1) f(x + x) f(x) + y = f(x) + f (x) x 13