MATH115. Indeterminate Forms and Improper Integrals. Paolo Lorenzo Bautista. June 24, De La Salle University

Transcription

1 MATH115 Indeterminate Forms and Improper Integrals Paolo Lorenzo Bautista De La Salle University June 24, 2014 PLBautista (DLSU) MATH115 June 24, / 25

2 Theorem (Mean-Value Theorem) Let f be a function that satisfies both of the following statements: i. f is continuous on the closed interval [a, b]. ii. f is differentiable on the open interval (a, b). Then there is a number c (a, b) such that f (c) = f (b) f (a). b a PLBautista (DLSU) MATH115 June 24, / 25

3 Theorem (Mean-Value Theorem) Let f be a function that satisfies both of the following statements: i. f is continuous on the closed interval [a, b]. ii. f is differentiable on the open interval (a, b). Then there is a number c (a, b) such that f (c) = f (b) f (a). b a Remark: When f (a) = f (b), we have a special case of the MVT, called Rolle s Theorem. PLBautista (DLSU) MATH115 June 24, / 25

4 Theorem (Cauchy s Mean-Value Theorem) Let f and g be two functions that satisfies the following statements: i. f and g are continuous on the closed interval [a, b]. ii. f and g are differentiable on the open interval (a, b). iii. For all x in the open interval (a, b), g (x) 0. Then there is a number z (a, b) such that f (b) f (a) g(b) g(a) = f (z) g (z). PLBautista (DLSU) MATH115 June 24, / 25

5 Example Find all values of z in the interval (0, 1) satisfying the conclusion of Cauchy s Mean-Value Thoerem for the functions f (x) = 2x 2 + 3x 4 and g(x) = 2x 3 8x + 3. PLBautista (DLSU) MATH115 June 24, / 25

6 Guillaume de l Hopital ( ) PLBautista (DLSU) MATH115 June 24, / 25

7 L Hopital s Rule Indeterminate Forms and L Hopital s Rule Theorem (L Hopital s Rule) Let f and g be functions differentiable on an open interval I, except possibly at the number a in I. Suppose that for all x a in I, g (x) 0. Suppose further that lim f (x) = 0 and lim g(x) = 0. x a x a f (x) f (x) If lim x a g = L, then lim (x) x a g(x) = L. PLBautista (DLSU) MATH115 June 24, / 25

8 Example Evaluate the following limits: 1. lim x 0 sin x x x lim x 1 x 1 tan x x 3. lim x 0 x sin x ln x 4. lim x 1 x 1 θ sin θ 5. lim θ 0 tan 3 θ e x 10 x 6. lim x 0 x PLBautista (DLSU) MATH115 June 24, / 25

9 Indeterminate Forms Definition If f and g are two functions such that lim x a f (x) = 0 and lim g(x) = 0, x a then f (x) g(x) has the indeterminate form 0 at a. 0 PLBautista (DLSU) MATH115 June 24, / 25

10 Remark Other indeterminate forms are the following: ± 1. ± 2. 0 ( ) (± ) ± PLBautista (DLSU) MATH115 June 24, / 25

11 Theorem (L Hopital s Rule) Let f and g be functions differentiable for all x > N, where N is a positive constant. Suppose that for all x > N, g (x) 0. Suppose further that f (x) = 0 and lim g(x) = 0. lim x + If x + f (x) lim x + g = L, then (x) lim f (x) x + g(x) = L. PLBautista (DLSU) MATH115 June 24, / 25

12 Example Evaluate the following limits: sin 2 x 1. lim x + 1 x 2. lim x + 3. lim x + 1 e 1/x 3 x 1 x tan 2 x PLBautista (DLSU) MATH115 June 24, / 25

13 Exercise Evaluate the following limits: 1. lim x 2 sin πx 2 x 2. lim x 0 sin 2 x sin x 2 3. lim x 0 tan 3x tan 2x ln(sin x) 4. lim x π/2 (π 2x) 2 5. lim x 0 (1 + x) 1/5 (1 x) 1/5 (1 + x) 1/3 (1 x) 1/3 PLBautista (DLSU) MATH115 June 24, / 25

14 Theorem (L Hopital s Rule) Let f and g be functions differentiable on an open interval I, except possibly at the number a in I. Suppose that for all x a in I, g (x) 0. Suppose further that lim f (x) is + or and lim g(x) is + or. x a x a f (x) f (x) If lim x a g = L, then lim (x) x a g(x) = L. PLBautista (DLSU) MATH115 June 24, / 25

15 Theorem (L Hopital s Rule) Let f and g be functions differentiable on an open interval I, except possibly at the number a in I. Suppose that for all x a in I, g (x) 0. Suppose further that lim f (x) is + or and lim g(x) is + or. x a x a Theorem (L Hopital s Rule) f (x) f (x) If lim x a g = L, then lim (x) x a g(x) = L. Let f and g be functions differentiable for all x > N, where N is a positive constant. Suppose that for all x > N, g (x) 0. Suppose further that f (x) is + or and lim g(x) is + or. lim x + If x + f (x) lim x + g = L, then (x) lim f (x) x + g(x) = L. PLBautista (DLSU) MATH115 June 24, / 25

16 Example Evaluate the following limits: x 2 1. lim x + e x 2. lim x 0 3. lim x 1 tan x(ln x) + ( 1 ln x 1 x 1 ) 4. lim x 0 + xsin x 5. lim x + (x2 6. lim x 0 (1 + 3x) 1/x x 4 x 2 + 2) PLBautista (DLSU) MATH115 June 24, / 25

17 Exercise Evaluate the following limits: 1. lim x 1/2 ln(1 2x) tan πx 2. lim x + (ex + x) 2/x 3. lim x)x2 x 0 +(sin 4. lim x 0 [(cos x)e x2 /2 ] 4/x4 5. lim x + [(x6 + 3x 5 + 4) 1/6 x] PLBautista (DLSU) MATH115 June 24, / 25

18 Improper Integrals Improper Integrals with Infinite Limits of Integration Definition If f is continuous for all x a, then if this limit exists. + a f (x)dx = b lim b + a f (x)dx PLBautista (DLSU) MATH115 June 24, / 25

19 Improper Integrals Improper Integrals with Infinite Limits of Integration Definition If f is continuous for all x a, then if this limit exists. b f (x)dx = b lim a a f (x)dx PLBautista (DLSU) MATH115 June 24, / 25

20 Improper Integrals Improper Integrals with Infinite Limits of Integration Remark If the aforementioned limits exist, then the improper integral is said to be convergent. Otherwise, the improper integral is divergent. PLBautista (DLSU) MATH115 June 24, / 25

21 Improper Integrals Example Evaluate the following improper integrals: dx (4 x) 2 xe x dx sin xdx PLBautista (DLSU) MATH115 June 24, / 25

22 Improper Integrals Exercise Evaluate the following improper integrals: e x/3 dx x5 x2 dx x2 x dx e xdx 3 9 x 2 e x dx dx x(ln x) 2 PLBautista (DLSU) MATH115 June 24, / 25

23 Improper Integrals Improper Integrals with an Infinite Discontinuity Definition If f is continuous for all x in the half open interval (a, b], and if lim f (x) = +, then x a + b a f (x)dx = lim t a + b t f (x)dx if this limit exists. PLBautista (DLSU) MATH115 June 24, / 25

24 Improper Integrals Improper Integrals with an Infinite Discontinuity Definition If f is continuous for all x in the half open interval [a, b), and if lim f (x) = +, then x b b a f (x)dx = lim t b t a f (x)dx if this limit exists. PLBautista (DLSU) MATH115 June 24, / 25

25 Improper Integrals Improper Integrals with an Infinite Discontinuity Definition If f is continuous for all x in the interval [a, b] except at c where a < c < b, and if lim x c f (x) = +, then b a f (x)dx = lim t a + if both these limits exist. b t f (x)dx + lim s b s a f (x)dx PLBautista (DLSU) MATH115 June 24, / 25

26 Improper Integrals Example Evaluate the following improper integrals: dx 1 x x ln xdx xdx x 2 9 dx x 3 PLBautista (DLSU) MATH115 June 24, / 25

27 Improper Integrals Exercise Evaluate the following improper integrals: /2 dw (w + 1) 1/3 dx x 3 dz z(ln z) 1/ π/ dy 1 sin y dx x x 2 4 PLBautista (DLSU) MATH115 June 24, / 25