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Consider the function a sin x if x < 0 (x-3)²-2 if x ≥ 0. Determine a, BER so that f is continuous and differentiable in R. f(x) =

Consider the function a sin x if x < 0 (x-3)²-2 if x ≥ 0. Determine a, BER so that f is continuous and differentiable in R. f(x) =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 54E
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I attached the exercise with solution can you please explain to me the solution and especially how they found alpha
SOLUTIONS Exercise 1. The function f is continuous and differentiable in R\ {0}. f is continuous at x = 0 if lim f(x) = lim f(x) = f(0) = 0. x+0+ x-0- Hence lim ((x-3)² - 2) = 8² - 2 = 0 = lim a sinx, Va € R. x-0 Therefore f is continuous in R if and only if ß = ±√2. To study the differentiability of f at x = 0, we compute f'. We have: f'(x) = If ß = ±√2, f is continuous at x = [a cos x 2(x-6) 0; hence, if بیر lim f'(x) = l ER x-0 Hence f is differentiable in R if and only if then f is differentiable at x = 0 and f'(0) = l. We have, lim f'(x) = a, H40 [a=-2√2 B = √2 if x < 0 if x > 0. V lim f'(x) = -26. x→0+ Ja = 2√2 B = -√2.
Transcribed Image Text:SOLUTIONS Exercise 1. The function f is continuous and differentiable in R\ {0}. f is continuous at x = 0 if lim f(x) = lim f(x) = f(0) = 0. x+0+ x-0- Hence lim ((x-3)² - 2) = 8² - 2 = 0 = lim a sinx, Va € R. x-0 Therefore f is continuous in R if and only if ß = ±√2. To study the differentiability of f at x = 0, we compute f'. We have: f'(x) = If ß = ±√2, f is continuous at x = [a cos x 2(x-6) 0; hence, if بیر lim f'(x) = l ER x-0 Hence f is differentiable in R if and only if then f is differentiable at x = 0 and f'(0) = l. We have, lim f'(x) = a, H40 [a=-2√2 B = √2 if x < 0 if x > 0. V lim f'(x) = -26. x→0+ Ja = 2√2 B = -√2.
Exercise 1. Consider the function a sin x if x < 0 (x-3)² -2 if x ≥ 0. Determine a, ß ER so that f is continuous and differentiable in R. f(x) =
Transcribed Image Text:Exercise 1. Consider the function a sin x if x < 0 (x-3)² -2 if x ≥ 0. Determine a, ß ER so that f is continuous and differentiable in R. f(x) =
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