MAC 2233 - Worksheet 7

Name(PRINT): Herline Michelin) MAC 2233 Worksheet 7 d/dx (x) = n * x ^ (n - 4) (uv)'='v+UV (a) ex = ex For each function f(z), find the domain. Then find the critical numbers. Recall: critical numbers are x-values in the domain where f' * (x) = 0 or f'(x) DNE. f(x) = (x + 2)/(x + 1) x+1≠0 x7-1 Domain = (-00,-1) (-1,00) Domain: (-00,-1) U (-1,00) critical numbers x = DINE (b) f(x) = x ^ 7 * e ^ x Domists: (-00,00) critical numbers z = x = 0 , x = - 7 a). f(x) = (x + 1)/(x + 1) 2+1+0 Domain (-00,-1) U (-1,00) (8(z) (z) )^ prime = (g' * (z) * h(z) - g(z) * h' * (z))/((h(z)] ^ 2) = (1(z + 1) - (z + 2) * 1)/((x + 1) ^ 2) g(x) = x + 2 = (x)[1(- x)] - 2 (2+1)2 9' * (x + 2) = 1 h(x) = x + 1 h' * (x + 1) = 1 = - 1/((x + 1) ^ 2) f' * (x) = (x + 1) ^ 2 = 0 x + 11 = 0 x = - 1 f(- 2) = (- 2 + 2)/(- 2 + 1) = 0/- 1 = 0 f(- 1) = (- 1 + 7)/(- 1 + 1) = - 1/0 = x f( 0 = (0 + 2)/(0 + 1) = 2/1 = 2 f(- 2) = 1/((- 2 + 1) ^ 2) = - 1/(- 1) = 1 f(- 1) = - 15/((- 1 + 1) ^ 2) = - 1/((0) ^ 2) = ank + xz b). f(x) = x ^ 7 * e ^ x (g(x)* h(x))*= 9' * (x) * h(x) + h(x) * g(x) g(x) = x ^ 7 g' * (x') = (7x ^ 6) f' * (x) = 7x ^ 6 * e ^ x + x ^ 7 * e ^ x = x ^ 6 * e ^ x * (7 + x) h(x) = ez n'(ez)=(x f(x) = 0 x ^ 6 * e ^ x * (7 + x) = 0 x ^ 6 = 0 = 7x = 0 ex=0 => no solution 7 + x = 0 = x = - 7 2. Let f(z) = 3 2+ (a) Find the first derivative. f'(x)= 3 or (2+2) -3- (2+x)-2 (b) Find open intervals where f(x) is increasing. Answer: No where (c) Find open intervals where f(z) is decreasing. A(-00,-2)U(-2,00) 3. Let g(2)223-322. Find the absolute extrema of g in the interval [-2,1/2]. Absolute Maximum: -(0,0) Absolute Minimum: (-2,-28) 4). f(x) = 3/(2 + x) 2+242 = 3 * (2 + x) ^ - 1 27-2 3 * d/dx ((2 + x) ^ - 1) = (- 1) * (2 + x) ^ - 2 = (3)(- 1) (2+x^ -2 = - 3 * (2 + x) ^ - 2 f(x) = - 3/((2 + x) ^ 2) 1 * (2 + x) ^ 2 = sqrt(0) 2 + x = 0 -2 -2 x = - 2 -3 => (2+2)² (-00,-2) U (-2,00) -3 03 0 2=-3=undefined - 3/(2 + (- 3) ^ 2) = - 3/(2 + 9) = - 3/11 decrease - 3/(2 + (0) ^ 2) =- 3 2 j decreas tau -3 2+(3)2=279=-y decrease