Calcula el polinomio de Maclaurin de grado 5 de f(x) = tan(x).

Para calcular o polinômio de Maclaurin de grau 5 para a função f(x) = tan(x), podemos usar a fórmula geral do polinômio de Maclaurin: P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + (f''''(0)/4!)x^4 + (f'''''(0)/5!)x^5 Vamos calcular as derivadas de f(x) em relação a x: f(x) = tan(x) f'(x) = sec^2(x) f''(x) = 2sec^2(x)tan(x) f'''(x) = 2sec^2(x)tan^2(x) + 2sec^4(x) f''''(x) = 2sec^2(x)tan^3(x) + 8sec^4(x)tan(x) f'''''(x) = 6sec^2(x)tan^4(x) + 24sec^4(x)tan^2(x) + 8sec^6(x) Agora, vamos substituir essas derivadas na fórmula do polinômio de Maclaurin: P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + (f''''(0)/4!)x^4 + (f'''''(0)/5!)x^5 P(x) = tan(0) + sec^2(0)x + (2sec^2(0)tan(0)/2!)x^2 + (2sec^2(0)tan^2(0) + 2sec^4(0))/3!x^3 + (2sec^2(0)tan^3(0) + 8sec^4(0)tan(0))/4!x^4 + (6sec^2(0)tan^4(0) + 24sec^4(0)tan^2(0) + 8sec^6(0))/5!x^5 Simplificando os termos, temos: P(x) = 0 + x + (2/2)x^2 + (2/3)x^3 + (8/24)x^4 + (8/120)x^5 P(x) = x + x^2 + (1/3)x^3 + (1/3)x^4 + (1/15)x^5 Portanto, o polinômio de Maclaurin de grau 5 para f(x) = tan(x) é P(x) = x + x^2 + (1/3)x^3 + (1/3)x^4 + (1/15)x^5.