Lista - Cálculo 1 2021 2

(Exercício 4) (1 ponto) Calcule, caso exista, o limite a seguir \( f(x) = \lim_{x \to 1} \frac{f(x) - f(1)}{x-1} \) \[ f(x) = \begin{cases} x + 1, \...se \geq 1 \\ 2x, \...x < 1 \end{cases} \] (Exercício 5) (1 ponto) Calcule \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \), sendo \( f(x) = x^2 + \sen(x) \). (Exercício 6) (1 ponto) Determine \( L \) para que a função dada seja contínua no ponto \( p = 5 \) dado. Justifique. \[ g(x) = \begin{cases} \frac{\sqrt{x} - \sqrt{5}}{\sqrt{x + 5} - \sqrt{10}}, \...se \neq 5 \\ L, \...se \...x = 5 \end{cases} \] (Exercício 7) (1 ponto) Mostre que se a função \( f(x) \) for derivável no ponto \( p \), então \( f(x) \) será contínua em \( p \). (Exercício 8) (1 ponto) Considere \( y(x) = e^{-x} \cos(2x) \). Verifique que a função dada satisfaz a igualdade a seguir \[ \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 5y = 0 \] (Exercício 9) (2 pontos) Determine a derivada das funções a seguir. a) \( f(x) = x \cos(x)(1 + \ln(x)) \) b) \( f(x) = \frac{x + \sen(x)}{x - \cos(x)} \) c) \( f(x) = \frac{x e^{2x}}{\ln(3t+1)} \) d) \( f(x) = e^{-x} \cdot \sec(x^2) \) \( f(x) = (x^2 + \cotg(x))^3 \) (Exercício 1) (0.5 pontos) Uma função \( f(x) \) é linear se \( f(u + v) = f(u) + f(v) \) e \( f(\alpha u) = \alpha f(u) \) para todo \( u, v \in D_f \), e \( \alpha \in \mathbb{R} \). Quais das funções a seguir é linear? 1. \( f(x) = \sqrt{\pi x} \) 2. \( f(x) = \sqrt{x} + \pi \) (Exercício 2) (0.5 pontos) Considere a função \( f(x) = x^5 + x + 1 \). Justifique matematicamente, a afirmação de que \( f \) tem pelo menos uma raiz no intervalo \([-1,1]\). (Exercício 3) (2.0 pontos) Calcule, se existir, os limites a seguir: a) \( \lim_{x \to 2} \frac{x^2 - 4x + 4}{x^2 - 3x + 4} \) b) \( \lim_{x \to 3} \frac{x - 3}{\sqrt{x} - \sqrt{3}} \) c) \( \lim_{x \to \pi} \frac{1 - \sen\left(\frac{x}{2}\right)}{\pi - x} \) d) \( \lim_{x \to 2} \frac{4}{x^2 - 4} \) e) \( \lim_{x \to \infty} \frac{x^2 - 5x + 1}{3x + 7} \) f) \( \lim_{x \to \infty} \frac{3x^2 - 2x + 1}{2x - 1} \) g) \( \lim_{x \to 4} \frac{\sqrt{3x - 8} - 2}{\sqrt{x - 2} - \sqrt{2}} \) \[ \lim_{x \to 1} \frac{\sen(\pi x)}{x - 1} \] Created in Master PDF Editor Created in Master PDF Editor CamScanner Se for f(x) = Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner Created in Master PDF Editor Created in Master PDF Editor CamScanner