UNIVERSIDAD DE CARABOBO FACULTAD DE CIENCIAS ECONÓMICAS Y SOCIALES CICLO BÁSICO CÁTEDRA DE MATEMÁTICA I (3) SEGUNDO PARCIAL (25%) NOMBRES Y APELLIDOS: Samuel Sánchez CI 31.395.601 ESCUELA ACCP SECCIÓN 32 FECHA: 29/01/2024 FIRMA: INSTRUCCIONES: Resuelva el problema en cada caso y simplifique. NO SE PERMITE INTERCAMBIO DE CALCULADORA, BORRADOR, LÁPICES ENTRE OTROS. APAGUE EL CELULAR. NO HAY CONSULTA. DURACIÓN 90MIN 1. Hallar frac{dy}{dx}, en cada una de las siguientes funciones: a) f(x) = e^{tang(cos3x)} (2 puntos) b) f(x) = arcosen(frac{2x+1}{x-3}) (2puntos) c) Dada f(x) = (sen8x - 1)(sen8x + 1) (2puntos) 2. Dada f(x) = log[sec(6x^2 - 3x + 5)], hallar frac{dy}{dx} (Aplique regla de la cadena) (3puntos) 3. Sea y = sqrt{(2e^{2x} + 2sen5x)} hallar frac{dx}{dy} (2puntos) 4. Dada f(x) = frac{x+5}{5x-2}, hallar frac{d^{400}y}{dx^{400}} en x = 3 (3puntos) 5. Hallar la ecuación de la recta tangente y normal a la curva e^{5y} + log(y + 1) - 3x + 3y = 4, en el punto de intersección con el eje de las abscisas. (3 puntos) 6. Sea f(x) = (lnx)^{sen(3x-1)} hallar f'(x) (3puntos)
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### 1a) Find \( \frac{dy}{dx} \) for \( f(x) = e^{\tan(\cos(3x))} \) Show more…
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CHAPTER 3 Complex Differentiation 1. For each following function, determine the singular points at which the function is not analytic. Determine the derivative at the points where the function is analytic using Cauchy-Riemann Equations. (a) f(z) = 1 / (z + 1) (b) f(z) = z^2 + z + 1 (c) f(z) = |z|^2 2. Verify that the Cauchy-Riemann equations hold for the following functions and find their derivatives: (a) f(z) = z^3 (b) f(z) = e^z (c) f(z) = sin z 3. Suppose u(x, y) = x^2 - y^2. (a) Show that u is harmonic. (b) Find a harmonic conjugate function v(x, y). (c) Express f(z) in terms of z. 4. Show that f(z) = z* (the complex conjugate of z) is not differentiable anywhere. 5. Given f(z) = u + iv is analytic, show that both u and v satisfy Laplace's equation.
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