Calculus Archive: Questions from January 11, 2023
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Evaluate the following integrals. a. \( \int_{C}(3 x-2 i)^{2} d x+d y \), where \( C \) is the line segment \( y=2 x, 0 \leq x \leq 2 \) b. \( \quad \int_{0}^{115 \pi} e^{i t} d t \)2 answers -
10. \( \int_{0}^{\sqrt{2} / 2} \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x= \) ? (a) \( \frac{\pi^{2}}{4} \) (b) \( \frac{\pi^{2}}{8} \) (c) \( \frac{\pi^{2}}{32} \) (d) \( \frac{\pi^{2}}{64} \) (e) \( \f2 answers -
Parte I: Determine la derivada dirección en dirección del vector unitario \( \vec{u}=\langle\cos \theta, \operatorname{sen} \theta\rangle \) 1. \( f(x, y)=x^{2}+y^{2}, \theta=\pi / 4 \) 2. \( f(x, y2 answers -
(1 point) Calculate \( y^{(k)}(0) \) for \( 0 \leq k \leq 5 \), where \( y=3 x^{4}+a x^{3}+b x^{2}+c x+d \) (with \( \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \) the constants) \[ \begin{array}{l1 answer -
Compruebe que la familia de funciones indicada es una solución de la ecuación diferencial dada. Suponga un intervalo I de definición adecuado para cada solución. d2y dx2 − 4 dy dx
Compruebe que la familia de funciones indicada es una solución de la ecuación diferencial dada. Suponga un intervalo \( I \) de definición adecuado para cada solución. \[ \begin{array}{l} \qquad \2 answers -
Ejercicio 3
Ejemplo 3: Halle el volumen de la región solida limitada por \( z=f(x, y)=e^{-x^{2}} \) y los planos, \( z=0, y=0, x=1 \) y \( y=x \) Grafica del sólido Proyeccion en el plano \( x y \)2 answers -
Calculate the circulation of the field F(x,y)=Power[x,2]yi+Divide[1,2]xPower[y,2]j on the curve defined between the intersections of y=x with y=x^2-x. Check with Green's theorem.
Calcule la circulación del campo \[ \vec{F}(x, y)=x^{2} y \hat{\imath}+\frac{1}{2} x y^{2} \hat{\jmath} \] en la curva definida entre las intersecciones de \( y=x \) con \( y=x^{2}-x \). Verifique co2 answers -
Calculate \( y^{(k)}(0) \) for \( 0 \leq k \leq 5 \), where \( y=3 x^{4}+a x^{3}+b x^{2}+c x+d \) (with a,b,c,d the constants) \[ \begin{array}{l} y^{(0)}(0)= \\ y^{(1)}(0)= \\ y^{(2)}(0)= \\ y^{(3)}(2 answers -
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Find the general solution to the following 2 nd Order O.D.E., where \( y^{\prime \prime}=\frac{d^{2} y(x)}{d x^{2}} \). \[ y^{\prime \prime}+y^{\prime}+y=0 \] \[ \begin{array}{l} y(x)=c_{1} \cdot e^{-2 answers -
Find the Solution to the following Non-Homogeneous O.D.E., where \( y^{\prime}=\frac{d y}{d t} \). \[ \begin{array}{l} y(0)=0 \\ \end{array} \] \[ \begin{array}{l} y(t)=\left(e^{-2 t}-1\right) \\ y(t)2 answers -
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Q] If \( \int_{1}^{3} f(x) d x=3 \) and \( \int_{1}^{3} g(x) d x=5 \) what is \( \int_{1}^{3} 2 f(x)+g(x) d x \) ? (A) 8 (B) 11 (C) 13 (D) 6 (E) 92 answers -
A box with a square base and no lid must contain a volume of 100u3. Use the method of Lagrange multipliers to determine the dimensions (side, x, of the square base of the box; height, y, of the box) t
(20 pts.) Un caja de base cuadrada y sin tapa ha de contener un volumen de \( 100 \mathrm{u}^{3} \). Utilice el método de los multiplicadores de Lagrange para determinar las dimensiones (lado, \( x \2 answers -
A Norman window is composed of a semicircle mounted on a rectangle (the diameter of the circumference measures the same as the base of the rectangle). Use the Lagrange multipliers method to obtain the
(15 pts.) Una ventana tipo Norman está compuesta por una semicircunferencia montada sobre un rectángulo (el diametro de la circunferencia mide lo mismo que la base del rectángulo). Utilice el méto2 answers -
a) Determine if F is conservative, and if so, find a potential function U for F. b) Set up the line integral of F along the helix r(t) = <cos(t),sin(t),t>
Sea \[ \mathbf{F}(x, y, z)=\left(2 x e^{x^{2}+y^{2}+z^{2}}+y+\operatorname{sen}(z)\right) \mathbf{i}+\left(2 y e^{x^{2}+y^{2}+z^{2}}+x\right) \mathbf{j}+\left(2 z e^{x^{2}+y^{2}+z^{2}}+x \cos (z)\righ2 answers -
Let D be the region bounded by the curves (make a sketch of the region). Consider the double integral. Write this integral as: a) two repeated integrals in Cartesian coordinates. b) two repeated i
Sea \( \mathscr{D} \) la región limitada por las curvas \( x^{2}+y^{2}=1, y=1, y=x \) (haga un dibujo de la región). Considere la integral doble \[ \int_{\mathscr{D}} \frac{x y}{x^{2}+y^{2}} d A . \2 answers -
a) Find a potential function for F. b) Write the integral as an ordinary integral, where C is the ellipse in the z=0 plane centered at the origin, with a semi-minor axis of length 3 along the x-axis,
\( \mathbf{F}(x, y, z)=(y+\operatorname{sen}(z)) \mathbf{i}+x \mathbf{j}+x \cos (z) \mathbf{k} \) Sea \( \mathbf{F}(x, y, z)=(y+\operatorname{sen}(z)) \mathbf{i}+x \mathbf{j}+x \cos (z) \mathbf{k2 answers -
Let D be the region bounded by the curves a) Make a sketch of region D. Let Where C is the frontier curve of D traversed counterclockwise and . Write, in terms of repeated integrals (also called ite
Sea \( \mathscr{D} \) la región limitada por las curvas \( (x-1)^{2}+(y-1)^{2}=1, y=x-1, y=2 \). (a) (5 pts.) Haga un dibujo de la región \( \mathscr{D} \). (b) (15 pts.) Sea \( \int_{\mathscr{C}} \2 answers -
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