Calculus Archive: Questions from October 13, 2022
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1. \( t^{2} y^{\prime \prime}+2 t y^{\prime}-1=0 \) 2. \( t y^{\prime \prime}+y^{\prime}=1 \),2 answers -
1. Find the derivative of the function. a) \( f(x)=\sqrt{4 x+\csc 7 x} \) b) \( y=\frac{\sec 3 \theta}{6 \theta+\cos \left(5 \theta^{2}\right)} \) c) \( y=\cot ^{3}(\sin 6 \theta) \) d) \( y=(2 x-3)^{2 answers -
Evaluate the following integrals a) \( \int_{0}^{2 \pi} \frac{1+\cos \theta}{2-\sin } d \theta \) b) \( \int_{0}^{\pi} \frac{\sin ^{2} \theta}{1+\cos ^{2} \theta} d \theta \)2 answers -
The problem is asking for the integral that represents the volume by rotating the x-axis. The region is y=sqrt(x) ; y=0; x=0 ; x=1. *Whole procedure is required, not just choosing the right answer*
1. La integral que representa el volumen del que se obtiene al rotar la región acotada por \( y=\sqrt{x} ; y=0 \); \( x=0 ; x=1 \) con respecto el eje de \( x \) es a. \( \int_{0}^{1} \pi x^{2} d x \2 answers -
El interés compuesto bancario crece un \( 100 k \% \) anual. Llamaremos \( y(t) \) a la cantidad de dinero en una cuenta bancaria en el tiempo \( t \). Si después de la inversión inicial \( y(0) \)0 answers -
For problems 14-18, solve the boundary-value problem 14. \( y^{\prime \prime}+5 y^{\prime}-6 y=0 \) \[ \begin{array}{l} y(0)=0 \\ y(2)=1 \end{array} \] 15. \( y^{\prime \prime}+4 y^{\prime}+4 y=0 \) \2 answers -
For problems 15-18, find the unique solution to the non-homogeneous initial value problem 15. \( y^{\prime \prime}+y=4 e^{x}+x^{2}-x \) \[ \begin{array}{l} y(0)=2 \\ y^{\prime}(0)=0 \end{array} \] \[1 answer -
1. \( y^{\prime \prime}+3 y^{\prime}+2 y=3 x^{2} \) 2. \( y^{\prime \prime}+2 y^{\prime}=\cos 2 x \) 3. \( y^{\prime \prime}-2 y^{\prime}=\sin 3 x \) 4. \( y^{\prime \prime}+6 y^{\prime}+9 y=2 x-1 \)1 answer -
a) Compute \[ \iiint_{D} z^{2} d x d y d z \] where \( D=\left\{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}+z^{2} \leq 2, x \leq 0, y \geq 0, z \leq 0\right\} \).1 answer -
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Find \( f^{\prime}(x) \) using the QUOTIENT RULE if \[ \begin{array}{l} f(x)=\frac{6-x^{2}}{3+x^{2}} \\ f^{\prime}(x)= \end{array} \]2 answers -
Suppose that \( w=f(x, y) \) satisfies \[ \frac{\partial^{2} w}{\partial x^{2}}-\frac{\partial^{2} w}{\partial y^{2}}=1 \] Put \( x=u+v, y=u-v \), and show that \( w \) satisfies \( \partial^{2} w / \2 answers -
Given \( f(x, y)=3 x y^{3}-8 x^{4} y \) \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}= \\ \frac{\partial^{2} f}{\partial y^{2}}= \end{array} \]2 answers -
Solve the initial value problem below: \[ y^{\prime}+\frac{1}{x} y=\frac{\sin x}{x}, \quad y(\pi)=1 \]2 answers -
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2. \( y^{\prime \prime}+2 y^{\prime}=\cos 2 x \) 3. \( y^{\prime \prime}-2 y^{\prime}=\sin 3 x \) 4. \( y^{\prime \prime}+6 y^{\prime}+9 y=2 x-1 \) 5. \( y^{\prime \prime}-4 y^{\prime}+5 y=5 e^{-x} \)2 answers -
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As an exact equation: a) Show that the given differential equation is exact, if it is not, find the integration factor b) Propose and solve one of the alternatives for the "almost solution" solution:
\[ (x+y+2) d x+d y=0 \] Como ecuación exacta: a) (3pts) Muestre que la ecuación diferencial dada es exacta, si no lo es busque el factor de integración. b) (6pts) Proponga y solucione una de las al0 answers -
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Question #26 and #30
Find the derivative of each of the given functions. (See Examples 4-7.) 21. \( y=(7 x-12)^{5} \) 22. \( y=(13 x-3)^{4} \) 23. \( y=7(4 x+3)^{4} \) 24. \( y=-4(3 x-2)^{7} \) 25. \( y=-4\left(10 x^{2}+81 answer -
II. Determine el área de superficie para \( f(x, y)=13+x^{2}-y^{2} \) sobre la región \( R=\left\{(x, y): x^{2}+y^{2} \leq 4\right\} \).2 answers -
III. Reescriba el integral utilizando el orden dxdzdy. Luego comente si es o no es conveniente hacer cambio de coordenadas a cilíndricas o esféricas y trabaje su evalaución. \[ \int_{0}^{4} \int_{02 answers -
(i) Evaluate sin m/n cos3 xdx, with m, n positive integers. (ii) Verify that sin m/n cos3 xdx < 2/3
(i) Evalúe \( \int \sin ^{m / n} \cos ^{3} x d x \), con \( m, n \) enteros positivos. (ii) Verifique que \[ \int \sin ^{m / n} \cos ^{3} x d x2 answers -
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Find \( \lim _{x \rightarrow \infty} h(x) \), if possible. \[ f(x)=-9 x^{2}+4 x-6 \] (a) \( h(x)=\frac{f(x)}{x} \) \[ \lim _{x \rightarrow \infty} h(x)= \] (b) \( h(x)=\frac{f(x)}{x^{2}} \) \[ \lim _{2 answers -
Evaluate the parcial derivatives of each function:
2. Evaluar las derivadas parciales de cada función en el punto dado: a. \( f(x, y)=\frac{x y}{x-y} \) en el punto \( (2,-2) \). b. \( q(x, y)=\frac{\sqrt{u \wedge y}}{4 x^{2}+5 y^{2}} \) en el punto2 answers -
Solve this PVI (Problem of initial value) using the method of reduction to separable explained in class. Substitute the values of your parameter b before solving. Express the answer in explicit form y
\( \frac{d y}{d x}=\sin ^{2}\left(b^{2} x-b^{2} y\right), \quad y(0)=\frac{\pi}{4 b^{2}} \) Resolver este PVI usando el método de reducción a separable explicado en clase. Sustituir los valores de2 answers -
\( \int_{C} \mathbf{F} \cdot d \mathbf{r} \). \( F(x, y)=x \mathbf{i}+y \mathbf{j} \) \( C: \mathbf{r}(t)=(9 t+1) \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1 \)2 answers -
Suppose that a population P of bacteria satisfies the hypothesis of the exponential population model: "The rate of change of the population is proportional to the population." Suppose that the initial
Suponer que una población \( P \) de bacteria satisface la hipótesis del modelo exponencial de población: "La razón de cambio de la población es proporcional a la población." Suponer que la pobl2 answers -
Solve this PVI (Problem of initial value) using the method of first order linear equations explained in class. Substitute the values of your parameters a and b before solving. Express the answer in th
\( \frac{d y}{d x}+a \tan (a x) y=\sec (a x), \quad y(0)=\frac{1}{b} \) Resolver este PVI usando el método de ecuaciones lineales de primer orden explicado en clase. Sustituir los valores de tus par2 answers -
Find the derivative
g. \( \left.f(x)=\tan ^{-1}(3 x)+\ln (\ln x)\right) \) i. \( f(x)=\ln (3 x+1) 2^{5 x+2} \) 2. Find \( y^{\prime} \). a) \( y=4 x^{3} \ln (\cos x+1) \) b) \( y=3 \sin ^{-1}(\ln x) \)2 answers -
3. Use logarithmic differentiation to find \( y^{\prime} \). a) \( y=\frac{\left(3 x^{2}+4\right)^{3}}{(2 x+1)^{2}} \) b) \( y=\frac{\left(3 x^{2}+e^{2 x}\right)^{2}}{\sqrt{(3 x+1)}} \)2 answers -
Solve this PVI (Problem of initial value) using the method of separable equations of first order explained in class. Substitute the values of your parameters c and d before solving. Express the answer
\( \frac{d y}{d x}=\frac{x^{c}}{y^{d}}, \quad y(0)=1 \) Resolver este PVI usando el método de ecuaciones separables de primer orden explicado en clase. Sustituir los valores de tus parámetros c y \2 answers -
Solve this PVI (Problem of initial value) using the Bernoulli equations method explained in class. Substitute the values of your parameters a and c before solving. Express the answer in the explicit f
\( \frac{d y}{d x}+\frac{a}{2 x} y=\frac{x^{c}}{y}, \quad y(1)=1 \) Resolver este PVI usando el método de ecuaciones de Bernoulli explicado en clase. Sustituir los valores de tus parámetros \( a \)2 answers -
Explain all the steps.
If \( f(x)=\left(x^{2}+3 x+4\right)^{2} \) \( f^{\prime}(x)= \) \[ f^{\prime}(5 \]2 answers -
\( \iiint_{E} e^{z / y} d V \), where \[ E-\{(x, y, z) \mid 0 \leqslant y \leqslant 1, y \leqslant x \leqslant 1,0 \leqslant z \leqslant x y\} \]1 answer -
Find all possible functions with the given derivative. 1. If \( y^{\prime}=\sin (8 t) \), then \( y= \) 2. If \( y^{\prime}=\cos \left(\frac{t}{8}\right) \), then \( y= \) 3. If \( y^{\prime}=\sin (82 answers -
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I. Considere \( w=x^{2}-2 x y+y^{2}, x=r+\theta, y=r-\theta \) para determinar \( \frac{\partial w}{\partial r} \& \frac{\partial w}{\partial \theta} \) II. Considere \( w=x y \cos (z), x=t, y=t^{2} \2 answers -
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Need help with 22 and 28 please
17-28. Calculating derivatives Find \( d y / d x \) for the following functions. 17. \( y=\sin x+\cos x \) 18. \( y=5 x^{2}+\cos x \) 19. \( y=e^{-x} \sin x \) 20. \( y=\sin x+4 e^{0.5 x} \) 21. \( y=2 answers -
Need help with 60 please
56-61. Calculating derivatives Find \( d y / d x \) for the following functions. 56. \( y=\frac{\sin x}{1+\cos x} \) 57. \( y=x \cos x \sin x \) 58. \( y=\frac{1}{2+\sin x} \) 59. \( y=\frac{2 \cos x}2 answers -
Compute the gradient vector fields of the following functions: A. \( f(x, y)=1 x^{2}+2 y^{2} \) \( \nabla f(x, y)= \) B. \( f(x, y)=x^{9} y^{2} \), \( \nabla f(x, y)= \) C. \( f(x, y)=1 x+2 y \) \( \n2 answers -
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2. (15\%) (i) Evalúe \( \int \sin ^{m / n} \cos ^{3} x d x \), con \( m, n \) enteros positivos. (ii) Verifique que \[ \int \sin ^{m / n} \cos ^{3} x d x0 answers -
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Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ \begin{array}{c} y=\cos (\sin (6 \theta)) \\ y^{\prime}= \\ y^{\prime \prime}=36\left[\cos ^{2}(60) \cos (\sin (60))-\sin (60)(\sin (\sin (60)))\r2 answers -
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Evaluate the triple integral. \[ \iiint_{E} 2 x d V \text {, where } E=\left\{(x, y, z) \mid 0 \leq y \leq 2,0 \leq x \leq \sqrt{4-y^{2}}, 0 \leq z \leq 2 y\right\} \]2 answers -
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