Calculus Archive: Questions from September 05, 2022
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(Gradable). Solve the following ODEs using the Laplace transform (Taken from Boyce DiPrima Chapter 6.2: Solution of Initial Value Problems). (a) \( y^{(4)}-y=0 ; y(0)=1, y^{\prime}(0)=0, y^{\prime \pr1 answer -
d to g
(Gradable). Solve the following ODEs using the Laplace transform (Taken from Boyce DiPrima Chapter 6.2: Solution of Initial Value Problems). (a) \( y^{(4)}-y=0 ; y(0)=1, y^{\prime}(0)=0, y^{\prime \pr1 answer -
8. Differentiate (using the formulas, Documents \( -> \) Formulas) (a) \( y=\frac{1}{t} \), (b) \( y=\sin (2 x+\pi) \), (c) \( y=\sin (2 t)+3 \cos (2 t)-t \), (d) \( y= \) \( \sqrt{x}+\ln (x) \)1 answer -
Solve the differential equation \( y^{\prime}+y=2 e^{-x} \) and \( y(0)=2 \). A. \( y=2 e^{-x}(x+1) \) B. \( y=e^{-x}(2 x+c) \) C. \( y=e^{x}(2 x+c) \) D. \( y=2 e^{x}(x+1) \)2 answers -
Which below function is one of the solutions of the differential equation \( y^{\prime}-y=1 \) ? A. \( y=e^{2} \) B. \( y=2 e^{x}+1 \) C. \( y=2 e^{x}-1 \) D. \( y=y^{\prime}-1 \)1 answer -
Solve the differential equation \( y^{\prime}=y \cos x \) A. \( y=c e^{-\sin x} \) B. \( y=c e^{\sin x} \) C. \( y=\frac{y^{2}}{2} \sin x+c \) D. \( y=-\frac{y^{2}}{2} \sin x+c \)1 answer -
Solve the differential equation \( 4 y^{\prime \prime}-4 y^{\prime}+y=0 \). A. \( y=e^{\frac{1}{2} t}(A \cos t+B \sin t) \) B. \( y=A e^{\frac{1}{2} t}+B e^{t} \) C. \( y=(A+B t) e^{\frac{1}{2} t} \)1 answer -
find \( y^{\prime} \) using Product Rule, \( Q \) Rule, 4 Step Rule 1) \( y=\frac{\sqrt{x}}{3 x} \). 2.) \( y=\frac{4 x \sqrt{x}}{x^{2}} \)1 answer -
Find \( \sin \theta \) and \( \tan \theta \) if \( \cos \theta=\frac{9}{41} \), assuming that \( 0 \leq \theta1 answer -
Use la gráfica para hallar el límite indicado: \[ \lim _{x \rightarrow 0} f(x) \] a. \( \lim _{x \rightarrow 0} f(x)=2 \) b. \( \lim _{x \rightarrow 0} f(x)=0 \) c. \( \lim _{x \rightarrow 0} f(x)=-1 answer -
Use la gráfica para hallar el limite indicado: \[ \lim _{x \rightarrow-2} f(x) \] a. \( \lim _{x \rightarrow-2} f(x) \) NO EXISTE b. \( \lim f(x)=2 \) \( x \rightarrow-2 \) c. \( \lim _{x \rightarrow1 answer -
Use la gráfica para hallar el límite indicado: \[ \lim _{x \rightarrow 2^{+}} f(x) \] a. \( \lim _{x \rightarrow 2^{+}} f(x)=3 \) b. \( \lim _{x \rightarrow 2^{+}} f(x)=2 \) c. \( \lim f(x)=-2 \)2 answers -
Solve differential equation:
Resolver esta ecuación diferenci al \[ \left(\frac{x+6 y+1}{x+2 y}\right) d x+\left(\frac{4 x+8 y+2}{x+2 y}\right) d y=0 \]1 answer -
Sketch the region close by the given curves and find its area. Problems #21, 25, 27, 33
25. \( y=\sqrt{x}, \quad y=\frac{1}{3} x, \quad 0 \leqslant x \leqslant 16 \) 26. \( y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi \) 27. \( y=\cos x, \quad y=\sin 2 x, \quad 0 \leqs1 answer -
Use the chain rule to find \( \frac{\partial z}{\partial s} \) and \( \frac{\partial z}{\partial t} \). \[ \begin{array}{r} z=\tan ^{-1}\left(x^{4}+y^{4}\right), \quad x=s \ln (t), \quad y=t e^{s} \\1 answer -
1 answer
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Find the general solution to the differential equation: \[ \begin{array}{l} y^{\prime \prime}+y-12=0 \\ y=C_{1} \cos (x)+C_{2} \sin (x)+12 \\ y=e^{x}\left(C_{1} \cos (3 i \sqrt{2} x)+C_{2} \sin (3 i \1 answer -
analyze the continuity of a function
II. Analice la continuidad de la función a) \( f(x, y, z)=\frac{z}{x^{2}+y^{2}-4} \) b) \( f(x, y)=\left\{\begin{array}{c}\frac{\operatorname{sen}(x y)}{x y}, x y \neq 0 \\ 1, x y=0\end{array}\right.1 answer -
1 answer
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Find the limit (if it exists). \( \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-13(x+\Delta x)-5-\left(x^{2}-13 x-5\right)}{\Delta x} \) \( x^{3}-13 x^{2}-5 x \) \( \frac{1}{3} x^{3}-\frac{13}2 answers -
Match the functions and their derivatives: 1. \( y=\sin (x) \tan (x) \) 2. \( y=\cos (\tan (x)) \) 3. \( y=\cos ^{3}(x) \) 4. \( y=\tan (x) \) A. \( y^{\prime}=\sin (x)+\tan (x) \sec (x) \) B. \( y^{\1 answer -
1 answer
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Find the length of the curve over the given interval. 1. \( y=x^{3 / 2}, 1 \leq x \leq 4 \) 2. \( y=1+6 x^{3 / 2}, 0 \leq x \leq 1 \) 3. \( y=\frac{1}{3}\left(x^{2}+2\right)^{3 / 2}, 0 \leq x \leq 1 \2 answers -
1) [6 pts.] Verifica que \( y=x+4 \sqrt{x+2} \) es solución de la ecuación diferencial: \( (y-x) y^{\prime}=y-x+8 \) ¿Es esta ecuación diferencial es pura, autónoma o no-autónoma?1 answer -
2) [5 pts.] ¿Será que el P.V.I. \( \frac{d y}{d x}=\sqrt{x y}, y(0)=1 \) tendrá solución unica? Explica.1 answer