Calculus Archive: Questions from August 31, 2023
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Determine la longitud del arco en el intervalo dado a) \( r(t)=i+t^{2} j+t^{3} k ;[0,2] \) b) \( r(t)=\langle 4 t,-\cos t, \operatorname{sen} t\rangle ;\left[0, \frac{3 \pi}{2}\right] \)1 answer -
Determine e interprete la curvatura \( \mathrm{K} \) de la curva en el valor del parámetro dado a) \( r(t)=t^{2} i+j ; t=2 \) b) \( r(t)=\left\langle 3 t, 2 t^{2}\right\rangle \) en el punto \( (-3,21 answer -
Given \( f(x, y)=2 x^{3} \cos \left(y^{8}\right) \), find \[ \begin{array}{l} f_{x y}(x, y)= \\ f_{y y}(x, y)= \end{array} \]1 answer -
Problem 8
Evaluate the limit assuming that \( \lim _{x \rightarrow-2} g(x)=2 \) : \[ \lim _{x \rightarrow-2} \frac{g(x)}{x^{2}}= \]1 answer -
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Metodo de regla falsa. Utiliza el metodo de regla falsa para hallar la raiz de \( f(x)=2+\operatorname{sen}(x)-x \), en el intervalo \( [2,3] \), con \( \varepsilon=10^{-4} \) a) Realiza las iteracion1 answer -
1. En cada caso se define una sucesión por la fórmula dada. Determinar si lasu cesion \( \left\{a_{n}\right\} \) converge o diverge a) \( a_{n}=\frac{n}{n+1}-\frac{n+1}{n} \) c) \( a_{n}=\frac{n}{2^1 answer -
2. En cada caso demostrar que la serie converge y que la suma es la indicada. a) \( \sum_{n=1}^{\infty} \frac{2^{n}+3^{n}}{6^{n}}=\frac{3}{2} \) c) \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(2 n+1)}{n(n+1 answer -
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For f(x, y), find all values of x and y such that fx(x, y) = 0 and f(x, y) = 0 simultaneously. f(x, y) = x² + 3xy + y² - 22x - 23y + 33 (x, y) =
For \( f(x, y) \), find all values of \( x \) and \( y \) such that \( f_{x}(x, y)=0 \) and \( f_{y}(x, y)=0 \) simultaneously. \[ f(x, y)=x^{2}+3 x y+y^{2}-22 x-23 y+33 \] \[ (x, y)=( \]1 answer -
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Find the indicated limit by using the limits \( \lim _{(x, y) \rightarrow(a, b)} f(x, y)=2 \) and \( \lim _{(x, y) \rightarrow(a, b)} g(x, y)=-5 \). \[ \lim _{(x, y) \rightarrow(a, b)} \frac{3 f(x, y)1 answer -
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Resuelva la ecuación diferencial dada por separación de variables. 19. \( \frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8} \) Determine la solución general de la ecuación diferencial dada. Indiqu1 answer -
1. Encontrar el tiempo que se empleará en saldar una deuda de \( \$ 8800 \) pagando \( \$ 250 \) el primer mes, \( \$ 270 \) el segundo, \( \$ 290 \) el tercero, etc. 2. Calcular la suma de todos los1 answer -
7. Determinar si la sucesión dada converge o diverge. En el caso de la convergencia calcular el límite. (a) \( \left\{(2 n-1)^{2}\right\}_{n \in \mathbb{N}} \) (b) \( \left\{\frac{1}{3 n}\right\}_{n1 answer -
9. Sea \( P=\left\{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right\} \) una partición uniforme de tamaño \( n \) del intervalo cerrado \( [a, b] \). (a) Si \( \Delta x_{k}=\left|x_{k}-x_{k-1}\right| \) den1 answer -
6. Demostrar por inducción matemática, que todo entero positivo \( n \) se cumple \[ \frac{1}{(1)(3)}+\frac{1}{(3)(5)}+\frac{1}{(5)(7)}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1} \]1 answer -
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Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ y=\ln (\sec (7 x)+\tan (7 x)) \] \[ y^{\prime}= \]1 answer -
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#7, 19, 23, 29
\( 7-36 \) Find the derivative of the function. 7. \( F(x)=\left(x^{4}+3 x^{2}-2\right)^{5} \) 8. \( F(x)=\left(4 x-x^{2}\right)^{100} \) 9. \( F(x)=\sqrt{1-2 x} \) 10. \( f(x)=\left(1+x^{4}\right)^{21 answer -
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3-16 Find \( d y / d x \) by implicit differentiation. 3. \( x^{3}+y^{3}=1 \) 4. \( 2 \sqrt{x}+\sqrt{y}=3 \) 5. \( x^{2}+x y-y^{2}=4 \) 6. \( 2 x^{3}+x^{2} y-x y^{3}=2 \) 7. \( x^{4}(x+y)=y^{2}(3 x-y)1 answer -
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17-29 Find the derivative of the function. Simplify where possible. 17. \( y=\left(\tan ^{-1} x\right)^{2} \) 18. \( y=\tan ^{-1}\left(x^{2}\right) \) 19. \( y=\sin ^{-1}(2 x+1) \) 20. \( F(\theta)=\a1 answer -
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2-20 Differentiate the function. 2. \( f(x)=x \ln x-x \) 3. \( f(x)=\sin (\ln x) \) 4. \( f(x)=\ln \left(\sin ^{2} x\right) \) 5. \( f(x)=\log _{2}(1-3 x) \) 6. \( f(x)=\log _{5}\left(x e^{x}\right) \1 answer -
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