Calculus Archive: Questions from September 29, 2022
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Find \( \lim _{x \rightarrow 0} \frac{\sin (3 x)}{x} \) Find \( \lim _{x \rightarrow 0} \frac{\sin (4 x)}{\sin (2 x)} \). Find \( \lim _{x \rightarrow 0} \frac{\sin (16 x)}{\sin (2 x)} \)2 answers -
2 answers
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Resuelva la ecuación como ecuación separable \( (x+3) d y+(-y+1) d x=0 \) (2 pts) Escríbala como separable para identificar las funciones (8 pts) Use la formula correspondiente para resolver la ecu2 answers -
For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \( x \)-axis or \( y \)-axis, whichever seems more conv2 answers -
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esuelva \( y^{\prime \prime}+y=\cos ^{2} x \) \[ y=c_{1} \cos x+c_{2} \sin x+\frac{1}{3} \cos ^{4} x-\frac{1}{3} \sin ^{2} x \] \[ y=c_{1} \cos x+c_{2} \sin x+\frac{1}{3}+\frac{1}{3} \sin ^{2} x \] \[2 answers -
Problem 1: Find \( \mathrm{f}^{\prime}(\mathrm{x}) \) and simplify if possible: (a) \( f(x)=\left(3 x^{2}-12 x+5\right)^{7} \) (b) \( f(x)=\frac{\sin x}{x} \) (c) \( \mathrm{f}(\mathrm{x})=\tan \left(2 answers -
Calculate \[ \iint_{\Omega} y d x d y \] When \( \Omega=\left\{(x, y) \mid x=r \cos (\theta), y=r \sin (\theta), 0 \leq r \leq 7, \frac{\pi}{4} \leq \theta \leq \frac{3 \pi}{4}\right\} \)2 answers -
PLEASE SOCVE THE INITIAL VALUE PROBLEM: \[ \begin{array}{l} \frac{d y}{d x}+\frac{y(x)}{x+1}=6 x \\ y(0)=1 \end{array} \] ANSWER SHOVLD BE: \[ y(x)=\frac{2 x^{3}+3 x^{2}+1}{x+1} \]2 answers -
Find indefinite integral \[ \int \sec ^{5}(\pi x) \tan (\pi x) d x \] \[ \int \frac{x+7}{x^{2}+x} d x \]2 answers -
Find indefinite integral \[ \int x \sin \left(x^{2}\right) \cos \left(x^{2}\right) d x \] \[ \int \pi x \cos \pi x d x \]2 answers -
In the following exercises, evaluate the triple integrals \( \iint_{E} f(x, y, z) d V \) over the solid \( E \). \[ \begin{aligned} f(x, y, z)=z \end{aligned} \]2 answers -
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show all woek please
Find the general solution of each differential equation. (1) \( y^{\prime \prime}+3 y^{\prime}-10 y=0 \) Attempt all (2) \( y^{\prime \prime}-4 y=0 \) 8 problems (3) \( y^{\prime \prime}+6 y^{\prime}+2 answers -
2. If \( F(2)=1, F^{\prime}(2)=5, F(4)=3, F^{\prime}(4)=7, G(4)=2, G^{\prime}(4)=6, G(3)=4, G^{\prime}(3)=8 \), find the following: (a) \( H^{\prime}(1) \) if \( H(x)=\cos (3 x) \) (b) \( K^{\prime}(22 answers -
Training: 1. Determine the derivative (choose 3 of the exercises) 1) f(x)=arctan (ex) 2) y= arcsin(3x) 3) y=sin(arccos(x)) 4) Determine the equation of the line that stops tangent to the curve
I. Determine la derivada (escoja 3 de los ejercicios) 1) \( f(x)=\arctan \left(e^{2 x}\right) \) 2) \( y=\frac{\operatorname{arcsen}(3 x)}{x} \) 3) \( y=\operatorname{sen}(\arccos (x)) \) 4) Determine2 answers -
II. Work the integrals (choose 3 of the exercises)
II. Trabaje los integrales (escoja 3 de los ejercicios) 1) \( \int \frac{1}{x \sqrt{1-(\ln x)^{2}}} d x \) 2) \( \int \frac{\operatorname{sen}(x)}{4+\cos ^{2}(x)} d x \) 3) \( \int_{0}^{\sqrt{2}} \fra2 answers -
Explain if the following integral can be solved with the integration formulas and techniques studied:
Situación: Explique si la siguiente integral se puede resolver con las fórmulas y técnicas de integración estudiadas: \[ \int_{2}^{3} \frac{2 x-3}{\sqrt{4 x-x^{2}}} d x \] Puede utilizar técnicas2 answers -
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Find the following limit. \[ \lim _{(x, y) \rightarrow(0,13)} \arctan \left(\frac{x^{2}+13}{x^{2}+(y-13)^{2}}\right) \]2 answers -
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If \( y=\left(x^{2}+2\right)^{6} \), find \( \frac{d^{2} y}{d x^{2}} \) \( \frac{d^{2} y}{d x^{2}}= \)2 answers -
2 answers
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Find the derivative of the function. \[ y=\frac{6 x-5}{8 x+1} \] \[ \begin{array}{l} y^{\prime}=(8 x+1) \cdot \frac{d}{d x}(6 x-5)-(6 x-5) \cdot \frac{d}{d x}(8 x+1) \\ y^{\prime}=\frac{(8 x+1) \cdot2 answers -
1) Halle \( \frac{d y}{d x} \) para \( y=x^{2} \ln \left(\frac{x}{x+3}\right) \) 2) Utilice diferenciación logaritmica para determinar \( \frac{d y}{d x} \) para \( y=\frac{x}{\sqrt{3 x+2}} \) 3) Det0 answers -
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Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ \begin{array}{c} y=\sqrt{\sin (x)} \\ y^{\prime}=\frac{\cos (x)}{2} \\ y^{\prime \prime}=\frac{\left(-\sin ^{2} x-\cos ^{2}(x)\right)}{\frac{(\sin2 answers -
5. Compute the path integral \( \int_{\mathcal{C}} f(x, y, z) d s \) for (a) \( f(x, y, z)=x \cos z, \quad \mathcal{C}: \overrightarrow{\mathbf{r}}(t)=t \widehat{\imath}+t^{2} \widehat{\jmath}, 0 \leq2 answers -
9,11 and 19
5. \( y=x^{3} e^{x} \) 6. \( y=\left(e^{x}+2\right)\left(2 e^{x}-1\right) \) 7. \( f(x)=\left(3 x^{2}-5 x\right) e^{x} \) 8. \( g(x)=(x+2 \sqrt{x}) e^{x} \) 9. \( y=\frac{x}{e^{x}} \) 10. \( y=\frac{e2 answers -
15, 18 and 19
13. \( f(\theta)=\frac{\sin \theta}{1+\cos \theta} \) 14. \( y=\frac{\cos x}{1-\sin x} \) 15. \( y=\frac{x}{2-\tan x} \) 16. \( f(t)=\frac{\cot t}{e^{t}} \) 17. \( f(w)=\frac{1+\sec w}{1-\sec w} \) 182 answers -
21, 23 and 27 find the derivative of the functions
19. \( f(t)=e^{a t} \sin b t \) 20. \( A(r)=\sqrt{ } \) 21. \( F(x)=(4 x+5)^{3}\left(x^{2}-2 x+5\right)^{4} \) 22. \( G(z)=(1-4 z)^{2} \sqrt{z^{2}+1} \) 23. \( y=\sqrt{\frac{x}{x+1}} \) 24. \( y=(x+ \2 answers -
Find all the second partial derivatives. \[ \begin{array}{l} f(x, y)=\sin ^{2}(m x+n y) \\ f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \\ f_{y y}(x, y)= \end{array} \]2 answers