Calculus Archive: Questions from September 20, 2022
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Problem Set # \( 1.3 \) (Page 18): Solve the following IVP 5) \( y^{\prime}=(x+y-2)^{2}, y(0)=2 \quad(\operatorname{set} v=x+y-2) \)1 answer -
1. Determine, with reasons, if following limits \( \lim _{(x, y) \rightarrow(0,0)} f(x, y) \) exist. (a) \( f(x, y)=\frac{-x^{2}+x y^{2}}{x^{2}+y^{2}} \). (b) \( f(x, y)=\left(-x^{2}+y^{2}\right) \sin1 answer -
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8. If \( x=a \cos ^{3} \theta, y=b \sin ^{3} \theta \), show that \( a \frac{d y}{d x}+b \tan \theta=0 \)1 answer -
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Determinar cuando la serie es absolutamente convergente, condicionalmente convergente o divergente. 1. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2 n^{2}} \) 2. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\s2 answers -
Find \( G^{\prime}(w) \) if \( G(w)=\frac{8}{9 w^{5}}+5 \sqrt{w} \) \[ G^{\prime}(w)= \] \( y=4 x^{-3}-7 x^{-1} \)1 answer -
Dibuje la región y encuentre suárea: \[ S=\left\{(x, y) / x \leq 1,0 \leq y \leq e^{x}\right\} \] 4) Para la serie de potencias \( \sum_{n=0}^{\infty} \frac{(x-2)^{n+1}}{(n+1) 4^{n+1}} \), halla el1 answer -
\( 4 x \leq g(x) \leq 2 x^{4}-2 x^{2}+4 \) for all \( x \), evaluate \( \lim _{x \rightarrow 1} g(x) \)3 answers -
The domain of \[ f(x, y)=\sqrt{y+\sin x} \] is \[ \begin{array}{l} x0, y>\sin x\} \\ \{(x, y) \mid y \geq-\sin x\} \\ \{(x, y) \mid x>0, y>0\} \end{array} \] Question 2 The range of \[ f(x, y)=\sqrt{y1 answer -
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Use the quotient rule to find the derivative of the following. 11. \( f(x)=\frac{6 x+1}{3 x+10} \) 12. \( f(x)=\frac{8 x-11}{7 x+3} \) 13. \( y=\frac{5-3 t}{4+t} \) 14. \( y=\frac{9-7 t}{1-t} \) 15. \1 answer -
If \( \cos ^{2} x+\sin ^{2} y=y \), then \( \frac{d y}{d x} \) (A) \( \frac{2 \cos x \sin x}{2 \cos y \sin y+1} \) (B) \( \frac{2 \cos x \sin x}{2 \cos y \sin y-1} \) (C) \( \frac{\sin y \cos y}{1-\co1 answer -
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Evaluate the integral. \[ \int \frac{7+x}{\sqrt{49-x^{2}}} d x \] A. \( 7 \tan ^{-1} \frac{x}{7}-\sqrt{49-x^{2}}+C \) B. \( 7 \sin ^{-1} \frac{x}{7}-\sqrt{49-x^{2}}+C \) C. \( \frac{1}{7} \tan ^{-1} \1 answer -
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Given \( \mathbf{v}=\mathrm{v}_{\mathrm{x}} \mathbf{i}+\mathrm{v}_{\mathrm{y}} \mathbf{j}+\mathrm{v}_{\mathrm{z}} \mathbf{k} \), show that \( \cos ^{2} \theta_{\mathrm{x}}+\cos ^{2} \theta_{\mathrm{y}1 answer -
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