Calculus Archive: Questions from July 26, 2022
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Evaluate the indefinite integral. \[ \int \sqrt{16 x} \sin \left(1+x^{3 / 2}\right) d x \] a) \( -\frac{4}{3}(\cos (\sqrt{x}))^{3} \) b) \( \frac{4}{3} \cos (1+(\sqrt[3]{x})) \) C) \( -\frac{8}{3} \co1 answer -
\( \lim _{(x, y) \rightarrow(0,0)} \frac{y^{4}}{x^{4}+3 y^{4}} \) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}} \)1 answer -
Evaluate \( \iiint_{E} x y d V \), where \( E=\{(x, y, z) \mid 0 \leq x \leq 3,0 \leq y \leq x, 0 \leq z \leq x+y\} \).1 answer -
If \( \vec{u}=(2,4,-4), \vec{v}=-\hat{j}-2 \hat{k}, \vec{w}=\hat{i}-3 \hat{k} \), determine: a) \( \vec{u} \cdot \vec{v} \) b) \( \vec{w} \times \vec{v} \)1 answer -
a) b)
For each problem, (a) graph and shade the region enclosed by the curves (b) find the volume of the solid that results when the region enclosed by the curves is revolved about the \( y \)-axis. \( x=91 answer -
a) b) c) please answer all parts, thank you
1. For each problem, (a) graph and shade the region enclosed by the curves (b) find using the disk/washer method the volume of the solid that results when the region enclosed by the curves is revolved1 answer -
Differentiate \( y=(4 x-1)(3 x+1)^{4} \) \( y=(2 x+1)^{\frac{5}{2}}(4 x-1)^{\frac{3}{2}} \) \( 1 c \). \[ y=\frac{x^{2}-6 x}{x-2} \]1 answer -
1 answer
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Differentiate. (12 pts) 1. \( y=(4 x-1)(3 x+1)^{4} \) 2. \( y=(2 x+1)^{5 / 2}(4 x-1)^{3 / 2} \) 3. \( y=\frac{x^{2}-6 x}{x-2} \)3 answers -
c) Solve: \[ \left(\mathrm{x}^{2} \mathrm{D}^{2}-\mathrm{xD}+2\right) \mathrm{y}=\mathrm{x} \ln \mathrm{x} \]1 answer -
Find a general solutian of \( y(4)-2 y^{\prime \prime}+y=0 \). (a) \( y=c_{1}+c_{2} x+e^{-2 x}\left(c_{3}+c_{4} x\right) \) (b) \( y=e^{-x}\left(c_{1}+c_{2} x\right)+e^{x}\left(c_{3}+c_{4} x\right) \)1 answer -
#17, 19, 21, 23 25 27 29 31 and 33
7-36 Find the derivative of the function. 7. \( F(x)=\left(x^{4}+3 x^{2}-2\right)^{5} \) 9. \( F(x)=\sqrt{1-2 x} \) 11. \( f(z)=\frac{1}{z^{2}+1} \) 13. \( y=\cos \left(a^{3}+x^{3}\right) \) 15. \( h(1 answer -
#3, 5, 7, 9, 11, 13
3-16 Find \( d y / d x \) by implicit differentiation. 3. \( x^{3}+y^{3}=1 \) 4. \( 2 \sqrt{x} \) 5. \( x^{2}+x y-y^{2}=4 \) 6. \( 2 x^{3}+ \) 7. \( x^{4}(x+y)=y^{2}(3 x-y) \) 9. \( x^{2} y^{2}+x \sin1 answer -
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(32 pts) In Problems 6-9, find all second-order partial derivatives. 6. \( f(x, y)=x+y+x y \) 7. \( f(x, y)=x^{2} y+\cos y+y \sin x \) 8. \( f(x, y)=x e^{y}+y+1 \) 9. \( f(x, y)=\ln (x+y) \)1 answer -
(b) Solve: \[ \left(\mathrm{D}^{3}+2 \mathrm{D}^{2}+\mathrm{D}\right) \mathrm{y}=\mathrm{e}^{2 \mathrm{x}}+\mathrm{x}^{2}+\mathrm{x} \]1 answer -
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Which of the following gives \[ \int_{-1}^{0} \int_{-y}^{\sqrt{-y}} f(x, y) d x d y \] with the order of integration reversed? Hint: Sketch the region and use it to reverse the order of integration a)1 answer -
Which of the following gives \[ \int_{-1}^{0} \int_{-y}^{\sqrt{-y}} f(x, y) d x d y \] with the order of integration reversed? Hint: Sketch the region and use it to reverse the order of integration a1 answer -
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(1 point) Find the Jacobian. \( \frac{\partial(x, y, z)}{\partial(s, t, u)} \), where \( x=-(3 s+t+u), y=s-3 t-4 u, z=s-4 t+u \). \[ \frac{\partial(x, y, z)}{\partial(s, t, u)}= \]3 answers -
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11. Find the derivative for each of the following. \( \begin{array}{ll}\text { a. } & y=\cos (-7 x) \\ \text { d. } & y=4 x \sin ^{4}(5-9 x)^{-1}\end{array} \) b. \( \quad y=-\frac{1}{2} \sin (4+5 x)1 answer -
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1. Solve the following linear systems. a. \( \quad \begin{aligned} {[x, y] } &=[1,6]+s[3,-2] \\ {[x, y] } &=[4,4]+t[-6,4] \end{aligned} \) b. \( \quad[x, y]=[-12,-7]+s[8,-5] \) C. \( \quad[x, y]=[16,13 answers -
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\( f(x)=\left\{\begin{array}{ll}\frac{\left|x^{2}-x-2\right|}{x-2} & \text { if } x \neq 2 \\ 4 & \text { if } x=2\end{array}\right. \)1 answer