Calculus Archive: Questions from April 18, 2023
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7.If \( f(x, y)=x^{3} y^{2}+3 x^{4}+4 y^{3} \ln x \) Find \( f_{x}(x, y), f_{y}(x, y), f_{y y}(x, y), f_{x y}(x, y) \)2 answers -
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Determinar cuál de las siguientes series converge. \[ \begin{array}{l} \sum_{n=1}^{\infty}\left(4+(-1)^{n}\right)^{n} \\ \sum_{n=0}^{\infty} 5\left(\frac{3}{2}\right)^{n} \\ \sum_{n=1}^{\infty} \frac2 answers -
Find \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} \). \[ \mathbf{a}=-\mathbf{i}-2 \mathbf{j}+5 \mathbf{k}, \mathbf{b}=6 \mathbf{i}-3 \mathbf{j}-2 \mathbf{k} \]2 answers -
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16. Evaluate the integral using the given substitution. \[ \int 9\left(y^{6}+4 y^{3}+1\right)^{3}\left(2 y^{5}+4 y^{2}\right) d y, \quad u=y^{6}+4 y^{3}+1 \] A. \( \frac{9}{4}\left(y^{6}+4 y^{3}+1\rig2 answers -
Solve the following initial value problem: a. \( y \frac{d y}{d x}=8 x e^{2 x}\left(y^{2}+1\right) \quad y(0)=0 \) b. \( \quad \frac{d y}{d x}=(y-1)^{2} e^{x-1} \quad y(1)=2 \)2 answers -
i need help with #36 please!
In Problems 31-42, find \( \frac{d y}{d x} \) for the indicated function \( y \). 31. \( y=\log _{2} x \) 32. \( y=3 \log _{5} x \) 33. \( y=3^{x} \) 34. \( y=4^{x} \) 35. \( y=2 x-\log x \) 36. \( y=2 answers -
Let \( y=\ln \left(x^{2}+y^{2}\right) \). Determine the derivative \( y^{\prime} \) at the point \( \left(\sqrt{e^{6}-36}, 6\right) \). \[ y^{\prime}\left(\sqrt{e^{6}-3 t}\right. \]3 answers -
183. \( \iiint_{B}(x \cos y+z) d V \), where \( B=\{(x, y, z) \mid 0 \leq x \leq 1,0 \leq y \leq \pi,-1 \leq z \leq 1\} \)2 answers -
Find \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} \). \[ \mathbf{a}=-\mathbf{i}-2 \mathbf{j}+9 \mathbf{k}, \mathbf{b}=6 \mathbf{i}-3 \mathbf{j}-2 \mathbf{k} \]2 answers -
2. Evaluate the following (a) \( f(x, y, z)=\frac{x}{y+z}, \quad f_{x y z}= \) ? (b) \( f(x, y)=\sin \left(x^{2}-y\right) \quad f_{y}(0, \pi)= \) ? (c) \( f(x, y)=e^{x z-x^{2} z^{3}} \quad f_{z}(3,0)=2 answers -
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\( \begin{array}{l}y: \frac{d y}{d x}=e^{y} \sin x \\ y=e^{\cos x}+C \\ y=e^{-\cos x}+C \\ y=\ln |\cos x+C| \\ y=-\ln |\cos x+C| \\ y=-\ln |\cos x|+C\end{array} \)2 answers -
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6. Evaluate \( \int_{C} \mathbf{F} \cdot \mathrm{d} \mathbf{r} \) if \( \mathbf{F}(x, y)=2 y^{2} \boldsymbol{i}+x y \mathbf{j} \) and \( \mathbf{r}(t)=e^{t} \mathbf{i}+2 \mathrm{e}^{-t} \mathbf{j},-12 answers -
stuck on f) and c)
PART A 1. Determine \( \frac{d y}{d x} \) for each of the following: a. \( y=\tan 3 x \) d. \( y=\frac{x^{2}}{\tan \pi x} \) b. \( y=2 \tan x-\tan 2 x \) e. \( y=\tan \left(x^{2}\right)-\tan ^{2} x \)2 answers -
stuck on f)
3. Determine \( y^{\prime} \) for each of the following: a. \( y=\tan (\sin x) \) d. \( y=(\tan x+\cos x)^{2} \) b. \( y=\left[\tan \left(x^{2}-1\right)\right]^{-2} \) e. \( y=\sin ^{3} x \tan x \) c.2 answers -
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1904. Find \( \frac{\partial^{2} u}{\partial x^{2}} \) if \( u=f(x, y, z) \), where \( z=\varphi(x, y) \).2 answers -
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