Calculus Archive: Questions from July 16, 2022
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3 answers
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3 answers
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9. Find \( y \) for the following condition: \[ \boldsymbol{y}^{\prime \prime}=12 x^{2}-6 x+3, \quad \mathrm{y}^{\prime}(1)=5 \text { and } y(1)=7 \]1 answer -
#9
Match the functions and their derivatives: 1. \( y=\cos ^{3}(x) \) 2. \( y=\sin (x) \tan (x) \) 3. \( y=\cos (\tan (x)) \) 4. \( y=\tan (x) \) A. \( y^{\prime}=-\sin (\tan (x)) / \cos ^{2}(x) \) B. \(1 answer -
#14
14 Find \( d y / d x \) in terms of \( x \) and \( y \) if \( \sqrt{x}=10 \sqrt{y} \). \[ \frac{d y}{d x}= \]1 answer -
#15
5 Find \( d y / d x \) in terms of \( x \) and \( y \) if \( \cos ^{2}(6 y)+\sin ^{2}(6 y)=y+12 \) \[ \frac{d y}{d x}= \]3 answers -
Find the Derivative of the following: \[ y=\log _{3}(3 x-7) \] A. \[ y^{\prime}=\frac{3}{(3 x-7) \ln 3} \] B. \[ y^{\prime}=\frac{3 \ln 3}{(3 x-7)} \] C. \[ y^{\prime}=\frac{3}{3 x-7} \] D. \[ y^{\pri1 answer -
Find the Derivative of the following: \( y=8^{4 x} \) A. \( y^{\prime}=4\left(8^{4 x}\right) \) B. \( y^{\prime}=4 x\left(8^{4 x-1}\right) \) C. \( y^{\prime}=4(\ln 8)\left(8^{4 x}\right) \) D. \( y^{1 answer -
Find the Derivative of the following: \( y=x e^{5 x}-e^{5 x} \) A. \( y^{\prime}=5 x e^{5 x}-4 e^{5 x} \) B. \( y^{\prime}=(x-1) e^{5 x} \) C. \( y^{\prime}=x e^{5 x} \) D. \( y^{\prime}=5 x e^{5 x}-53 answers -
Find the Derivative of the following: \[ y=\cot ^{-1}(\sqrt{x}) \] A. \[ y^{\prime}=\frac{-\sqrt{x}}{(1+x)} \] B. \[ y^{\prime}=\frac{-2 \sqrt{x}}{1+x} \] C. \[ y^{\prime}=\frac{-1}{1+x} \] \[ y^{\pri1 answer -
Find the Derivative of the following: \( y=(x)^{3 x} \) A. \( y^{\prime}=x^{3 x}\left[\frac{3}{x}\right] \) B. \( y^{\prime}=3 x(x)^{3 x-1} \) C. \( y^{\prime}=3(x)^{3 x} \) D. \( y^{\prime}=x^{3 x}[31 answer -
Find the Derivative of the following: \( y=\ln \left(\sin ^{3} x\right) \) A. \( y^{\prime}=3 \cot x \) B. \( y^{\prime}=\tan x \) C. \( y^{\prime}=-3 \tan x \) D. \( y^{\prime}=\frac{-3}{\cos x} \)1 answer -
Integrate \( \mathbf{h}(x, y, z)=\cos (x) \mathbf{i}+\sin (y) \mathbf{j}+y z \mathbf{k} \) over the path: \[ \mathbf{r}(u)=2 u^{2} \mathbf{i}-2 u^{3} \mathbf{j}+u \mathbf{k} \] \( u \in[0,1] \) a) \(1 answer -
using quotient rule
\( y=\frac{6 x-x^{2}}{\sqrt{1-3 x}} \) \( y=\frac{6 x-x^{2}}{(1-3 x)^{\frac{1}{2}}} \)1 answer -
3 answers
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\[ \int d x / \sqrt{ }\left(2 a x-x^{2}\right)=? \] A \[ \sin ^{-1} \frac{x}{a}+C \] \( \frac{1}{a} \tan a x-x+C \) C \[ \sin ^{-1} \frac{x-a}{a}+C \]1 answer -
Evaluate the triple integral. \[ \iiint_{E} y d V, \text { where } E=\{(x, y, z) \mid 0 \leq x \leq 8,0 \leq y \leq x, x-y \leq z \leq x+y\} \] \[ 15 \sin ^{-1}\left(\frac{2}{3}\right) \]1 answer -
Evaluate the double integral. \[ \iint_{D}(2 x+y) d A, \quad D=\{(x, y) \mid 1 \leq y \leq 2, y-1 \leq x \leq 1\} \]1 answer -
\( \underline{2}, \underline{3}, \underline{4}, \underline{5}, \underline{6}, \underline{7}, \underline{8}, \underline{9}, \underline{10}, \underline{11}, \underline{12}, \underline{13}, \underline{141 answer -
Find the gradient vector field \( \nabla f \) of \( f \). \[ f(x, y, z)=7 \sqrt{x^{2}+y^{2}+z^{2}} \]1 answer -
(1 point) Let \( f(x, y)=6 x^{2} y^{2} \) \[ \begin{aligned} f_{x}(x, y) &=\\ f_{x}(5, y) &=\\ f_{x}(x,-4) &=\\ f_{x}(5,-4) &=\\ f_{y}(x, y) &=\\ f_{y}(5, y) &=\\ f_{y}(x,-4) &=\\ f_{y}(5,-4) &= \end{3 answers -
(1 point) Let \( f(x, y, z)=\frac{x^{2}-3 y^{2}}{y^{2}+5 z^{2}} \). Then \[ \begin{array}{l} f_{x}(x, y, z)= \\ f_{y}(x, y, z)= \\ f_{z}(x, y, z)= \end{array} \]1 answer -
Given \( f(x, y)=9 x y^{5}-4 x^{6} y \) \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}= \\ \frac{\partial^{2} f}{\partial y^{2}}= \end{array} \]1 answer -
\( \left(1\right. \) point) Let \( f(x, y)=4 x^{4} y^{3} \) \( f_{x}(x, y)= \) \( f_{x}(1, y)= \) \( f_{x}(x, 1)= \) \( f_{x}(1,1)= \) \( f_{y}(x, y)= \) \( f_{y}(1, y)= \) \( f_{y}(x, 1)= \) \( f_{y}1 answer -
Solve
\( y^{\prime \prime \prime}-y^{\prime}=-2 x ; y(0)=0 \quad ; \quad y^{\prime}(0)=1 ; \quad y^{\prime \prime}(0)=2 \)3 answers -
Solve
\( y^{\prime \prime \prime \prime}-y=8 e^{x} ; y(0)=1 ; y^{\prime}(0)=0 ; y^{\prime \prime}(0)=1 ; y^{\prime \prime \prime}(0)=0 \)1 answer