Calculus Archive: Questions from August 01, 2022
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If \( f(x, y)=x^{3} y+2 x y^{4} \) \[ \left.\frac{\partial^{2}}{\partial x^{2}} f(x, y)\right|_{(-1,1)}= \]3 answers -
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Gradient fields in \( \mathbb{R}^{3} \) Find the vector field \( \mathbf{F}=\nabla \varphi \) \( \varphi(x, y, z)=1 /|\mathbf{r}| \), where \( \mathbf{r}=\langle x, y, z\rangle \)1 answer -
Calculate the derivative of the following functions: - \( y=\sqrt{2-\sqrt{2-x}}+\frac{x}{x^{2}-3} \) - \( y=\sin ^{3}\left(2 x-\tan ^{2} x^{2}\right)^{2} \) - \( y=\frac{x+\sqrt{x}}{1-\frac{x}{x+\sqrt1 answer -
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Find the derivative of the function. \[ \begin{array}{c} \left.g(x)=\int_{6 x}^{7 x} \frac{u^{2}-5}{u^{2}+5} d u \quad \text { Hint: } \int_{6 x}^{7 x} f(u) d u=\int_{6 x}^{0} f(u) d u+\int_{0}^{7 x}1 answer -
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Calculate the derivative of the following functions: - \( y=\sqrt{2-\sqrt{2-x}}+\frac{x}{x^{2}-3} \) - \( y=\sin ^{3}\left(2 x-\tan ^{2} x^{2}\right)^{2} \) - \( y=\frac{x+\sqrt{x}}{1-\frac{x}{x+\sqrt1 answer -
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15. \( Y(s)=\frac{2 s-3}{(s-1)^{2}+5} \) 17. \( Y(s)=\frac{3 s+2}{s^{2}+4 s+29} \) \( Y(s)=2 \cdot \frac{s}{s^{2}+4}+\frac{5}{2} \cdot \frac{2}{s^{2}+4} \) Use the technique developed in Exercises \1 answer -
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(1 point) Match each vector field with its graph. 1. \( \vec{F}(x, y)=\langle-y, x\rangle \) 2. \( \vec{F}(x, y)=\langle y, x\rangle \) 3. \( \vec{F}(x, y)=\langle x, y\rangle \) 4. \( \vec{F}(x, y)=\1 answer -
Find the function \( y=y(x) \), given \[ \frac{d y}{d x}+y \cos (x)=2 \cos (x), \quad y(0)=4 \] Answer: \( \quad y(x)= \)3 answers -
Find the derivative of the function \( y=x^{x} \). A) \( y^{\prime}=x^{x}(\ln (x)+1) \) B) \( y^{\prime}=\ln (x)+1 \) C) \( y^{\prime}=\ln (x)\left(x^{x}+1\right) \) D) \( y^{\prime}=x^{x-1} \)1 answer -
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Solve via laplace transform method.
\( y^{(4)}-4 y=0 ; y(0)=y^{\prime}(0)=1, y^{\prime \prime}(0)=-2, y^{(3)}(0) \)1 answer -
Find the partial derivatives of \( f(x, y)=5 x^{3} y^{2} \). (a) \( f_{x}(x, y)= \) (b) \( f_{y}(x, y)= \) (c) \( f_{x}(5, y)= \) (d) \( f_{x}(x, 1)= \) (e) \( f_{y}(5, y)= \) (f) \( f_{y}(x, 1)= \) (1 answer