Calculus Archive: Questions from August 05, 2022
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3 answers
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Challenge: Given \[ \mathrm{f}(x, y)=\min (x, 5 y) . \] For example, \( \mathrm{f}(3,1)=\min (3,5)=3 \). Compute \[ \int_{x=0}^{1} \int_{y=0}^{1} \mathrm{f}(x, y) \mathrm{dy} \mathrm{dx}=\int_{x=0}^{11 answer -
Find the derivative \[ y=\frac{e^{5 x}}{x^{4 / 3}+4} \] \( y^{\prime}=\frac{5 e^{5 x}}{\frac{4}{3} x^{1 / 3}} \) b. \[ y^{\prime}=\frac{5 e^{5 x}\left(x^{4 / 3}+4\right)+e^{5 x}\left(\frac{4}{3} x^{13 answers -
c. Given a vector \( f(x, y)=x^{2} y \), where \( x=t^{3}-t+1, y=2-t^{2} \). Find i. \( \frac{\partial}{\partial x} f(1,1) \) ii. \( \quad \frac{\partial}{\partial y} f(1,1) \)1 answer -
\[ \int_{0}^{1} \int_{0}^{x} f(x, y) d y d x=\int_{0}^{x} \int_{0}^{1} f(x, y) d x d y \] True False3 answers -
Challenge: Given \[ f(x, y)=\min (x, 5 y) . \] For example, \( \mathrm{f}(3,1)=\min (3,5)=3 \). Compute \[ \int_{x=0}^{1} \int_{y=0}^{1} \mathrm{f}(x, y) \mathrm{dy} \mathrm{dx}=\int_{x=0}^{1} \int_{y1 answer -
Evaluate each integral. \[ \text { . } \int_{0}^{x} \int_{x-y}^{x} y d y d z= \] Now evaluate \( \iiint_{E} y d V \), where \( E=\{(x, y, z) \mid 0 \leq x \leq 2,0 \leq y \leq x, x-y \leq z \leq x+y\}1 answer -
Find the Jacobian of the transformation. \[ x=7 v+7 w^{2}, \quad y=6 w+6 u^{2}, \quad z=5 u+5 v^{2} \] \[ \frac{\partial(x, y, z)}{\partial(u, v, w)}= \]1 answer -
Evaluate the triple integral. \[ \iiint_{E} y d V \text {, where } E=\{(x, y, z) \mid 0 \leq x \leq 2,0 \leq y \leq x, x-y \leq z \leq x+y\} \]1 answer -
(1 point) Find the partial derivatives of the function \( f(x, y)=e^{7 x-4 y} \) : \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \]1 answer -
(1 point) Find all first- and second-order partial derivatives of the function \( f(x, y)=8 y e^{4 x} \) : \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y y}(x,1 answer -
For the given parametric equations, find the points \( (x, y) \) corresponding to the parameter values \( t=-2,-1,0,1,2 \). \[ \begin{array}{ll} & x=9 t^{2}+9 t, \quad y=3^{t+1} \\ t=-2 & (x, y)=( \\3 answers -
1 answer
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integrate
\( \int_{0}^{\frac{\pi}{3}} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta \)1 answer -
Let \( a \) and \( b \) be nonzero constants. Find \( \mathrm{dy} / \mathrm{d} x \). \[ y=(x+1)^{a}\left(x^{2}+1\right)^{-3 b} y=(x+1)^{a}\left(x^{2}+1\right)^{-3 b} \]1 answer -
Evaluate each integral. \[ \begin{array}{r} \int_{x-y}^{x+y} y d z= \\ \int_{0}^{x} \int_{x-y}^{x+y} y d z d y= \end{array} \] Now evaluate \( \iiint_{E} y d V \), where \( E=\{(x, y, z) \mid 0 \leq x1 answer -
Find the derivative of the functions below
\( y=\sin \left(t^{2}+t\right) \) \( y=\sin (2 x) \) \( y=\sin x \cos x \) \( f(t)=\ln \left(t^{2}-1\right) \)1 answer -
If \( \iint_{D} f(x, y) d A \geq 0 \), then \( f(x, y) \geq 0 \) for all \( (x, y) \) in \( D \). True False1 answer -
\( \int_{0}^{5} \int_{0}^{\sqrt{25-x^{2}}} \int_{0}^{\sqrt{25-x^{2}-y^{2}}} \frac{1}{1+x^{2}+y^{2}+z^{2}} d z d y d x \)1 answer -
Find the derivative \( y=\sqrt{x^{5}+7 x^{4}+6 x} \) a. \( y^{\prime}=\frac{1}{2}\left(x^{5}+7 x^{4}+6 x\right)^{1 / 2}\left(5 x^{4}+28 x^{3}+6\right) \) b. \( y^{\prime}=\frac{1}{2}\left(x^{5}+7 x^{41 answer -
Find the derivative \[ y=\left(x^{-4}+x^{-2}+4\right) \ln \left(x^{4}+5\right) \] a. \[ y=\left(-4 x^{-5}-2 x^{-3}\right) \ln \left(x^{4}+5\right)+\left(x^{-4}+x^{-2}+4\right)\left(\frac{4 x^{3}}{x^{41 answer -
Find the derivative of the function \[ y=x^{3}+x+6 \] a. \( y^{\prime}=3 x^{2}+1+6 \) b. \( y^{\prime}=3 x^{2}+x \) c. \( y^{\prime}=3 x^{2}+1 \) d. \( y^{\prime}=\frac{x^{4}}{4}+\frac{x^{2}}{2}+6 x+C1 answer -
Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ x y+y x^{2}=2 \] \[ y^{\prime}= \] \[ y^{\prime \prime}= \]3 answers -
1 answer