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Questions and Answers
State and prove the Mean Value Theorem for a function of two variables.
State and prove the Mean Value Theorem for a function of two variables.
The Mean Value Theorem states that if a function is continuous on a closed disk and differentiable on the interior, there exists a point in the interior where the gradient is proportional to the direction between two boundary points. Proving involves using the concept of differentiable functions and the fundamental theorem of calculus.
Change the order of integration for the integral $\int_{0}^{2} \int_{x^{2} / 8}^{x} xy , dx , dy$.
Change the order of integration for the integral $\int_{0}^{2} \int_{x^{2} / 8}^{x} xy , dx , dy$.
The order of integration changes to $\int_{0}^{2} \int_{0}^{\sqrt{8y}} xy , dy , dx$.
Using triple integration, find the volume of a sphere of radius.
Using triple integration, find the volume of a sphere of radius.
The volume of a sphere of radius r is $V = \frac{4}{3}\pi r^{3}$.
If $u = log(x^3 + y^3 - x^2y - xy^2)$, prove that: $x^{2}\frac{\partial^{2}u}{\partial x^{2}} + 2xy\frac{\partial^{2}u}{\partial x\partial y} + y^{2}\frac{\partial^{2}u}{\partial y^{2}} = -3$
If $u = log(x^3 + y^3 - x^2y - xy^2)$, prove that: $x^{2}\frac{\partial^{2}u}{\partial x^{2}} + 2xy\frac{\partial^{2}u}{\partial x\partial y} + y^{2}\frac{\partial^{2}u}{\partial y^{2}} = -3$
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Evaluate $\iint y , dx , dy$ over the area bounded by $y = x^{2}$.
Evaluate $\iint y , dx , dy$ over the area bounded by $y = x^{2}$.
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Study Notes
Mean Value Theorem for Two Variables
- The Mean Value Theorem for a function of two variables states that if $f(x,y)$ is continuous on a closed and bounded region $R$ and differentiable on the interior of $R$, then there exists a point $(c,d)$ in the interior of $R$ such that:
$$ f(a,b) - f(c,d) = \nabla f(c,d) \cdot (a-c, b-d) $$
- The proof involves using the Fundamental Theorem of Calculus for functions of one variable and applying it to each of the partial derivatives of $f$.
Changing the Order of Integration
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To change the order of integration for the integral $\int_{0}^{2} \int_{x^{2} / 8}^{x} xy , dx , dy$, we need to:
- Identify the limits of integration for $x$ and $y$ from the original integral
- Sketch the region of integration
- Determine the new limits of integration for $x$ and $y$ after changing the order of integration
- Evaluate the integral with the new limits of integration
The Volume of A Sphere
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The volume of a sphere is $\frac{4}{3}\pi r^{3}$ and can be found using triple integration.
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We use spherical coordinates and integrate over the volume enclosed by the sphere.
Partial Derivatives
- Start with the function $u = log(x^3 + y^3 - x^2y - xy^2)$.
- Calculate the partial derivatives: $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, $\frac{\partial^2 u}{\partial x^2}$, $\frac{\partial^2 u}{\partial y^2}$, and $\frac{\partial^2 u}{\partial x \partial y}$.
- Substitute these partial derivatives into the equation and simplify to show that it equals $-3$.
Double Integrals
- The given integral $\iint, y,dx , dy$ is over an area defined by the curves $y = x^2$ and $x = ...$
- We need to identify the limits of integration that define the region of integration.
- Integrate with respect to $x$ first, then with respect to $y$ or vice versa depending on the limits of integration found.
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Description
This quiz covers key concepts related to the Mean Value Theorem for two variables, including its proof and applications. It also explores techniques for changing the order of integration in double integrals, as well as calculating the volume of a sphere.
