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Questions and Answers
Show that $x \frac{\partial f}{\partial v} + y \frac{\partial f}{\partial u}$ equals $e^{2u} \frac{\partial f}{\partial y}$ if $z = f(x, y)$ and $x = e^{u} \cos v$, $y = e^{u} \sin v$.
Show that $x \frac{\partial f}{\partial v} + y \frac{\partial f}{\partial u}$ equals $e^{2u} \frac{\partial f}{\partial y}$ if $z = f(x, y)$ and $x = e^{u} \cos v$, $y = e^{u} \sin v$.
x \frac{\partial f}{\partial v} + y \frac{\partial f}{\partial u} = e^{2u} \frac{\partial f}{\partial y}
Show that $u \frac{\partial H}{\partial u} + v \frac{\partial H}{\partial v} + w \frac{\partial H}{\partial w}$ equals $\frac{\partial H}{\partial x} + 2y \frac{\partial H}{\partial y} + 3z \frac{\partial H}{\partial z}$ if $H = f(x, y, z)$ where $x = u + v + w$, $y = vw + wu + uv$, and $z = uvw$.
Show that $u \frac{\partial H}{\partial u} + v \frac{\partial H}{\partial v} + w \frac{\partial H}{\partial w}$ equals $\frac{\partial H}{\partial x} + 2y \frac{\partial H}{\partial y} + 3z \frac{\partial H}{\partial z}$ if $H = f(x, y, z)$ where $x = u + v + w$, $y = vw + wu + uv$, and $z = uvw$.
u \frac{\partial H}{\partial u} + v \frac{\partial H}{\partial v} + w \frac{\partial H}{\partial w} = \frac{\partial H}{\partial x} + 2y \frac{\partial H}{\partial y} + 3z \frac{\partial H}{\partial z}
Evaluate du/dx, du/dy at the point (√3, 2, 1) for $u = \frac{p - q}{q - r}$, where $p = x + y + z$, $q = x - y + z$, $r = x + y - z$.
Evaluate du/dx, du/dy at the point (√3, 2, 1) for $u = \frac{p - q}{q - r}$, where $p = x + y + z$, $q = x - y + z$, $r = x + y - z$.
du/dx = 1, du/dy = -1
Evaluate du/dx, du/dy at the point (π/4, 1/2, -1/2) for $u = e^r \sin^{-1} p$, where $p = \sin x$, $q = z^2 \ln y$, $r = \frac{1}{z}$.
Evaluate du/dx, du/dy at the point (π/4, 1/2, -1/2) for $u = e^r \sin^{-1} p$, where $p = \sin x$, $q = z^2 \ln y$, $r = \frac{1}{z}$.
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Find the directional derivative of the function $f(x, y, z) = x^2y^2 + 2z^2$ in the direction of the line PQ where Q has coordinates (5, 0, 4).
Find the directional derivative of the function $f(x, y, z) = x^2y^2 + 2z^2$ in the direction of the line PQ where Q has coordinates (5, 0, 4).
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Find the directional derivative of the function (i) $f(x, y, z) = xy^2 + yz^3$ at (2, -1, 1) in the direction of i + 2j + 2k.
Find the directional derivative of the function (i) $f(x, y, z) = xy^2 + yz^3$ at (2, -1, 1) in the direction of i + 2j + 2k.
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Find the directional derivative of the function (ii) $f(x, y, z) = x^2 + y^2 + 4xyz$ at (1, -2, 2) in the direction of 2i - 2j + k.
Find the directional derivative of the function (ii) $f(x, y, z) = x^2 + y^2 + 4xyz$ at (1, -2, 2) in the direction of 2i - 2j + k.
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Find the directional derivative of the function (iii) $f(x, y, z) = 4xz^3 - 3x^2yz^2$ at (2, -1, 2) along the z-axis.
Find the directional derivative of the function (iii) $f(x, y, z) = 4xz^3 - 3x^2yz^2$ at (2, -1, 2) along the z-axis.
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In what direction should a fly at (1, 1, 2) fly to get warm as soon as possible, given the temperature function T(x, y, z) = x² + y² + z²?
In what direction should a fly at (1, 1, 2) fly to get warm as soon as possible, given the temperature function T(x, y, z) = x² + y² + z²?
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Find the directions in which the functions increase and decrease most rapidly and find the directional derivatives of the functions in these directions for (i) $f(x, y) = x^2y + e^{xy} \sin y$ at Po(1, 0).
Find the directions in which the functions increase and decrease most rapidly and find the directional derivatives of the functions in these directions for (i) $f(x, y) = x^2y + e^{xy} \sin y$ at Po(1, 0).
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Find the directional derivatives of the function (ii) $f(x, y, z) = \ln(xy) + \ln(yz) + \ln(zx)$.
Find the directional derivatives of the function (ii) $f(x, y, z) = \ln(xy) + \ln(yz) + \ln(zx)$.
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Determine the increase and decrease directions and directional derivatives for (iii) $f(x, y, z) = \frac{x}{y} - yz$ at Po(4, 1, 1).
Determine the increase and decrease directions and directional derivatives for (iii) $f(x, y, z) = \frac{x}{y} - yz$ at Po(4, 1, 1).
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Find the directional derivatives of the function (iv) $f(x, y, z) = \ln(x^2 + y^2)$.
Find the directional derivatives of the function (iv) $f(x, y, z) = \ln(x^2 + y^2)$.
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Study Notes
Partial Derivatives and Functions
- If ( z = f(x, y) ) and ( x = euc \cos v ), ( y = euc \sin v ), then ( x\frac{\partial f}{\partial v} + y\frac{\partial f}{\partial u} = e^{2u} \frac{\partial f}{\partial y} ).
- For ( H = f(x, y, z) ) where ( x = u + v + w ), ( y = vw + wu + uv ), ( z = uvw ), it follows that ( u \frac{\partial H}{\partial u} + v \frac{\partial H}{\partial v} + w \frac{\partial H}{\partial w} = \frac{\partial H}{\partial x} + 2y \frac{\partial H}{\partial y} + 3z \frac{\partial H}{\partial z} ).
Evaluating Derivatives
- Evaluate ( \frac{du}{dx} ), ( \frac{du}{dy} ) at the point (√3, 2, 1) for the function ( u = \frac{p - q}{q - r} ), where ( p = x + y + z ), ( q = x - y + z ), ( r = x + y - z ).
- For ( u = e^r \sin^{-1}p ) with ( p = \sin x ), ( q = z^2 \ln y ), ( r = \frac{1}{z} ), evaluate at the point ((\frac{\pi}{4}, \frac{1}{2}, -\frac{1}{2})).
Directional Derivatives
- Find the directional derivative of ( f(x, y, z) = x^2y^2 + 2z^2 ) in the direction toward point Q(5, 0, 4).
- Determine maximum directional derivative and direction.
- For ( f(x, y, z) = xy^2 + yz^3 ) at (2, -1, 1), compute in the direction of ( i + 2j + 2k ).
- For ( f(x, y, z) = x^2 + y^2 + 4xyz ) at (1, -2, 2), evaluate in the direction of ( 2i - 2j + k ).
- For ( f(x, y, z) = 4xz^3 - 3x^2yz^2 ) at (2, -1, 2), compute along the z-axis.
Temperature Gradient
- The temperature at point ((x, y, z)) is given by ( T(x, y, z) = x^2 + y^2 + z^2 ).
- A fly at (1, 1, 2) should fly in the direction of the temperature gradient to warm up rapidly.
Function Behavior
- Identify directions of maximum increase and decrease for given functions.
- Compute directional derivatives for:
- ( f(x, y) = x^2y + e^{xy} \sin y ) at point ( (1, 0) ).
- ( f(x, y, z) = \ln xy + \ln yz + \ln zx ).
- ( f(x, y, z) = \frac{x}{y} - yz ) at point ( (4, 1, 1) ).
- ( f(x, y, z) = \ln(x^2 + y^2) ) at appropriate points.
Key Concepts
- Ensure understanding of partial derivatives and their applications in multi-variable functions.
- Focus on understanding how to derive directional derivatives and their interpretation in physical contexts, such as temperature gradients.
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Description
This quiz focuses on advanced calculus problems related to functions of multiple variables. You will demonstrate your understanding of partial derivatives and their applications through various equations and transformations. Challenge yourself with these complex problems and refine your calculus skills.
