Calculus Problems and Answers

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 $x = e^{u} \cos v$, $y = e^{u} \sin v$.

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 $z = uvw$.

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 $p = x + y + z$, $q = x - y + z$, $r = x + y - z$.

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 $p = \sin x$, $q = z^2 \ln y$, $r = \frac{1}{z}$.

du/dx = 0, du/dy = \frac{-e^{-2}}{2} Signup and view all the answers

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).

The maximum direction is towards Q and the value is calculated to be a specific numerical derivative. Signup and view all the answers

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.

Calculated directional derivative value is dependent on the input values and directional vector. Signup and view all the answers

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.

Calculated directional derivative value based on the provided point and direction. Signup and view all the answers

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.

Calculated directional derivative value specifically along the z-axis direction. Signup and view all the answers

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?

The fly should head in the direction of the gradient of the temperature function. Signup and view all the answers

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).

Calculated directional derivative values for increase and decrease directions. Signup and view all the answers

Find the directional derivatives of the function (ii) $f(x, y, z) = \ln(xy) + \ln(yz) + \ln(zx)$.

Specific values derived based on the behavior of the given function. Signup and view all the answers

Determine the increase and decrease directions and directional derivatives for (iii) $f(x, y, z) = \frac{x}{y} - yz$ at Po(4, 1, 1).

Results depend on calculated values for the specified point. Signup and view all the answers

Find the directional derivatives of the function (iv) $f(x, y, z) = \ln(x^2 + y^2)$.

Specific directional derivative value determined at a specific point. Signup and view all the answers

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.