Partial Differentiation Explained

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Questions and Answers

Given $f(x, y) = x^4y^2 + 3x - y$, which expression correctly represents $\frac{\partial f}{\partial x}$?

$4x^3y^2 + 3$ (correct)
$2x^4y - 1$
$4x^3y^2 - 1$
$4x^3y^2 + 3 - 1$

If $f(x, y) = e^{x^2 + y}$, determine $\frac{\partial f}{\partial y}$.

$e^{x^2 + y}$ (correct)
0
$xe^{x^2 + y}$
$2xe^{x^2 + y}$

For $f(x, y) = \sin(xy)$, find $\frac{\partial^2 f}{\partial x \partial y}$.

$\cos(xy) - x^2y^2\sin(xy)$
$\cos(xy) - y^2\sin(xy)$ (correct)
$\cos(xy) - x^2\sin(xy)$
$\cos(xy) - xy\sin(xy)$

Given $f(x, y) = x^3 + y^3 - 3xy$, determine the critical points by solving the system of equations derived from setting the first partial derivatives to zero.

$(0, 0)$ and $(1, 1)$ (C)

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If $z = f(x, y)$, $x = g(t)$, and $y = h(t)$, which of the following expresses the chain rule for $\frac{dz}{dt}$?

$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$ (C)

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Given the surface $z = f(x, y)$, what represents the normal vector to the tangent plane at the point $(x_0, y_0, z_0)$?

$\mathbf{n} = \left\langle \frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0), -1 \right\rangle$ (A)

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What condition must be met for Clairaut's Theorem to apply to a function $f(x, y)$?

The second-order mixed partial derivatives must be continuous. (B)

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Given $z = f(x, y)$, which expression defines the total differential $dz$?

$dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ (B)

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To find the local maxima and minima of a function $f(x, y)$, what is the first step?

Set the first partial derivatives equal to zero and solve for critical points. (C)

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If $D(a, b) > 0$ and $\frac{\partial^2 f}{\partial x^2}(a, b) < 0$ at a critical point $(a, b)$, what does this indicate according to the second derivative test?

A local maximum (A)

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What is the purpose of using Lagrange multipliers?

To find the maximum and minimum values of a function subject to a constraint. (D)

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What system of equations needs to be solved when using Lagrange multipliers to optimize $f(x, y)$ subject to the constraint $g(x, y) = k$?

$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x}$, $\frac{\partial f}{\partial y} = \lambda \frac{\partial g}{\partial y}$, $g(x, y) = k$ (A)

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Find $\frac{\partial z}{\partial s}$ if $z = f(x, y)$, where $x = g(s, t)$ and $y = h(s, t)$.

$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$ (B)

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What does the linear approximation $L(x, y)$ of a function $f(x, y)$ at a point $(a, b)$ represent?

An approximation of $f(x, y)$ for points $(x, y)$ near $(a, b)$. (D)

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Given a function $f(x, y)$, how is the discriminant $D(x, y)$ defined for the second derivative test?

$D(x, y) = \left( \frac{\partial^2 f}{\partial x^2} \right) \left( \frac{\partial^2 f}{\partial y^2} \right) - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2$ (A)

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Flashcards

Partial Differentiation

Rate of change of a multivariable function with respect to one variable, keeping others constant.

∂ (Partial Derivative Symbol)

Symbol used to denote a partial derivative, distinguishing it from ordinary derivatives.

Calculating ∂f/∂x

To find ∂f/∂x, treat y as constant and differentiate f(x,y) with respect to x.

Higher-Order Partial Derivatives

Derivatives of partial derivatives. E.g., ∂²f/∂x² or f_xy.

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Clairaut's Theorem

If mixed partial derivatives are continuous, the order of differentiation doesn't matter.

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Chain Rule (Partial Derivatives)

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

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Tangent Plane Equation

z - z₀ = (∂f/∂x)(x₀,y₀)(x - x₀) + (∂f/∂y)(x₀,y₀)(y - y₀)

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Normal Vector to Tangent Plane

<∂f/∂x(x₀, y₀), ∂f/∂y(x₀, y₀), -1>

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Linear Approximation

L(x, y) = f(a, b) + (∂f/∂x)(a, b)(x - a) + (∂f/∂y)(a, b)(y - b)

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Total Differential dz

dz = (∂f/∂x)dx + (∂f/∂y)dy

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Finding Critical Points

Set ∂f/∂x = 0 and ∂f/∂y = 0

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Discriminant D

D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²

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Second Derivative Test

If D > 0 and ∂²f/∂x² > 0, then local minimum. If D > 0 and ∂²f/∂x² < 0, then local maximum. If D < 0, saddle point.

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Lagrange Multipliers

Method to find max/min of f(x,y) subject to constraint g(x,y) = k.

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Lagrange Multiplier Equations

Solve: ∂f/∂x = λ∂g/∂x, ∂f/∂y = λ∂g/∂y and g(x, y) = k.

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Study Notes

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Podcast

Questions and Answers

Given $f(x, y) = x^4y^2 + 3x - y$, which expression correctly represents $\frac{\partial f}{\partial x}$?

If $f(x, y) = e^{x^2 + y}$, determine $\frac{\partial f}{\partial y}$.

For $f(x, y) = \sin(xy)$, find $\frac{\partial^2 f}{\partial x \partial y}$.

Given $f(x, y) = x^3 + y^3 - 3xy$, determine the critical points by solving the system of equations derived from setting the first partial derivatives to zero.

If $z = f(x, y)$, $x = g(t)$, and $y = h(t)$, which of the following expresses the chain rule for $\frac{dz}{dt}$?

Given the surface $z = f(x, y)$, what represents the normal vector to the tangent plane at the point $(x_0, y_0, z_0)$?

What condition must be met for Clairaut's Theorem to apply to a function $f(x, y)$?

Given $z = f(x, y)$, which expression defines the total differential $dz$?

To find the local maxima and minima of a function $f(x, y)$, what is the first step?

If $D(a, b) > 0$ and $\frac{\partial^2 f}{\partial x^2}(a, b) < 0$ at a critical point $(a, b)$, what does this indicate according to the second derivative test?

What is the purpose of using Lagrange multipliers?

What system of equations needs to be solved when using Lagrange multipliers to optimize $f(x, y)$ subject to the constraint $g(x, y) = k$?

Find $\frac{\partial z}{\partial s}$ if $z = f(x, y)$, where $x = g(s, t)$ and $y = h(s, t)$.

What does the linear approximation $L(x, y)$ of a function $f(x, y)$ at a point $(a, b)$ represent?

Given a function $f(x, y)$, how is the discriminant $D(x, y)$ defined for the second derivative test?

Flashcards

Partial Differentiation

∂ (Partial Derivative Symbol)

Calculating ∂f/∂x

Higher-Order Partial Derivatives

Clairaut's Theorem

Chain Rule (Partial Derivatives)

Tangent Plane Equation

Normal Vector to Tangent Plane

Linear Approximation

Total Differential dz

Finding Critical Points

Discriminant D

Second Derivative Test

Lagrange Multipliers

Lagrange Multiplier Equations

Study Notes

Studying That Suits You

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