Partial Derivatives and Continuity Concepts
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Questions and Answers

What does the expression $f(a + \Delta x, b) - f(a, b)$ measure?

  • Increment of $f$ with respect to $y$
  • Constant value of $f$ at $(a, b)$
  • Total increment of $f$ at the point $(a, b)$
  • Increment of $f$ with respect to $x$ (correct)

What does the increment $\Delta f$ represent at the point $(a, b)$?

  • The change in $f$ due to small changes in $x$ and $y$ (correct)
  • The total area under the curve of $f$
  • The value of $f$ at point $(a, b)$
  • The average rate of change of $f$

When is a function $f$ said to be differentiable at a point $(a, b)$?

  • When it maintains a constant value throughout the domain
  • When the total increment $\\Delta f$ can be expressed in a specific form (correct)
  • When the partial derivatives are not continuous
  • When it has no discontinuities in the region

What are $\epsilon_1$ and $\epsilon_2$ in the context of the total increment $\Delta f$?

<p>Errors that vanish as increments approach zero (C)</p> Signup and view all the answers

What happens to $\epsilon_1$ and $\epsilon_2$ as $\Delta x$ and $\Delta y$ both approach zero?

<p>They both approach zero (A)</p> Signup and view all the answers

Under which condition can $f$ be represented as $\Delta f = f_x(a, b) \Delta x + f_y \Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y$?

<p>When the partial derivatives are continuous (D)</p> Signup and view all the answers

What does the expression $f(a, b + \Delta y) - f(a, b)$ measure?

<p>Increment of $f$ in the direction of $y$ (C)</p> Signup and view all the answers

Which of the following is a requirement for the Increment Theorem to apply?

<p>The function must have continuous partial derivatives (B)</p> Signup and view all the answers

Which of the following expressions is equivalent to the total increment $\Delta f$?

<p>$f(a + h, b + k) - f(a, b)$ (C)</p> Signup and view all the answers

What is the consequence of having continuous partial derivatives for the function 𝑓(𝑥, 𝑦) in the Chain Rule 1?

<p>It ensures that 𝑓(𝑥, 𝑦) can be differentiated. (B)</p> Signup and view all the answers

What is the form of the differential expressed in Chain Rule 1?

<p>$ rac{dz}{dt} = rac{ackslash partial z}{ackslash partial x} rac{dx}{dt} + rac{ackslash partial z}{ackslash partial y} rac{dy}{dt}$ (C)</p> Signup and view all the answers

In which situation can implicit differentiation be applied according to the content?

<p>When 𝐹(a, b, c) = 0 and 𝐹z(a, b, c) ≠ 0. (A)</p> Signup and view all the answers

How is $z_x$ calculated from a function defined implicitly by 𝐹(x, y, z) = 0?

<p>$z_x = - rac{F_x}{F_z}$ (B)</p> Signup and view all the answers

What does the expression $z(t)$ = $ rac{1}{2}sin(2t)$ illustrate in the context of the Chain Rule?

<p>It is a direct application of Chain Rule 2. (B)</p> Signup and view all the answers

In the example with $z = e^x sin(y)$, what does the term $(e^x sin(y))t^2$ indicate?

<p>It correlates the changes in $z$ to changes in $t$. (D)</p> Signup and view all the answers

What is the derivative expression for $z$ when $z = e^x sin(y)$ and $x = st^2, y = s^2t$?

<p>t * e^s(t sin(2st) + 2s cos(2st)) (B)</p> Signup and view all the answers

Which of the following correctly describes what happens as $ riangle t ightarrow 0$ in the proof?

<p>It implies that all $ riangle x$ must also approach 0. (B)</p> Signup and view all the answers

Which condition is necessary for a function to be defined as $z = f(x, y)$ through the implicit function theorem?

<p>The function must be defined in an open ball around a point. (B)</p> Signup and view all the answers

What does it imply if both partial derivatives $f_x(a, b)$ and $f_y(a, b)$ exist at a point?

<p>The function $f(x, y)$ can be discontinuous at $(a, b)$. (D)</p> Signup and view all the answers

Under which condition is the function $f(x, y)$ declared continuous at $(a, b)$?

<p>If at least one of the partial derivatives is continuous in a region around $(a, b)$. (C)</p> Signup and view all the answers

What does it mean if the limit of $f(x, y)$ as $(x,y) o (0,0)$ does not exist?

<p>The function may still have a defined value at $(0, 0)$. (D)</p> Signup and view all the answers

What is the significance of the increment theorem for defining the partial derivative $f_x(a, b)$?

<p>It provides a formula for finding limits in terms of $h$ as it approaches zero. (A)</p> Signup and view all the answers

Which of the following statements is true regarding the continuity of the function $f(x, y)$ at the origin?

<p>The function $f(x, y)$ is continuous if the limits along different paths are equal. (C)</p> Signup and view all the answers

What conclusion can be drawn from the example of the function defined piecewise at the origin?

<p>The function can be discontinuous even with existing partial derivatives. (D)</p> Signup and view all the answers

Which of the following is an implication of the statement 'if $f_x(a, b)$ exists, then $f(x, b)$ is continuous at $x = a$'?

<p>Existence of the partial derivative implies local linearity. (C)</p> Signup and view all the answers

What does the existence of the limit $ ext{lim}_{(x,y) o (0,0)} f(x,y)$ depend on for the function to be continuous?

<p>It must equal the value of $f(0, 0)$. (C)</p> Signup and view all the answers

What condition must at least one of the partial derivatives meet for continuity of $f(x, y)$ around $(a, b)$?

<p>Be continuous at $(a, b)$ or be bounded in a neighborhood surrounding $(a, b)$. (D)</p> Signup and view all the answers

What is the formula for the directional derivative represented mathematically?

<p>$(D_u f)(x_0, y_0) = (∇f)(x_0, y_0) · u$ (D)</p> Signup and view all the answers

What are the conditions for the formula of the directional derivative to be valid?

<p>ƒ_x and ƒ_y are continuous at (x₀, y₀). (A), ƒ is differentiable at (x₀, y₀). (C)</p> Signup and view all the answers

In the calculation of the directional derivative for the function $f(x, y) = xe^y + cos(xy)$ at (2, 0), what is the value of $(∇f)(2, 0)$?

<p>$1 · old{i} + 2 · old{j}$ (B)</p> Signup and view all the answers

If a point (x₀, y₀) moves in the direction of a unit vector $u$ by a small amount $h$, how is the change in $f(x, y)$ approximated?

<p>$f(x₀ + a h, y₀ + b h) ≈ f(x₀, y₀) + h(D_u f)(x₀, y₀)$ (D)</p> Signup and view all the answers

What does the directional derivative represent concerning the gradient and unit vector?

<p>It is the projection of the gradient in the direction of u. (D)</p> Signup and view all the answers

What is the value of the directional derivative $(D_u f)(P)$ if $(∇f)(0, 1, 0) · u = -2$ and the unit vector in direction v is $(2; 1; -2)$?

<p>-0.067 (D)</p> Signup and view all the answers

How much does the function $f(x, y, z) = y sin x + 2yz$ change when moving 0.1 units from (0, 1, 0) toward (2, 2, -2)?

<p>-0.067 units (B)</p> Signup and view all the answers

What is the significance of the formula $(D_u f) = ∇f · old{u}$ in differential calculus?

<p>It describes the rate of change of a function at a particular point. (C)</p> Signup and view all the answers

What happens to the value of the directional derivative if the unit vector u is not normalized?

<p>The directional derivative may not accurately represent the rate of change. (C)</p> Signup and view all the answers

What is the form of the remainder term 𝑅𝑛(𝑥) in Taylor's Formula?

<p>$\frac{f^{(n+1)}(t)}{(n+1)!} (x - a)^{n+1}$ (B)</p> Signup and view all the answers

At which point is the derivative taken as the left-hand derivative in Taylor's formula?

<p>At 𝑥 = 𝑏 (B)</p> Signup and view all the answers

What is the significance of continuously differentiable functions in Taylor's formula?

<p>It ensures the existence of derivatives up to order n. (A)</p> Signup and view all the answers

What does the term $f^{(k)}(a)$ represent in Taylor's formula?

<p>The k-th derivative at point 𝑎 (A)</p> Signup and view all the answers

How is the estimate for the remainder term $R_n(x)$ bounded?

<p>$\frac{m(x - a)^{n + 1}}{n!} \leq R_n(x) \leq \frac{M(x - a)^{n + 1}}{(n + 1)!}$ (A)</p> Signup and view all the answers

In induction proofs for Taylor's formula, what is assumed true for 𝑛 = 𝑘?

<p>That Taylor's formula holds for the k-th order polynomial (B)</p> Signup and view all the answers

Which of the following terms contributes to the Taylor series expansion?

<p>All of the above (D)</p> Signup and view all the answers

Which expression corresponds to the integral remainder term for an (n + 1)-times continuously differentiable function?

<p>$R_n(x) = \int_a^x (x - t)^n f^{(n+1)}(t) dt$ (D)</p> Signup and view all the answers

Which of the following statements about Taylor’s formula is true?

<p>The function must be continuously differentiable over the interval. (C)</p> Signup and view all the answers

What is the formula for the directional derivative of a function at a point in a given direction?

<p>$(D_u f)(x, y) = f_x(x, y) a + f_y(x, y) b$ (C)</p> Signup and view all the answers

Which expression correctly defines a unit vector in the direction of $( rac{ ext{cos } rac{ ext{π}}{6}}{ ext{sin } rac{ ext{π}}{6}})$?

<p>$u = rac{ ext{√3}}{2} extbf{i} + rac{1}{2} extbf{j}$ (B)</p> Signup and view all the answers

What role do continuous partial derivatives play in the theorem regarding directional derivatives?

<p>They guarantee the existence of the directional derivative. (C)</p> Signup and view all the answers

In the proof, what function is defined to show the existence of the directional derivative?

<p>$g(h) = f(x_0 + a h, y_0 + b h)$ (B)</p> Signup and view all the answers

What is the significance of the gradient symbol ∇ in the context of the directional derivative?

<p>It is the vector of first partial derivatives of the function. (C)</p> Signup and view all the answers

What would be the directional derivative of the function $f(x, y) = x^3 - 3xy + 4y^2$ in the direction of the unit vector $( rac{ ext{√3}}{2}, rac{1}{2})$ evaluated at point $(1, 2)$?

<p>5 (C)</p> Signup and view all the answers

What does the notation $D_u f$ represent in the context of multivariable calculus?

<p>The directional derivative of function f in the direction of unit vector u. (A)</p> Signup and view all the answers

How does the Chain rule apply in the context of finding the directional derivative?

<p>It provides a framework for relating partial derivatives to the function's values. (A)</p> Signup and view all the answers

In a directional derivative, what does the vector $ abla f$ represent?

<p>The maximum rate of change of the function. (B)</p> Signup and view all the answers

What does the limit expression in the proof indicate about the definition of directional derivatives?

<p>It shows how the function behaves as the direction approaches zero. (C)</p> Signup and view all the answers

Flashcards

Partial Derivative Existence

A partial derivative 𝑓𝑥 (𝑎, 𝑏) exists if the limit of the ratio of 𝑓 (𝑎+ℎ, 𝑏) − 𝑓 (𝑎, 𝑏) to h as h → 0 exists.

Partial Derivative (fx(a,b))

The rate of change of a function 𝑓(𝑥, 𝑦) with respect to x, considered at a specific point (a,b).

Continuity of f(x,y) at (a,b)

f(x,y) is continuous at (a,b) if the approached value equals the true value when x and y values at (a,b) approach from x and y.

Partial Derivatives and Continuity

Existence of both partial derivatives (fx and fy) at a point does not guarantee continuity of the function at that point.

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Example of fx,fy but discontinuous function

The function f(x,y) = 1 if xy ≠ 0 and f(x,y) = 0 if xy = 0 demonstrates discontinuous function with both partial derivatives existing.

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Continuity Implication

If a partial derivative (e.g., 𝑓𝑥 (𝑎, 𝑏)) exists at a point, then the function is continuous along that specific direction, in x direction at this example.

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Partial Derivatives and Disk

If both partial derivatives (fx and fy) exist and are continuous or bounded in a disk around a point, then the function is continuous at that point.

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Increment Theorem

The partial derivative 𝑓𝑥 (𝑎, 𝑏) is determined by the ratio change of a multivariable function considering approaching h → 0.

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Limit of f(x,y)

The limit measures the function's value as (x,y) are approach to (a,b).

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1 variable function implication

If a derivative of a single variable function exists at a point, its function is continuous at that point.

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Total Increment (Δf)

The change in a function f(x, y) when both x and y change from a point (a, b)

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Increment in x (Δx)

Change in the independent variable x.

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Increment in y (Δy)

Change in the independent variable y.

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Partial Derivative 𝑓x(𝑎, 𝑏)

Rate of change of f with respect to x, when y is held constant at b.

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Partial Derivative 𝑓y(𝑎, 𝑏)

Rate of change of f with respect to y, when x is held constant at a.

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Increment Theorem

A theorem stating that if partial derivatives are continuous, the total increment can be expressed as a linear combination of partial derivatives and small error terms.

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Differentiable at (𝑎, 𝑏)

A function where the total increment can be described by its partial derivatives plus smaller error terms.

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Continuous Partial Derivatives

Partial derivatives are continuous at a point.

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Function f: D → R

A function that maps points in an open region D(in the plane) to the real numbers.

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Open Region D (in the plane)

A set of points where each point has an open neighborhood completely contained within the set D.

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Directional Derivative

The rate of change of a function in a specific direction.

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Directional Derivative Formula

(Du f)(x, y) = fx(x, y)a + fy(x, y)b, where a and b are components of the unit vector.

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Gradient

A vector operator that combines the partial derivatives of a function.

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Gradient Formula

∇f = ∂f/∂x i + ∂f/∂y j

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Gradient of f

The vector of partial derivatives of a function with respect to x and y.

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Unit vector

A vector with a magnitude of 1.

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Directional derivative in terms of gradient

Du f = ∇f · u, where u is the unit vector.

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Partial Derivatives

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

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fx(x,y)

Partial derivative of f with respect to x at (x,y)

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fy(x,y)

Partial derivative of f with respect to y at (x,y)

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Chain Rule (1 variable)

If f(x, y) has continuous partial derivatives and x(t) and y(t) are differentiable, then the derivative of f with respect to t is the sum of partial derivatives of f with respect to x and y, each multiplied by the derivative of the corresponding function with respect to t.

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Chain Rule (2 variables)

If f(x, y), x(s, t), and y(s, t) have continuous partial derivatives, then the partial derivatives of f with respect to s and t are found by applying the chain rule.

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Implicit Differentiation of z=f(x,y)

If F(x,y,z)=0 defines z implicitly, then the partial derivatives of z with respect to x and y can be found using the formula for implicit differentiation.

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Implicit Differentiation Formula (zx)

The partial derivative of z with respect to x is given by -Fx/Fz.

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Implicit Differentiation Formula (zy)

The partial derivative of z with respect to y is given by -Fy/Fz.

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Increment Theorem

A theorem that relates change in a function f(x,y) to changes in x and y with an error term.

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Continuous Partial Derivatives

Partial derivatives must exist and be continuous for the chain rule to apply.

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Implicit function

A function where the connection between variables is not directly expressed through an algebraic equation.

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Chain rule 1 variables- application

Used when we want to find dz/dt, where z = f(x, y), and x and y depend on t.

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Partial Derivatives (2 variables)

Rate of change of f with respect to s and t when looking to f(x,y).

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Directional Derivative

The rate of change of a function in a specific direction at a given point.

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Gradient of f

A vector containing the partial derivatives of f with respect to x and y. (∇f)

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Unit Vector (𝑢̂)

A vector with a magnitude of 1, used to specify a direction.

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Directional Derivative Formula

(𝐷𝑢 𝑓 )(𝑥0, 𝑦0 ) = (∇𝑓 )(𝑥0, 𝑦0 ) · 𝑢.

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Approximating f at neighboring points

Using the directional derivative to estimate a function's value near a point.

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Function Approximation

𝑓 (𝑥 0 + 𝑎ℎ, 𝑦0 + 𝑏ℎ) ≈ 𝑓 (𝑥 0, 𝑦0 ) + ℎ (𝐷𝑢 𝑓 )(𝑥 0, 𝑦0 )

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Change in multivariable function

Estimate of how much a function value changes when moving in a direction from a given point.

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Directional Derivative in 3D

Extension of directional derivative concept to three variables, using gradient in 3D space.

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Gradient in 3D

∇𝑓 = (𝑦 cos 𝑥)𝚤 + (sin 𝑥 + 2𝑧) 𝚥 + 2𝑦 𝑘 (in the example)

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Directional Derivative Meaning

The length of the gradient vector's projection onto the direction of the unit vector 𝑢̂.

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Taylor's Formula (Integral Form)

Represents a function as a sum of its derivatives at a point, plus a remainder term expressed as a definite integral.

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Remainder Term (Rn(x))

The difference between the function and the Taylor polynomial approximation.

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Continuous differentiability

The function and all its derivatives up to a certain order exist and are continuous.

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Right-hand derivative

Derivative at an endpoint of the interval, calculated from the limit approaching that endpoint from the right (i.e values greater than a).

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Left-hand derivative

Derivative at an endpoint of the interval calculated from the limit approaching that endpoint from the left (i.e values less than b).

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Taylor's Formula (Basis)

Taylor's formula is proven by induction on the order of the derivative

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Fundamental Theorem of Calculus

Establishes a relationship between differentiation and integration.

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Integration by parts

A technique for evaluating definite or indefinite integrals by expressing the integral as a combination of simpler integrals.

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Inductive proof

A method of proof that involves showing a statement is true for a base case, then showing that if it is true for some value, it must be true for the next.

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Estimate for Rn(x)

An upper and lower bound for the remainder term in Taylor's formula.

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Study Notes

Partial Derivatives and Continuity

  • Partial derivatives can exist at a point even if the function isn't continuous at that point.
  • If a partial derivative exists and is continuous at a point, the function is continuous at that point in that direction.
  • If both partial derivatives exist at a point, the function may or may not be continuous there. A counterexample is given by a function that's 1 if xy ≠ 0 and 0 otherwise, at the origin (0,0).
  • If both partial derivatives are continuous within a region around a point, or one is continuous and bounded, the function is continuous at the point.

Increment Theorem

  • The partial derivative 𝑓𝑥(𝑎, 𝑏) represents the rate of change of 𝑓(𝑥, 𝑦) at (𝑎, 𝑏) in the 𝑥-direction.
  • Δ𝑥 and Δ𝑦 represent increments in 𝑥 and 𝑦, respectively.
  • The total increment Δ𝑓 at (𝑎, 𝑏) is given by Δ𝑓 = 𝑓(𝑎 + Δ𝑥, 𝑏 + Δ𝑦) − 𝑓(𝑎, 𝑏).
  • If the partial derivatives are continuous, the increment theorem gives a formula for Δ𝑓: Δ𝑓 = 𝑓𝑥(𝑎, 𝑏)Δ𝑥 + 𝑓𝑦(𝑎, 𝑏)Δ𝑦 + 𝜖1Δ𝑥 + 𝜖2Δ𝑦, where 𝜖1 and 𝜖2 approach 0 as Δ𝑥 and Δ𝑦 approach 0.
  • A function with continuous partial derivatives is continuous itself.
  • Differentiability implies the function satisfies the increment theorem formula.

Chain Rule

  • Chain rules for functions of two or more variables are provided with their respective formulas.

Directional Derivatives

  • Directional derivative measures the rate of change of a function in a specific direction.
  • Given a unit vector û = â𝚤̂ + b𝚥̂, the directional derivative at (x,y) is (𝐷𝑢𝑓)(x,y) = 𝑓𝑥(𝑥, 𝑦)𝑎 + 𝑓𝑦(𝑥, 𝑦)𝑏.
  • The gradient of a function 𝑓(𝑥, 𝑦) is a vector defined as ∇𝑓 = grad𝑓 = 𝑓𝑥𝚤̂ + 𝑓𝑦𝚥̂.
  • The directional derivative can be expressed as the dot product of the gradient and the unit vector: (𝐷𝑢𝑓)(𝑥, 𝑦) = (∇𝑓)(𝑥, 𝑦) · û.
  • Directional derivatives can be used to approximate function values at nearby points.

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Description

This quiz covers essential concepts of partial derivatives and their relationship with continuity. Key topics include the conditions under which partial derivatives exist and the implications for function continuity. Understand the Increment Theorem and its application in calculating total increments of functions.

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