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What does the expression $f(a + \Delta x, b) - f(a, b)$ measure?
What does the expression $f(a + \Delta x, b) - f(a, b)$ measure?
What does the increment $\Delta f$ represent at the point $(a, b)$?
What does the increment $\Delta f$ represent at the point $(a, b)$?
When is a function $f$ said to be differentiable at a point $(a, b)$?
When is a function $f$ said to be differentiable at a point $(a, b)$?
What are $\epsilon_1$ and $\epsilon_2$ in the context of the total increment $\Delta f$?
What are $\epsilon_1$ and $\epsilon_2$ in the context of the total increment $\Delta f$?
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What happens to $\epsilon_1$ and $\epsilon_2$ as $\Delta x$ and $\Delta y$ both approach zero?
What happens to $\epsilon_1$ and $\epsilon_2$ as $\Delta x$ and $\Delta y$ both approach zero?
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Under which condition can $f$ be represented as $\Delta f = f_x(a, b) \Delta x + f_y \Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y$?
Under which condition can $f$ be represented as $\Delta f = f_x(a, b) \Delta x + f_y \Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y$?
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What does the expression $f(a, b + \Delta y) - f(a, b)$ measure?
What does the expression $f(a, b + \Delta y) - f(a, b)$ measure?
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Which of the following is a requirement for the Increment Theorem to apply?
Which of the following is a requirement for the Increment Theorem to apply?
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Which of the following expressions is equivalent to the total increment $\Delta f$?
Which of the following expressions is equivalent to the total increment $\Delta f$?
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What is the consequence of having continuous partial derivatives for the function 𝑓(𝑥, 𝑦) in the Chain Rule 1?
What is the consequence of having continuous partial derivatives for the function 𝑓(𝑥, 𝑦) in the Chain Rule 1?
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What is the form of the differential expressed in Chain Rule 1?
What is the form of the differential expressed in Chain Rule 1?
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In which situation can implicit differentiation be applied according to the content?
In which situation can implicit differentiation be applied according to the content?
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How is $z_x$ calculated from a function defined implicitly by 𝐹(x, y, z) = 0?
How is $z_x$ calculated from a function defined implicitly by 𝐹(x, y, z) = 0?
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What does the expression $z(t)$ = $ rac{1}{2}sin(2t)$ illustrate in the context of the Chain Rule?
What does the expression $z(t)$ = $ rac{1}{2}sin(2t)$ illustrate in the context of the Chain Rule?
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In the example with $z = e^x sin(y)$, what does the term $(e^x sin(y))t^2$ indicate?
In the example with $z = e^x sin(y)$, what does the term $(e^x sin(y))t^2$ indicate?
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What is the derivative expression for $z$ when $z = e^x sin(y)$ and $x = st^2, y = s^2t$?
What is the derivative expression for $z$ when $z = e^x sin(y)$ and $x = st^2, y = s^2t$?
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Which of the following correctly describes what happens as $ riangle t
ightarrow 0$ in the proof?
Which of the following correctly describes what happens as $ riangle t ightarrow 0$ in the proof?
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Which condition is necessary for a function to be defined as $z = f(x, y)$ through the implicit function theorem?
Which condition is necessary for a function to be defined as $z = f(x, y)$ through the implicit function theorem?
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What does it imply if both partial derivatives $f_x(a, b)$ and $f_y(a, b)$ exist at a point?
What does it imply if both partial derivatives $f_x(a, b)$ and $f_y(a, b)$ exist at a point?
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Under which condition is the function $f(x, y)$ declared continuous at $(a, b)$?
Under which condition is the function $f(x, y)$ declared continuous at $(a, b)$?
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What does it mean if the limit of $f(x, y)$ as $(x,y) o (0,0)$ does not exist?
What does it mean if the limit of $f(x, y)$ as $(x,y) o (0,0)$ does not exist?
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What is the significance of the increment theorem for defining the partial derivative $f_x(a, b)$?
What is the significance of the increment theorem for defining the partial derivative $f_x(a, b)$?
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Which of the following statements is true regarding the continuity of the function $f(x, y)$ at the origin?
Which of the following statements is true regarding the continuity of the function $f(x, y)$ at the origin?
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What conclusion can be drawn from the example of the function defined piecewise at the origin?
What conclusion can be drawn from the example of the function defined piecewise at the origin?
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Which of the following is an implication of the statement 'if $f_x(a, b)$ exists, then $f(x, b)$ is continuous at $x = a$'?
Which of the following is an implication of the statement 'if $f_x(a, b)$ exists, then $f(x, b)$ is continuous at $x = a$'?
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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?
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?
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What condition must at least one of the partial derivatives meet for continuity of $f(x, y)$ around $(a, b)$?
What condition must at least one of the partial derivatives meet for continuity of $f(x, y)$ around $(a, b)$?
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What is the formula for the directional derivative represented mathematically?
What is the formula for the directional derivative represented mathematically?
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What are the conditions for the formula of the directional derivative to be valid?
What are the conditions for the formula of the directional derivative to be valid?
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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 $(∇f)(2, 0)$?
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 $(∇f)(2, 0)$?
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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?
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?
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What does the directional derivative represent concerning the gradient and unit vector?
What does the directional derivative represent concerning the gradient and unit vector?
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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 $(2; 1; -2)$?
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 $(2; 1; -2)$?
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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)?
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)?
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What is the significance of the formula $(D_u f) = ∇f · old{u}$ in differential calculus?
What is the significance of the formula $(D_u f) = ∇f · old{u}$ in differential calculus?
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What happens to the value of the directional derivative if the unit vector u is not normalized?
What happens to the value of the directional derivative if the unit vector u is not normalized?
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What is the form of the remainder term 𝑅𝑛(𝑥) in Taylor's Formula?
What is the form of the remainder term 𝑅𝑛(𝑥) in Taylor's Formula?
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At which point is the derivative taken as the left-hand derivative in Taylor's formula?
At which point is the derivative taken as the left-hand derivative in Taylor's formula?
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What is the significance of continuously differentiable functions in Taylor's formula?
What is the significance of continuously differentiable functions in Taylor's formula?
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What does the term $f^{(k)}(a)$ represent in Taylor's formula?
What does the term $f^{(k)}(a)$ represent in Taylor's formula?
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How is the estimate for the remainder term $R_n(x)$ bounded?
How is the estimate for the remainder term $R_n(x)$ bounded?
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In induction proofs for Taylor's formula, what is assumed true for 𝑛 = 𝑘?
In induction proofs for Taylor's formula, what is assumed true for 𝑛 = 𝑘?
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Which of the following terms contributes to the Taylor series expansion?
Which of the following terms contributes to the Taylor series expansion?
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Which expression corresponds to the integral remainder term for an (n + 1)-times continuously differentiable function?
Which expression corresponds to the integral remainder term for an (n + 1)-times continuously differentiable function?
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Which of the following statements about Taylor’s formula is true?
Which of the following statements about Taylor’s formula is true?
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What is the formula for the directional derivative of a function at a point in a given direction?
What is the formula for the directional derivative of a function at a point in a given direction?
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Which expression correctly defines a unit vector in the direction of $( rac{ ext{cos } rac{ ext{π}}{6}}{ ext{sin } rac{ ext{π}}{6}})$?
Which expression correctly defines a unit vector in the direction of $( rac{ ext{cos } rac{ ext{π}}{6}}{ ext{sin } rac{ ext{π}}{6}})$?
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What role do continuous partial derivatives play in the theorem regarding directional derivatives?
What role do continuous partial derivatives play in the theorem regarding directional derivatives?
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In the proof, what function is defined to show the existence of the directional derivative?
In the proof, what function is defined to show the existence of the directional derivative?
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What is the significance of the gradient symbol ∇ in the context of the directional derivative?
What is the significance of the gradient symbol ∇ in the context of the directional derivative?
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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 $(1, 2)$?
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 $(1, 2)$?
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What does the notation $D_u f$ represent in the context of multivariable calculus?
What does the notation $D_u f$ represent in the context of multivariable calculus?
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How does the Chain rule apply in the context of finding the directional derivative?
How does the Chain rule apply in the context of finding the directional derivative?
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In a directional derivative, what does the vector $
abla f$ represent?
In a directional derivative, what does the vector $ abla f$ represent?
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What does the limit expression in the proof indicate about the definition of directional derivatives?
What does the limit expression in the proof indicate about the definition of directional derivatives?
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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.
