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Questions and Answers
What is an explicit function?
What is an explicit function?
Why is the chain rule important in the context of implicit functions?
Why is the chain rule important in the context of implicit functions?
What are mixed partial derivatives?
What are mixed partial derivatives?
What is the contribution of Isaac Newton to implicit differentiation?
What is the contribution of Isaac Newton to implicit differentiation?
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How is the chain rule applied to explicit functions?
How is the chain rule applied to explicit functions?
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What characterizes an implicit function?
What characterizes an implicit function?
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What role do partial derivatives play in implicit differentiation?
What role do partial derivatives play in implicit differentiation?
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Who independently developed implicit differentiation alongside Isaac Newton?
Who independently developed implicit differentiation alongside Isaac Newton?
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What is one primary advantage of using implicit functions in calculus?
What is one primary advantage of using implicit functions in calculus?
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In what scenario is implicit differentiation particularly useful?
In what scenario is implicit differentiation particularly useful?
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Why is implicit differentiation a common technique in multivariable calculus?
Why is implicit differentiation a common technique in multivariable calculus?
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Which of the following examples typically requires implicit differentiation?
Which of the following examples typically requires implicit differentiation?
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What aspect of real-world problems makes implicit functions particularly relevant?
What aspect of real-world problems makes implicit functions particularly relevant?
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What is an explicit function defined as?
What is an explicit function defined as?
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What is a limitation of explicit functions compared to implicit functions?
What is a limitation of explicit functions compared to implicit functions?
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The chain rule is primarily used to find the derivative of what type of function?
The chain rule is primarily used to find the derivative of what type of function?
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What type of mathematical relationship do implicit functions often represent?
What type of mathematical relationship do implicit functions often represent?
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Which mathematician is credited with introducing the modern notation f(x)?
Which mathematician is credited with introducing the modern notation f(x)?
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What is one key benefit of working with explicit functions in calculus?
What is one key benefit of working with explicit functions in calculus?
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Which fundamental skill is enhanced by understanding implicit functions in calculus?
Which fundamental skill is enhanced by understanding implicit functions in calculus?
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Which of the following operations rely heavily on the use of explicit functions?
Which of the following operations rely heavily on the use of explicit functions?
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How did Leibniz contribute to the concept of the chain rule?
How did Leibniz contribute to the concept of the chain rule?
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What is a fundamental operation in calculus that is facilitated by explicit functions?
What is a fundamental operation in calculus that is facilitated by explicit functions?
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In what century did mathematicians like Cauchy rigorously develop function theory?
In what century did mathematicians like Cauchy rigorously develop function theory?
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What is the derivative of the function y = x^2 + sin x - x + 4?
What is the derivative of the function y = x^2 + sin x - x + 4?
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Which expression correctly represents y in the equation xy - y = 0?
Which expression correctly represents y in the equation xy - y = 0?
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What is the result of differentiating the implicit function xy + 2x - tan(xy) + y^2 = 0 with respect to x?
What is the result of differentiating the implicit function xy + 2x - tan(xy) + y^2 = 0 with respect to x?
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What does the chain rule help to compute?
What does the chain rule help to compute?
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Given z = x^2 + y^2 with x = cos(t) and y = sin(t), what is the derivative dz/dt?
Given z = x^2 + y^2 with x = cos(t) and y = sin(t), what is the derivative dz/dt?
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What is the derivative of the function y with respect to x for the expression y = 1/(x - 1)?
What is the derivative of the function y with respect to x for the expression y = 1/(x - 1)?
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Which of the following describes an explicit function?
Which of the following describes an explicit function?
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In the expression dy/dx = -1/(x - 1)^2, what does the term -1/(x - 1)^2 represent?
In the expression dy/dx = -1/(x - 1)^2, what does the term -1/(x - 1)^2 represent?
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How is the derivative $ heta ' (x)$ derived according to the Implicit Function Theorem I?
How is the derivative $ heta ' (x)$ derived according to the Implicit Function Theorem I?
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In the context of the Implicit Function Theorem II, what is the significance of $F_z(x_0, y_0, z_0)
eq 0$?
In the context of the Implicit Function Theorem II, what is the significance of $F_z(x_0, y_0, z_0) eq 0$?
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What does the slope of the tangent line to the circle $x^2 + y^2 = 25$ at the point (3, 4) represent?
What does the slope of the tangent line to the circle $x^2 + y^2 = 25$ at the point (3, 4) represent?
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What condition must be satisfied for the Implicit Function Theorem I to apply?
What condition must be satisfied for the Implicit Function Theorem I to apply?
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In Implicit Function Theorem II, what is the set where the function $ heta$ has continuous first order partial derivatives?
In Implicit Function Theorem II, what is the set where the function $ heta$ has continuous first order partial derivatives?
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What is the role of the Chain Rule in deriving the formulas for $ heta_1 (x, y)$ and $ heta_2 (x, y)$?
What is the role of the Chain Rule in deriving the formulas for $ heta_1 (x, y)$ and $ heta_2 (x, y)$?
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Which of the following statements about the implicit function theorem is true?
Which of the following statements about the implicit function theorem is true?
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What happens when partial derivatives are not continuous in the context of the Implicit Function Theorem?
What happens when partial derivatives are not continuous in the context of the Implicit Function Theorem?
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Study Notes
Implicit Function
- Is a function where the dependent variable is not isolated, defined by an equation involving both the dependent and independent variables.
- For example, x^2 + y^2 = r^2
- Critical in calculus as many real-world problems involve complex relationships between variables
- Implicit differentiation is used to find the derivative of the dependent variable with respect to the independent variable
- Importance in calculus:
- Broader application: allows us to work with equations without isolating a variable
- Implicit Differentiation: finds derivatives of functions when they are not explicitly defined
- Multivariable Calculus: essential for working with functions that depend on several variables at once
- Real-World Problems : many real-world problems, such as constraints in physics and engineering, naturally occur in implicit form
- History:
- Developed by Isaac Newton and Gottfried W.Leibniz, applied for physics problems, particularly in the 17th and 18th centuries
Explicit Function
- Expresses the dependent variable directly in terms of the independent variable.
- For example, y = f(x)
- Importance in calculus:
- Ease of Differentiation and Integration: allows for direct mathematical manipulations
- Simplification of Problem Solving: easier to find critical points, slopes, and areas
- Clear Representation of Relationships: easy to understand relationships between variables
- Foundation of Calculus: serves as the base for many calculus operations like power rule, chain rule, and product rule
- History: dates back to ancient civilizations, formalized by mathematicians like Euler, Cauchy, and Weierstrass in the 18th and 19th centuries.
Chain Rule
- A fundamental rule for finding the derivative of a composite function, a function that depends on an intermediate variable.
- States that the derivative of a composite function is the product of the derivative of the outer function with respect to the intermediate variable and the derivative of the intermediate variable with respect to the original variable.
- Chain Rule for a Function of Two Variables:
- If z = f(x, y), and x = g(t) and y = h(t)
- Then ⅆ𝑧 / ⅆ𝑡 = (ⅆ𝑧 / ⅆ𝑥) (ⅆ𝑥 / ⅆ𝑡) + (ⅆ𝑧 / ⅆ𝑦) (ⅆ𝑦 / ⅆ𝑡)
Implicit Function Theorem
-
Used for finding derivatives of implicitly defined functions.
-
Uses the chain rule to derive a formula to identify the derivatives of implicitly defined functions.
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Implicit Function Theorem I
- Let F : D ⊂ ℝ2 → R and let (𝑥0 , 𝑦0 ) be an interior point of D with F(𝑥0 , 𝑦0 ) = 0.
- Suppose both first-order partial derivatives of F exist in D and are continuous at (𝑥0 , 𝑦0 ) with 𝐹𝑌 (𝑥0 , 𝑦0 ) ≠ 0.
- Then there is an interval I ⊂ R with 𝑥0 an interior point of I and a function φ : I → R such that φ is differentiable on I, φ(𝑥0 ) = 𝑦0 and F(x, φ(x)) = 0 for each x ∈ I.
-
Implicit Function Theorem II
- Let F : D ⊂ ℝ3 → R and let (𝑥0 , 𝑦0 ,𝑧0 ) be an interior point of D with F(𝑥0 , 𝑦0 ,𝑧0 ) = 0.
- Suppose all first-order partial derivatives of F exist in D and are continuous at (𝑥0 , 𝑦0 ,𝑧0 ) with 𝐹𝑧 (𝑥0 , 𝑦0 ,𝑧0 )≠0.
- Then there is a rectangle R ⊂ ℝ2 with (𝑥0 , 𝑦0 ) an interior point of R and a function φ : R → R such that φ has continuous first-order partial derivatives on R, φ(𝑥0 , 𝑦0 ) =𝑧0 and F((x, y, φ(x, y)) = 0 for each (x, y) ∈ R.
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The Chain Rule for Functions of Several Variables is then used to compute a formula for 𝜑 ′ (x), 𝜑1 (x, y) and for 𝜑2 (x, y).
Examples
- Example 1: Find the slope of the tangent line to the circle 𝒙𝟐 + 𝒚𝟐 = 25 at the point (3, 4).
-
Example 2: Find the derivative of the explicit function y = 𝑥 2 + sin x - x + 4
- dy/dx = 2x + cos x - 1
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Example 3: Find the derivative of the function xy - y = 0
- Express the function explicitly as y = 1/(x - 1)
- dy/dx = -1(𝑥 − 1)2
Summary Table
| Feature | Implicit Function | Explicit Function |
|---|---|---|
| Definition | Dependent variable is not isolated, defined by equation involving both variables | Dependent variable is directly expressed in terms of the Independent variable |
| General Form | f(x, y) = 0 | y = f(x) |
| Example | xy + 2x - tan (xy) + 𝑦 2 = 0 | y = x + 2 |
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Description
Explore the concept of implicit functions and their significance in calculus. This quiz covers implicit differentiation, its applications, and its historical background. Understand how these functions relate to real-world problems and their mathematical implications.
