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Questions and Answers
Implicit differentiation works by remembering that y = y(x) for the same function of x and applying which differentiation rule anytime we differentiate y in terms of x?
Implicit differentiation works by remembering that y = y(x) for the same function of x and applying which differentiation rule anytime we differentiate y in terms of x?
- Product rule
- Chain rule (correct)
- Constant rule, dy/dx = 0 always.
- Quotient rule
- Sum/difference rule
A function is called one-to-one (1-1) if it never takes the same value ______.
A function is called one-to-one (1-1) if it never takes the same value ______.
twice
The Horizontal Line Test states that a function is one-to-one if and only if no horizontal line intersects the graph more than once.
The Horizontal Line Test states that a function is one-to-one if and only if no horizontal line intersects the graph more than once.
True (A)
The function g(x) = x^3 with domain of all real numbers is a 1-1 function.
The function g(x) = x^3 with domain of all real numbers is a 1-1 function.
All functions are one-to-one.
All functions are one-to-one.
When we invert the process, we refer to it as the original operation.
When we invert the process, we refer to it as the original operation.
What are the two equations that describe the cancellation equations?
What are the two equations that describe the cancellation equations?
The function f(x) = x^2 with domain of all real numbers has an inverse.
The function f(x) = x^2 with domain of all real numbers has an inverse.
If a function is one-to-one, it must have an inverse.
If a function is one-to-one, it must have an inverse.
What are the two steps involved in finding the inverse of a function, assuming it has one?
What are the two steps involved in finding the inverse of a function, assuming it has one?
The inverse of f(x) = x^3 + 2 is f'(x) = (x - 2)^(1/3).
The inverse of f(x) = x^3 + 2 is f'(x) = (x - 2)^(1/3).
It is always possible to find the inverse of any function.
It is always possible to find the inverse of any function.
What is the inverse of the exponential function y =e^x?
What is the inverse of the exponential function y =e^x?
The logarithm approach for finding inverses is generally less common than using the algorithm.
The logarithm approach for finding inverses is generally less common than using the algorithm.
A function is determined solely by its rule.
A function is determined solely by its rule.
The square root function with a domain of all real numbers is the inverse of the squaring function with a domain of all real numbers.
The square root function with a domain of all real numbers is the inverse of the squaring function with a domain of all real numbers.
Restricting the domain of a function can make it one-to-one even if it was not one-to-one before.
Restricting the domain of a function can make it one-to-one even if it was not one-to-one before.
Flashcards
Implicit Differentiation
Implicit Differentiation
A technique to find the derivative of a function where y is defined implicitly in terms of x.
Chain Rule
Chain Rule
A rule for differentiating composite functions (functions within functions).
Product Rule
Product Rule
A rule for differentiating the product of two functions.
Quotient Rule
Quotient Rule
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Difference Rule
Difference Rule
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Sum Rule
Sum Rule
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Constant Rule
Constant Rule
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One-to-One Function (1-1)
One-to-One Function (1-1)
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Horizontal Line Test (HLT)
Horizontal Line Test (HLT)
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Domain
Domain
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Function
Function
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x-axis
x-axis
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y-axis
y-axis
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Coordinate Plane
Coordinate Plane
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Visual Representation
Visual Representation
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"if...then" statement
"if...then" statement
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x2
x2
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1-1
1-1
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IR
IR
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Study Notes
Lecture #11
- Bonus assignment due October 31
- Submitting solutions to homework problems is required.
Pop Quiz
- Implicit differentiation calculates derivatives of equations where y is a function of x.
- Chain rule, product rule, quotient rule, sum/difference rule, and constant rule are differentiation rules.
One-to-One Functions
- A function is one-to-one (1-1) if each output corresponds to exactly one input.
- The horizontal line test is used to determine whether a function is one-to-one. A function passes the horizontal line test if no horizontal line intersects the graph of the function more than once.
- Examples of one-to-one functions such as f(x) = 2x + 3 or g(x) = x³.
- To determine if a function is one to one: Use algebraic approach by letting f(x)=f(x₂).
Inverse Functions
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The inverse of a function reverses the input and output of the original function.
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The inverse function is denoted by f⁻¹(x).
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To find the inverse of a function, swap x and y and solve for y in terms of x.
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The domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function.
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Examples of inverse functions such as f(x) = x² or g(x) = eˣ.
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To generate an inverse function for example f(x) = x², it must be restricted, e.g., with the domain x ≥ 0.
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The inverse of f(x) = x² is written f⁻¹(x) = √x.
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Techniques to find the inverse function:
- Write y = f(x), then solve equation for x in terms of y
- Swap x and y
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Examples of functions that don't have inverses unless the domain is restricted.
Finding Inverse Functions
- Algorithm example: Write y = f(x). Solve for x in terms of y. Swap x and y. This is f⁻¹(x)
- Example: f(x)=x³+2. f⁻¹(x)=(x-2)⅓
Exponential/Logarithmic Functions
- The inverse of the exponential function (eˣ) is the natural logarithm (ln x).
- The inverse of a general exponential function (bˣ) is the logarithm with base b (logₐ x).
- Properties of logarithms such as:
- ln (eˣ) = x for all x ∈ ℝ
- ln (1/x) = -ln (x) for all x > 0
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Description
Test your knowledge on implicit differentiation, one-to-one functions, and inverse functions. This quiz covers essential rules of differentiation and the unique properties of functions in calculus. Get ready to apply what you've learned in the lecture!
