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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?
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
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.
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All functions are one-to-one.
All functions are one-to-one.
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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.
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What are the two equations that describe the cancellation equations?
What are the two equations that describe the cancellation equations?
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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.
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If a function is one-to-one, it must have an inverse.
If a function is one-to-one, it must have an inverse.
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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?
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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).
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It is always possible to find the inverse of any function.
It is always possible to find the inverse of any function.
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What is the inverse of the exponential function y =e^x?
What is the inverse of the exponential function y =e^x?
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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.
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A function is determined solely by its rule.
A function is determined solely by its rule.
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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.
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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.
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Study Notes
Lecture #11
- Bonus assignment due October 31
- Submitting solutions to homework problems is required.
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- 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!
