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Questions and Answers
What is the inverse function of $y = e^x$?
What is the inverse function of $y = e^x$?
- $x = e^y$ (correct)
- $x = rac{1}{y}$
- $y = rac{1}{x}$
- $y = e^x$
What is the derivative of the function $y = e^x$?
What is the derivative of the function $y = e^x$?
- $rac{dy}{dx} = e^x$ (correct)
- $rac{dy}{dx} = rac{1}{x}$
- $rac{dy}{dx} = 2e^x$
- $rac{dy}{dx} = x^2$
How is the inverse function related to the original function?
How is the inverse function related to the original function?
- It swaps $x$ and $y$ in the original equation (correct)
- It adds $x$ and $y$ in the original equation
- It multiplies $x$ and $y$ in the original equation
- It subtracts $x$ from $y$ in the original equation
What does the derivative of the inverse function of $y = e^x$ equal to?
What does the derivative of the inverse function of $y = e^x$ equal to?
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How is the reciprocal function related to the original function?
How is the reciprocal function related to the original function?
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Study Notes
Inverse Function of y = e^x
- The inverse function of y = e^x is y = ln(x), where ln is the natural logarithm
Derivative of y = e^x
- The derivative of y = e^x is y' = e^x, which means the function is its own derivative
Relationship between Inverse and Original Function
- The inverse function reverses the roles of x and y, i.e., if (a, b) is on the original function, then (b, a) is on the inverse function
- The inverse function reflects the original function over the line y = x
Derivative of Inverse Function
- The derivative of the inverse function y = ln(x) is y' = 1/x
Reciprocal Function
- The reciprocal function of y = e^x is y = 1/e^x
- The reciprocal function is not the same as the inverse function, although they are related
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Description
This quiz aims to clarify the distinction between 'inverse' and 'reciprocal' in mathematics, particularly in the context of calculus. It discusses how the two terms are used interchangeably and delves into their specific meanings and applications.
