How to graph y=x^x by using calculus | Summary and Q&A

TL;DR

Graphing the function x to the x power requires considering the undefined nature of 0 to the 0 power, the change in base as well as exponent, and the need for x to be greater than zero to avoid complex numbers.

Key Insights

• ✊ Zero to the zeroth power is undefined and poses a challenge in calculating the function at x=0.
• 💱 The function x to the x power is not an exponential function, as both the base and the exponent change.
• #️⃣ Negative values for x result in complex numbers, which are not desired when graphing with real numbers.
• ❓ The domain of the function consists of positive values of x.
• 😥 The function has a minimum point at x=1/e.
• 😥 The first derivative of the function determines its behavior and allows for the identification of critical points.
• 😥 The graph of the function starts from an undefined point at x=0, decreases until x=1/e, and then increases indefinitely.

Transcript

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Questions & Answers

Q: Why is zero to the zero power undefined?

Zero to the zero power is undefined for regular computation. It does not provide a clear answer within the context of usual calculations.

Q: Can x be a negative number in x to the x power?

It is preferable to have x as a positive value in order to avoid complex numbers. Negative values for x result in the square root of a negative number, which is a complex solution.

Q: What is the domain of the function x to the x power?

The domain consists of x values that are greater than zero. Zero and negative numbers are excluded from the domain because they lead to undefined or complex solutions.

Q: Is the function always increasing?

The function appears to be increasing for x values greater than one. However, to determine if there are any maximum or minimum points, the derivative of the function needs to be analyzed.

Summary & Key Takeaways

• Zero to the zeroth power is undefined, so the function is undefined at x=0.

• The function is not an exponential function because both the base and the exponent change.

• To avoid complex numbers, x must be greater than zero, as negative values result in imaginary solutions.

• The domain of the function is x greater than zero.

• The function has a relative minimum at x=1/e.