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Chapter 5: Problem 3

Is \(x^{12}\) an even or odd function? Is \(\sin x^{2}\) an even or odd function?

Short Answer

Expert verified
Question: Determine if \(x^{12}\) and \(\sin x^{2}\) are even or odd functions. Answer: Both \(x^{12}\) and \(\sin x^{2}\) are even functions.

Step by step solution

01

Determine if \(x^{12}\) is even or odd

To find if \(x^{12}\) is an even or odd function, we will check its behavior when we replace \(x\) with \(-x\): $$f(-x) = (-x)^{12}$$ Since the exponent is even, \((-x)^{12}\) will be equal to \(x^{12}\): $$f(-x) = x^{12}$$ Comparing with the original function, \(f(x) = x^{12}\), we can see that \(f(-x) = f(x)\). Therefore, \(x^{12}\) is an even function.
02

Determine if \(\sin x^{2}\) is even or odd

To find if the function \(\sin x^{2}\) is even or odd, we will replace \(x\) with \(-x\) and observe the behavior of the function: $$g(-x) = \sin((-x)^{2})$$ Squaring a negative number will give a positive result, so \((-x)^{2} = x^{2}\). Thus: $$g(-x) = \sin(x^{2})$$ Comparing with the original function, \(g(x) = \sin(x^{2})\), we can see that \(g(-x) = g(x)\). Therefore, \(\sin x^{2}\) is an even function.
03

Conclusion

In conclusion, the function \(x^{12}\) is an even function and the function \(\sin x^{2}\) is also an even function.

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Most popular questions from this chapter

Fill in the following table with either even or odd, and prove each result. Assume \(n\) is a nonnegative integer and \(f^{n}\) means the \(n\) th power of \(f\). $$\begin{array}{cccc} & f \text { is even } & f \text { is odd } \\\\\hline n \text { is even } & f^{n}\text { is } \underline{\quad}\underline{\quad} & f^{n} \text { is } \underline{\quad} \underline{\quad} \\\n \text { is odd } & f^{n} \text { is } \underline{\quad} \underline{\quad} & f^{n} \text { is } \underline{\quad} \underline{\quad} \\\\\hline\end{array}$$

Use a change of variables to evaluate the following integrals. $$\int_{-\pi}^{0} \frac{\sin x}{2+\cos x} d x$$

Use a change of variables to find the following indefinite integrals. Check your work by differentiating. $$\int x e^{x^{2}} d x$$

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{1} \sqrt{x-x \sqrt{x}} d x$$

Find the value of \(c\) such that the region bounded by \(y=c \sin x\) and the \(x\) -axis on the interval \([0, \pi]\) has area 1.
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