Question

List some examples of even functions in calculus.

Answer 1

In calculus, some examples of even functions include _f(x) =
`x^2`_, _g(x) = cos(x)_, and _h(x) = |x|_. Even functions are those where _f(x) = f(-x)_, exhibiting symmetry about the y-axis.

Step-by-step explanation:

Even functions in calculus possess a particular symmetry: _f(x) = f(-x)_. One of the simplest examples is the function _f(x) =
`x^2`_. For any value of _x_, _f(x)_ equals _f(-x)_, demonstrating symmetry about the y-axis. Another example is the cosine function, _g(x) = cos(x)_. Substituting _-x_ into the cosine function yields the same value as _x_, thus exhibiting even symmetry.

Interestingly, certain functions with absolute values can also be even functions. Take the function _h(x) = |x|_, which might not seem even at first glance due to its absolute value notation. However, when observing its behavior, it becomes clear that _h(x)_ equals _h(-x)_ for all values of _x_, indicating symmetry about the y-axis. Even functions are crucial in calculus as they simplify integrals across symmetric intervals; their properties help reduce calculations, making problem-solving more efficient.

Understanding even functions is fundamental in calculus as they relate to symmetry properties. Whether it's the quadratic function, _f(x) =
`x^2`_, the cosine function, _g(x) = cos(x)_, or the absolute value function, _h(x) = |x|_, recognizing their symmetry about the y-axis identifies them as examples of even functions. This symmetry aids in simplifying calculations, especially in integration, enabling mathematicians to efficiently solve complex problems involving symmetric intervals.