In today's video, we'll discuss even functions, whose curves have reflectional symmetry in the y-axis and odd functions, whose curves have a 180 degree rotational symmetry about the origin. If one forms definite integrals over intervals in the real line that are symmetric about the origin, then there are certain simplifications. In particular, the area under an odd function over such an interval evaluates automatically to zero. The notions of even and odd catch precisely certain types of symmetries. For example, the letters M, T, and Y have reflectional symmetry about a vertical line drawn down the middle. By contrast, letters N, S, and Z have a 180 degree rotational symmetry about the midpoints. From the point of view of reflectional and rotational symmetry, the most perfect figure in the plan is the circle, which has an infinite supplies of such symmetries. The use of an arbitrarily large number of symmetries in the circle was implicit in the argument that Greeks used to find the formula for the area of a circle. In an earlier video, we analyzed the curve known as The Witch of Maria Agnesi. Put g of x equal to one over x squared plus one. Its graph has perfect reflectional symmetry about the vertical y-axis. We used the curve sketching techniques to locate the two points of reflection and they become images of each other by reflecting in the y-axis. If we feed in minus x and the rule for function g, then we quickly reproduce the original rule for g. This property with the outcome for the rule remains the same after failing in minus x to x corresponds to the curve having reflectional symmetry in the y-axis. There's another curve closely related to the witch known as Newton's Serpentine. Put h of x equal to x over x squared plus one. It's a good exercise in curve sketching to verify that the graph of h looks like this, and it has a 180 degree rotational symmetry about the origin. For example, the turning points one half and minus one minus a half, rotate into each other. Notice that if you feed in minus x, in the rule for the function h, you end up with minus h of x. This algebraic relationship corresponds to a 180 degree rotational symmetry. To define Even functions in general, consider a function y equals f of x. We say that f is even, if the condition f of minus x equals f of x holds for all inputs x. This implicitly assumes x is in the domain, then minus x is also. Thus if you have the graph of an Even function, for example, this bell shaped curve, which is symmetric about the y-axis, then if you move up to the curve from some input x and reflect the points on the curve in the y-axis, then you land again on the curve at a point corresponding to the input minus x. Both f of x and f of minus x are equal. Thus in general, this algebraic criterion corresponds to reflectional symmetry in the y-axis. Simple examples of the quadratic function y equals x squared giving rise to this parabola. The constant function y equals c, the curve of y equals x to the fourth. Bowl-shape up like a parabola but much steeper and the cosine curve. Any polynomial function or rational function involving constant multiples with even powers of x only will be even. No matter how complicated, we don't even need to be able to visualize the curves to know for sure they all have reflectional symmetry in the y-axis. Note the constant functions are multiples of x to the zero and zero is even. In all cases, even powers cause the minus sign go away leaving the final value of the function unaffected. The connection with even powers of x is the reason for the terminology. To define odd functions in general, consider a function y equals f of x. We say that f is odd if condition f of minus x equals minus f of x holds for all inputs x. Thus, if you have the graph of an odd function, like we saw before with Newton's Serpentine which has rotational symmetry about the origin, then if you move up to the curve from some input x and rotate the point on the curve at 180 degrees about the origin, then you land again on the curve at a point corresponding to input minus x. But now with a value f of minus x being the negative of f of x. Thus in general, this algebraic criterion corresponds to a 180 degree rotational symmetry. Simple examples with this rotational symmetry are the cubic function y equals x cubed. The cube root function, even simpler, the identity function y equals x and the sine curve. Any polynomial function involving constant multiples of odd powers of x only will be odd. This is the case also for any rational function where the numerator is even and the denominator is odd or the other way around where the numerator is odd and the denominator is even. No matter how complicated any of these might be, and again, we don't even need to be able to visualize the curves. You know for sure, they all have a 180 degree rotational symmetry about the origin. The explanation for this claim about rational functions is a bit technical and more details appearing in the notes. If you look at any particular odd power of x, say x to 2k plus one where 2k plus one as a typical odd integer, then you see how one minus sign peels off and comes out the front and all of the other minus signs disappear within the even power. So you end up with a negative of x to the 2k plus one. The connection between the algebraic criterion and odd powers of x is the reason for the terminology. What connection, if any exists, between odd and even powers of x and the circular functions? y equals sine x, which is odd and y equals cos x which is even. There are no powers of x in sight, let alone although even powers. There is a connection, but it's well hidden. I'll give you a sneak preview of these series expansions, part of an important and extensive topic in advanced calculus. We can think of sine x is like a polynomial except that there are infinitely many terms that go on forever. This infinite polynomial starts off with x, an odd power and you subtract x cubed, the next odd power, divided by what is called three factorial. Which is an abbreviation for three times two times one which is 6, then you add x to the fifth divided by five factorial. Then you subtract x to the seventh divided by seven factorial. You repeat this pattern of alternating between plus and minus, always moving towards the next odd power of x and dividing by the factorial of the new exponent. Where n factorial indicated by an exclamation mark is an abbreviation for n times n minus one all the way down to two times one. It's an amazing fact that this polynomial goes on forever, accurately describes sine x for any real number x. To make sense of going on forever, one has to use limits in the sense that we've been talking about in this course. If you truncate the expression at any particular term, then you'll get an approximation of sine x. Amazingly, the theory enables one to predict in advance how good the approximation will be for any particular real number x. Notice that for sine x, only odd powers of x are used, connecting to the fact that y equals sine x is an odd function. There's a similar expansion of cos x as a polynomial goes on forever. But this time, it starts with the constant one, the smallest even powers of x and alternates between subtracting and adding successive even powers of x, dividing through by the factorial of the new exponent. The fact that this expression only involves even powers of x is connected to the fact that y equals cos x is an even function. This is a very beautiful and elegant ways of representing the circular functions and using them, you can see why the derivative of sine x ought to be cos x. If you differentiate the infinite polynomial for sine x term by term, you can check that you get the infinite polynomial for cos x. If you differentiate the infinite polynomial for cos x term by term, you get the infinite polynomial for negative sine x. The derivative of sine x is cos x and the derivative of cos x is minus sine x. Taking an odd function to an even function and an even function to an odd function. As you also know, the derivative of x squared is 2x and the derivative of 2x is two. This time taking even to odd and odd to even. These are not coincidences because of the connection with even and odd powers of x. The derivative of x to the end is n times x to the minus one. Subtracting one from the exponent converts an odd power into an even power and an even power into an odd power. We have the following general facts, the derivative of odd function is always even and the derivative of an even function is always odd. It's a good exercise to verify both of these facts directly from the limit definition of the derivative. It's a good exercise also to interpret these results geometrically. In terms of the effects on slopes and tangent lines caused by reflecting on the y-axis or rotating a curve a 180 degrees about the origin.
