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2_ A function f is called even if f(-x) = f(z) on the domain of f For example, f(x) = x2 and g(z) = cos(x) are even functions A function f is called odd if f(-x) = ~f(z) o the domain of f _ For example, f(z) = x3 and g(x) = sin() are odd functions. Most functions are: never odd O” even _ a) The integral of an even function over a symmetric interval is twice the integral over half the interval.

f(x) dx = 2

f(x) dx for even f

Prove this formula by using f-R f(x) dx = f’R f(z) dx + Jor f(x) dx and by using integration by substitution on f-R f(x) dx.

b) The integral of an odd function over a symmetric interval is zero

f(x) dx = 0 for odd f ~R R Prove this formula by using ~R f(x) dx = f-R f(r) dx + J” f(r) dx and by using integration by substitution on f’R f(r) dx:

Do not try to prove these formulas by drawing pictures. The problem is to prove these formulas by using integration by substitution:

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03:05

(a) Let $f$ be an odd function; that is, $f(-x)=-f(x) .$ Invent a theorem that makes a statement about the value of an integral of the form $$\int_{-a}^{a} f(x) d x$$(b) Confirm that your theorem works for the integrals $\int_{-1}^{1} x^{3} d x \quad$ and $\quad \int_{-\pi / 2}^{\pi / 2} \sin x d x$(c) Let $f$ be an even function; that is, $f(-x)=f(x) .$ Invent a theorem that makes a statement about the relationship between the integrals$$\int_{-a}^{a} f(x) d x \quad \text { and } \quad \int_{0}^{a} f(x) d x$$(d) Confirm that your theorem works for the integrals$$\int_{-1}^{1} x^{2} d x \quad \text { and } \quad \int_{-\pi / 2}^{\pi / 2} \cos x d x$$

01:23

Let $F$ be any function whose domain contains $-x$ whenever it contains $x$. Prove each of the following.(a) $F(x)-F(-x)$ is an odd function.(b) $F(x)+F(-x)$ is an even function.(c) $F$ can always be expressed as the sum of an odd and an even function.

02:55

A function $f$ is said to be even if $f(-x)=f(x)$ for all $x$ A function $f$ is said to be odd if $f(-x)=-f(x) .$ Suppose that $f$ is continuous for all $x$. Show that if $f$ is even, then $\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x .$ Also, if $f$ is odd, show that $\int_{-a}^{a} f(x) d x=0$

10:16

Recall that a function $ f $ is called \textit{even} if $ f(-x) = f(x) $ for all $ x $ in its domain and \textit{odd} if $ f(-x) = -f(x) $ for all such $ x $. Prove each of the following.(a) The derivative of an even function is an odd function.(b) The derivative of an odd function is an even function.

09:11

Recall that a function $f$ is called even if $f(-x)=f(x)$ for all $x$ in its domain and odd if $f(-x)=-f(x)$ for all such $x$ . Prove each of the following.(a) The derivative of an even function is an odd function.(b) The derivative of an odd function is an even function.

Transcript

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