Harrison Chapman

Math 317: Homework 11

1. (34.4) Let $f$ be the function,

   $$f(t) = \left\{\begin{array}{lr} t & t < 0\\ t^2+1 & 0 \le t \le 2\\ 0 & t > 2 \end {array}\right.$$

   1. Determine the function $F(x) = \int_0^x f(t)\; dt$.

   2. Sketch $F$. Where is $F$ continuous?

   3. Where is $F$ differentiable? Calculate $F'$ at the points of differentiability.

   1. $$F(x) = \left\{\begin{array}{lr} t^2 & t < 0\\ t^3/3+t & 0 \le t \le 2\\ 14/3 & t > 2 \end {array}\right.$$

   2. By the FTC II, $F(x)$ is continuous everywhere.

   3. By the FTC II, $F(x)$ is differentiable everywhere that $f(x)$ is continuous, hence at all $x$ except possibly $x=0$ and $x=2$.

      We can prove that $F(x)$ is not differentiable at $x=0$ and $x=2$ by showing that the left-hand and right-hand difference quotient limits disagree.

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2. (34.6) Let $f$ be a continuous function on $\mathbb R$ and define

   $$F(x) = \int_0^{\sin x}{f(t)\; dt}\quad\textrm{for } x \in \mathbb R.$$

   Show that $F$ is differentiable on $\mathbb R$ and compute $F'$.

   Let $G(x) = \int_0^x{f(t)\; dt}$. Then $G(x)$ is differentiable and $G'(x) = f(x)$ by the FTC II as $f$ is continuous. Notice that $F(x) = G(\sin x)$. So by the chain rule, $F$ is differentiable and $F'(x) = G'(\sin x)\cos x = f(\sin x)\cos x$.

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3. (34.11) Suppose $f$ is a continuous function on $[a, b]$. Show that if $\int_a^b{f(x)^2\; dx} = 0$, then $f(x) = 0$ for all $x \in [a, b]$.

   Proof: We have that $f^2(x) \ge 0$ and $f^2(x)$ is continuous as $f$ is, so by Theorem 33.4(ii) $f^2(x) = 0$. By factoring, this is only possible if $f(x) = 0$. $\Box$

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4. (34.12) Show that if $f$ is a continuous real-valued function on $[a, b]$ satisfying $\int_a^b{f(x)g(x)\; dx} = 0$ for every continuous function $g$ on $[a, b]$, then $f(x) = 0$ for all $x \in [a, b]$.

   Proof. As the statement is true for all $g$, it is true for $g = f$. Then this is true by question (3). $\Box$

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5. For this problem, you may use the results of questions (3) and (4) freely. Let $C([a, b])$ be the set of all continuous functions on the interval $[a, b]$. Define a function

   $$\langle \cdot, \cdot \rangle: C([a, b]) \times C([a, b]) \to \mathbb R$$

   by,

   $$\langle f, g \rangle = \int_a^b{fg}$$

   1. Let $f, g, h \in C([a,b])$. Show that $\langle \cdot, \cdot \rangle$ is an inner product. That is, show each of:

      • $\langle f,g \rangle = \langle g,f \rangle$.

      • $\langle af,g \rangle = a\langle f,g \rangle$.

      • $\langle f+h,g \rangle = \langle f,g \rangle + \langle h, g\rangle$.

      • $\langle f, f\rangle = 0$ if and only if $f = 0$.

   2. Let $f \in C([a,b])$. Show that if $\langle f, g \rangle = 0$ for all $g \in C([a,b])$, then $f = 0$. In other words, you are showing that the only function orthogonal to all other functions with this inner product is the zero function.

   1. • By commutativity of function multiplication.
      • By linearity of the integral.
      • By linearity of the integral.
      • By the result of question (3)
   2. By the result of question (4).
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