(34.4) Let \(f\) be the function,
\[f(t) = \left\{\begin{array}{lr} t & t < 0\\ t^2+1 & 0 \le t \le 2\\ 0 & t > 2 \end {array}\right.\]Determine the function \(F(x) = \int_0^x f(t)\; dt\).
Sketch \(F\). Where is \(F\) continuous?
Where is \(F\) differentiable? Calculate \(F'\) at the points of differentiability.
By the FTC II, \(F(x)\) is continuous everywhere.
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.
(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 \(F'(x) = G'(\sin x)\cos x = f(\sin x)\cos x\).
(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 \(f(x) = 0\). \(\Box\)
(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). \(\Box\)
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,
\[\langle f, g \rangle = \int_a^b{fg}\]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 \(f = 0\).
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.