1.
Show that the function \(f(x)=x|x|\) is differentiable on \(\R\text{.}\) Compute \(f'(x)\text{.}\)
Exercises 5.5 Exercises
2.
Give examples of functions \(f\) and \(g\) such that neither \(f\) nor \(g\) is differentiable at \(0\text{,}\) but:
-
\(f+g\) is differentiable at \(0\text{,}\)
-
\(fg\) is differentiable at \(0\text{,}\)
-
\(g \circ f\) is differentiable at \(0\text{.}\)
3.
Let \(f\) and \(g\) be differentiable at \(c\text{,}\) and assume \(g(c)\ne 0\text{.}\) Prove that \(f/g\) is differentiable at \(c\) and that
\begin{equation*}
\left(\frac{f}{g}\right)'(c)
=
\frac{f'(c)g(c)-f(c)g'(c)}{g(c)^2}.
\end{equation*}
4.
Use Proposition 5.1.6 and induction on \(n\) to prove that, for every \(n \in \N\text{,}\) the function \(x \mapsto x^n\) is differentiable on \(\R\) and satisfies
\begin{equation*}
\frac{d}{dx}x^n = nx^{n-1}.
\end{equation*}
5.
Consider the functions from Section 5.2 with \(n=4\text{:}\)
\begin{equation*}
f_4(x)=\begin{cases}
x^4 & \text{if } x \ge 0,\\
-x^4 & \text{if } x \lt 0
\end{cases}
\qquad\text{and}\qquad
g_4(x)=\begin{cases}
x^8\sin(1/x) & \text{if } x \ne 0,\\
0 & \text{if } x=0.
\end{cases}
\end{equation*}
Compute several derivatives of these functions and verify that \(f_4 \in C^3(\R)\setminus D^4(\R)\) and \(g_4 \in D^4(\R)\setminus C^4(\R)\text{.}\)
6.
Use Proposition 5.1.9 with \(f(x)=x^2\) on \((0,\infty)\) to prove that \(\sqrt{x}\) is differentiable on \((0,\infty)\) and that
\begin{equation*}
\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}.
\end{equation*}
7.
Let \(f(x)=x^2\) on \((-2,2)\text{.}\)
-
Find an open interval \(I_+\) containing \(1\) such that \(f|_{I_+}\) is injective, and describe its inverse explicitly.
-
Find an open interval \(I_-\) containing \(-1\) such that \(f|_{I_-}\) is injective, and describe its inverse explicitly.
-
Use Theorem 5.1.10 to compute the derivative of each local inverse.
8.
Give two examples showing that the hypotheses in Theorem 5.3.2 are necessary:
-
one function that is differentiable on \((a,b)\) and satisfies \(f(a)=f(b)\text{,}\) but is not continuous on \([a,b]\) and has no point \(c \in (a,b)\) with \(f'(c)=0\text{;}\)
-
one function that is continuous on \([a,b]\) and satisfies \(f(a)=f(b)\text{,}\) but is not differentiable on \((a,b)\) and has no point \(c \in (a,b)\) with \(f'(c)=0\text{.}\)
9.
Let \(f \colon [a,b] \to \R\) be continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) Assume that \(|f'(x)| \le M\) for every \(x \in (a,b)\text{.}\) Prove that
\begin{equation*}
|f(x)-f(y)| \le M|x-y|
\end{equation*}
for all \(x,y \in [a,b]\text{.}\)
10.
Let \(f,g \colon [a,b] \to \R\) be continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) Assume that \(g'(x)\ne 0\) for every \(x \in (a,b)\text{.}\) Use Theorem 5.3.6 to prove that there exists \(c \in (a,b)\) such that
\begin{equation*}
\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)}.
\end{equation*}
11.
Let \(f \colon [a,b] \to \R\) be continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) Prove directly from Theorem 5.3.3 that if \(f'(x)=0\) for every \(x \in (a,b)\text{,}\) then \(f\) is constant on \([a,b]\text{.}\)
12.
Use Proposition 5.4.6 to find the local extrema of
\begin{equation*}
f(x)=x^3-3x.
\end{equation*}
Also determine the absolute maximum and minimum of \(f\) on the interval \([-2,2]\text{.}\)
13.
Find all critical points of
\begin{equation*}
f(x)=x^4-4x^2.
\end{equation*}
Use Proposition 5.4.7 where applicable to classify them. If the second derivative test is inconclusive at some critical point, analyze that point by another method.
14.
Use Theorem 5.4.8 to show that there does not exist a differentiable function \(f \colon [0,1] \to \R\) such that \(f'(0) \lt 0 \lt f'(1)\) and \(f'(x)\ne 0\) for every \(x \in (0,1)\text{.}\)
15.
Assume the basic properties of the definite integral. Let \(f \colon [a,b] \to \R\) be continuous and suppose that
\begin{equation*}
\int_a^b f(x)\,dx = 0.
\end{equation*}
Prove that if \(f(x)\ge 0\) for every \(x \in [a,b]\text{,}\) then there exists \(c \in [a,b]\) such that \(f(c)=0\text{.}\)
16.
Assume the basic properties of the definite integral. Let \(f \colon [a,b] \to \R\) be continuous, and suppose that
\begin{equation*}
\int_a^x f(t)\,dt = 0
\end{equation*}
for every \(x \in [a,b]\text{.}\) Use Theorem 5.4.10 to prove that \(f(x)=0\) for every \(x \in [a,b]\text{.}\)
