Chapter 5

Let \(a>0\). Prove that for any numbers \(x_{1}\) and \(x_{2}\), a. \(a^{x_{1}} \cdot a^{x_{2}}=a^{x_{1}+x_{2}}\) b. \(\left(a^{x_{1}}\right)^{x_{2}}=a^{x_{1} x_{2}}\)

Find a formula for \(\sin 3 a\) in terms of \(\sin a\) and \(\cos a\). Use it to calculate \(\sin \pi / 3\) and \(\cos \pi / 3 .\) Also calculate \(\sin \pi / 6\) and \(\cos \pi / 4\)

Prove that \(\arcsin x+\arccos x=\pi / 2\) if \(-1 \leq x \leq 1\).

For \(a>0\), show that $$ \lim _{n \rightarrow \infty} n\left[a^{1 / n}-1\right]=\ln a $$

Find the unique solution of the differential equation $$ \left\\{\begin{array}{l} F^{\prime}(x)=x / \sqrt{1-x^{4}}, \quad-1

Derive formulas for \(\cos (a-b)\) and \(\sin (a-b)\) in terms of \(\sin a, \sin b, \cos a,\) and \(\cos b\)

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are periodic functions of period \(T .\) Under what conditions is the sum \(f+g: \mathbb{R} \rightarrow \mathbb{R}\) also periodic? Under what conditions is the composition \(f \circ g: \mathbb{R} \rightarrow \mathbb{R}\) periodic?

Let \(a>0\). Prove that there is a number \(k\) such that $$ a^{x}=e^{k x} \quad \text { for all } x $$

Show that there is a number \(c\) in the open interval \((1, e)\) such that $$ 1=\ln e-\ln 1=\frac{1}{c}(e-1) $$

Find the maximum and minimum points of the set \(\\{\sin x+\cos x \mid x\) in \(\mathbb{R}\\}\).

For a fixed number \(a\), how many solutions does the following equation have? $$ x \ln x=a, \quad x>0 $$

Suppose that \(h: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function having the property that \(h(a+b)=h(a) h(b) \quad\) for all \(a\) and \(b\) and that the function is not identically equal to \(0 .\) a. Using the definition of a derivative, prove that $$ h^{\prime}(x)=h^{\prime}(0) h(x) \quad \text { for all } x $$ b. Show that if \(k=h^{\prime}(0),\) then \(h(x)=e^{k x}\) for all \(x\).

Let \(k\) be a fixed number. Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a solution of the differential equation $$ \left\\{\begin{array}{lc} f^{\prime \prime}(x)+k^{2} f(x)=0 & \text { for all } x \\ f(0)=0 \quad \text { and } & f^{\prime}(0)=0 \end{array}\right. $$ Prove that \(f(x)=0\) for all \(x\).

The hyperbolic cosine of \(x,\) denoted by \(\cosh x,\) and the hyperbolic sine of \(x,\) denoted by \(\sinh x,\) are defined by $$ \cosh x \equiv \frac{e^{x}+e^{-x}}{2} \quad \text { and } \quad \sinh x \equiv \frac{e^{x}-e^{-x}}{2} \quad \text { for all } x $$ Given numbers \(a, \alpha,\) and \(\beta,\) find a solution of the equation $$ \left\\{\begin{array}{lr} f^{\prime \prime}(x)-a^{2} f(x)=0 & \text { for all } x \\ f(0)=\alpha \quad \text { and } & f^{\prime}(0)=\beta \end{array}\right. $$ that is of the form $$ f(x)=c_{1} \cosh a x+c_{2} \sinh a x $$ for all \(x\)

Show that $$ \lim _{n \rightarrow \infty}\left[\frac{\ln (1+1 / n)-\ln 1}{1 / n}\right]=\lim _{n \rightarrow \infty}\left[n \ln \left(1+\frac{1}{n}\right)\right]=1 $$ (Hint: Use the definition of the derivative of the logarithm at \(x=1 .)\)

Let \(a\) and \(b\) be numbers such that \(a^{2}+b^{2}=1 .\) Prove that there exists exactly one number \(\theta\) in the interval \([0,2 \pi)\) such that $$ \left\\{\begin{array}{l} \cos \theta=a \\ \sin \theta=b \end{array}\right. $$

Define $$ f(x)=\left\\{\begin{array}{ll} x^{2} \sin (1 / x)+x & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Prove that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that \(f^{\prime}(0)=1 .\) Also prove that there is no neighborhood \(I\) of 0 such that the function \(f: I \rightarrow \mathbb{R}\) is increasing.

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, \(g(0)>0,\) and at some positive number \(x_{0}, g\left(x_{0}\right)=0 .\) Prove that there is a smallest positive number \(p\) at which \(g(x)=0 .[\) Hint: Define \(p=\inf \\{x \mid x>0, g(x)=0\\}\) and prove that \(p>0\) and \(g(p)=0 .]\)