11.9: Problems

Exercise \(\PageIndex{1}\)

Prove the following identities using only the definitions of the trigonometric functions, the Pythagorean identity, or the identities for sines and cosines of sums of angles.

1. \(\cos 2 x=2 \cos ^{2} x-1\).
2. \(\sin 3 x=A \sin ^{3} x+B \sin x\), for what values of \(A\) and \(B\) ?
3. \(\sec \theta+\tan \theta=\tan \left(\frac{\theta}{2}+\frac{\pi}{4}\right)\).

Exercise \(\PageIndex{2}\)

Determine the exact values of

1. \(\sin \frac{\pi}{8}\).
2. \(\tan 15^{\circ}\)
3. \(\cos 105^{\circ}\).

Exercise \(\PageIndex{3}\)

Denest the following if possible.

1. \(\sqrt{3-2 \sqrt{2}}\).
2. \(\sqrt{1+\sqrt{2}}\).
3. \(\sqrt{5+2 \sqrt{6}}\).
4. \(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\)
5. Find the roots of \(x^{2}+6 x-4 \sqrt{5}=0\) in simplified form.

Exercise \(\PageIndex{4}\)

Determine the exact values of

1. \(\sin \left(\cos ^{-1} \frac{3}{5}\right)\).
2. \(\tan \left(\sin ^{-1} \frac{x}{7}\right)\)
3. \(\sin ^{-1}\left(\sin \frac{3 \pi}{2}\right)\).

Exercise \(\PageIndex{5}\)

Do the following.

1. Write \((\cosh x-\sinh x)^{6}\) in terms of exponentials.
2. Prove \(\cosh (x-y)=\cosh x \cosh y-\sinh x \sinh y\) using the exponential forms of the hyperbolic functions.
3. Prove \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\).
4. If \(\cosh x=\frac{13}{12}\) and \(x<0\), find \(\sinh x\) and \(\tanh x\).
5. Find the exact value of \(\sinh (\operatorname{arccosh} 3)\).

Exercise \(\PageIndex{6}\)

Prove that the inverse hyperbolic functions are the following logarithms:

1. \(\cosh ^{-1} x=\ln \left(x+\sqrt{x^{2}-1}\right)\).
2. \(\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}\).

Exercise \(\PageIndex{7}\)

Write the following in terms of logarithms:

1. \(\cosh ^{-1} \frac{4}{3}\).
2. \(\tanh ^{-1} \frac{1}{2}\).
3. \(\sinh ^{-1} 2\).

Exercise \(\PageIndex{8}\)

Solve the following equations for \(x\).

1. \(\cosh (x+\ln 3)=3\).
2. \(2 \tanh ^{-1} \frac{x-2}{x-1}=\ln 2\).
3. \(\sinh ^{2} x-7 \cosh x+13=0\).

Exercise \(\PageIndex{9}\)

Compute the following integrals.

1. \(\int x e^{2 x^{2}} d x\).
2. \(\int_{0}^{3} \frac{5 x}{\sqrt{x^{2}+16}} d x\).
3. \(\int x^{3} \sin 3 x d x\). (Do this using integration by parts, the Tabular Method, and differentiation under the integral sign.)
4. \(\int \cos ^{4} 3 x d x\).
5. \(\int_{0}^{\pi / 4} \sec ^{3} x d x\).
6. \(\int e^{x} \sinh x d x\)
7. \(\int \sqrt{9-x^{2}} d x\)
8. \(\int \frac{d x}{\left(4-x^{2}\right)^{2}}\), using the substitution \(x=2 \tanh u\).
9. \(\int_{0}^{4} \frac{d x}{\sqrt{9+x^{2}}}\), using a hyperbolic function substitution.
10. \(\int \frac{d x}{1-x^{2}}\), using the substitution \(x=\tanh u\).
11. \(\int \frac{d x}{\left(x^{2}+4\right)^{3 / 2}}\), using the substitutions \(x=2 \tan \theta\) and \(x=2 \sinh u\).
12. \(\int \frac{d x}{\sqrt{3 x^{2}-6 x+4}}\).

Exercise \(\PageIndex{10}\)

Find the sum for each of the series:

1. \(5+\frac{25}{7}+\frac{125}{49}+\frac{625}{343}+\cdots\).
2. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\).
3. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\).
4. \(\sum_{n=-1}^{\infty}(-1)^{n+1}\left(\frac{e}{\pi}\right)^{n}\).
5. \(\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)\).
6. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\).
7. What is \(0.56 \overline{9}\) ?

Exercise \(\PageIndex{11}\)

A superball is dropped from a \(2.00 \mathrm{~m}\) height. After it rebounds, it reaches a new height of \(1.65 \mathrm{~m}\). Assuming a constant coefficient of restitution, find the (ideal) total distance the ball will travel as it keeps bouncing.

Exercise \(\PageIndex{12}\)

Here are some telescoping series problems.

1. Verify that \[\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}=\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right) .\nonumber \] 
2. Find the \(n\)th partial sum of the series \(\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right)\) and use it to determine the sum of the resulting telescoping series.
3. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).

Exercise \(\PageIndex{13}\)

Determine the radius and interval of convergence of the following infinite series:

1. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{(x-1)^{n}}{n}\).
2. \(\sum_{n=1}^{\infty} \frac{x^{n}}{2^{n} n !}\).
3. \(\sum_{n=1}^{\infty} \frac{1}{n}\left(\frac{x}{5}\right)^{n}\).
4. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{\sqrt{n}}\).

Exercise \(\PageIndex{14}\)

Find the Taylor series centered at \(x=a\) and its corresponding radius of convergence for the given function. In most cases, you need not employ the direct method of computation of the Taylor coefficients.

1. \(f(x)=\sinh x, a=0\).
2. \(f(x)=\sqrt{1+x}, a=0\).
3. \(f(x)=\ln \frac{1+x}{1-x}, a=0\).
4. \(f(x)=x e^{x}, a=1\).
5. \(f(x)=\frac{1}{\sqrt{x}}, a=1\).
6. \(f(x)=x^{4}+x-2, a=2\).
7. \(f(x)=\frac{x-1}{2+x}, a=1\).

Exercise \(\PageIndex{15}\)

Consider Gregory’s expansion \[\tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1} x^{2 k+1} .\nonumber \]

1. Derive Gregory’s expansion by using the definition \[\tan ^{-1} x=\int_{0}^{x} \frac{d t}{1+t^{2}},\nonumber \] expanding the integrand in a Maclaurin series, and integrating the resulting series term by term. 
2. From this result, derive Gregory’s series for \(\pi\) by inserting an appropriate value for \(x\) in the series expansion for \(\tan ^{-1} x\).

Exercise \(\PageIndex{16}\)

In the event that a series converges uniformly, one can consider the derivative of the series to arrive at the summation of other infinite series.

1. Differentiate the series representation for \(f(x)=\frac{1}{1-x}\) to sum the series \(\sum_{n=1}^{\infty} n x^{n},|x|<1\).
2. Use the result from part a to sum the series \(\sum_{n=1}^{\infty} \frac{n}{5^{n}}\).
3. Sum the series \(\sum_{n=2}^{\infty} n(n-1) x^{n},|x|<1\).
4. Use the result from part c to sum the series \(\sum_{n=2}^{\infty} \frac{n^{2}-n}{5^{n}}\).
5. Use the results from this problem to sum the series \(\sum_{n=4}^{\infty} \frac{n^{2}}{5^{n}}\).

Exercise \(\PageIndex{17}\)

Evaluate the integral \(\int_{0}^{\pi / 6} \sin ^{2} x d x\) by doing the following:

1. Compute the integral exactly.
2. Integrate the first three terms of the Maclaurin series expansion of the integrand and compare with the exact result.

Exercise \(\PageIndex{18}\)

Determine the next term in the time dilation example, 11.8.2. That is, find the \(\frac{v^{4}}{c^{2}}\) term and determine a better approximation to the time difference of \(1 \mathrm{~ns}\).

Exercise \(\PageIndex{19}\)

Evaluate the following expressions at the given point. Use your calculator or your computer (such as Maple). Then use series expansions to find an approximation to the value of the expression to as many places as you trust.

1. \(\frac{1}{\sqrt{1+x^{3}}}-\cos x^{2}\) at \(x=0.015\).
2. \(\ln \sqrt{\frac{1+x}{1-x}}-\tan x\) at \(x=0.0015\).
3. \(f(x)=\frac{1}{\sqrt{1+2 x^{2}}}-1+x^{2}\) at \(x=5.00 \times 10^{-3}\).
4. \(f(R, h)=R-\sqrt{R^{2}+h^{2}}\) for \(R=1.374 \times 10^{3} \mathrm{~km}\) and \(h=1.00 \mathrm{~m}\).
5. \(f(x)=1-\frac{1}{\sqrt{1-x}}\) for \(x=2.5 \times 10^{-13}\).