11.9: Problems
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Exercise 11.9.1
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.
- cos2x=2cos2x−1.
- sin3x=Asin3x+Bsinx, for what values of A and B ?
- secθ+tanθ=tan(θ2+π4).
Exercise 11.9.2
Determine the exact values of
- sinπ8.
- tan15∘
- cos105∘.
Exercise 11.9.3
Denest the following if possible.
- √3−2√2.
- √1+√2.
- √5+2√6.
- 3√√5+2−3√√5−2
- Find the roots of x2+6x−4√5=0 in simplified form.
Exercise 11.9.4
Determine the exact values of
- sin(cos−135).
- tan(sin−1x7)
- sin−1(sin3π2).
Exercise 11.9.5
Do the following.
- Write (coshx−sinhx)6 in terms of exponentials.
- Prove cosh(x−y)=coshxcoshy−sinhxsinhy using the exponential forms of the hyperbolic functions.
- Prove cosh2x=cosh2x+sinh2x.
- If coshx=1312 and x<0, find sinhx and tanhx.
- Find the exact value of sinh(arccosh3).
Exercise 11.9.6
Prove that the inverse hyperbolic functions are the following logarithms:
- cosh−1x=ln(x+√x2−1).
- tanh−1x=12ln1+x1−x.
Exercise 11.9.7
Write the following in terms of logarithms:
- cosh−143.
- tanh−112.
- sinh−12.
Exercise 11.9.8
Solve the following equations for x.
- cosh(x+ln3)=3.
- 2tanh−1x−2x−1=ln2.
- sinh2x−7coshx+13=0.
Exercise 11.9.9
Compute the following integrals.
- ∫xe2x2dx.
- ∫305x√x2+16dx.
- ∫x3sin3xdx. (Do this using integration by parts, the Tabular Method, and differentiation under the integral sign.)
- ∫cos43xdx.
- ∫π/40sec3xdx.
- ∫exsinhxdx
- ∫√9−x2dx
- ∫dx(4−x2)2, using the substitution x=2tanhu.
- ∫40dx√9+x2, using a hyperbolic function substitution.
- ∫dx1−x2, using the substitution x=tanhu.
- ∫dx(x2+4)3/2, using the substitutions x=2tanθ and x=2sinhu.
- ∫dx√3x2−6x+4.
Exercise 11.9.10
Find the sum for each of the series:
- 5+257+12549+625343+⋯.
- ∑∞n=0(−1)n34n.
- ∑∞n=225n.
- ∑∞n=−1(−1)n+1(eπ)n.
- ∑∞n=0(52n+13n).
- ∑∞n=13n(n+3).
- What is 0.56¯9 ?
Exercise 11.9.11
A superball is dropped from a 2.00 m height. After it rebounds, it reaches a new height of 1.65 m. Assuming a constant coefficient of restitution, find the (ideal) total distance the ball will travel as it keeps bouncing.
Exercise 11.9.12
Here are some telescoping series problems.
- Verify that ∞∑n=11(n+2)(n+1)=∞∑n=1(n+1n+2−nn+1).
- Find the nth partial sum of the series ∑∞n=1(n+1n+2−nn+1) and use it to determine the sum of the resulting telescoping series.
- Sum the series ∑∞n=1[tan−1n−tan−1(n+1)] by first writing the N th partial sum and then computing limN→∞sN.
Exercise 11.9.13
Determine the radius and interval of convergence of the following infinite series:
- ∑∞n=1(−1)n(x−1)nn.
- ∑∞n=1xn2nn!.
- ∑∞n=11n(x5)n.
- ∑∞n=1(−1)nxn√n.
Exercise 11.9.14
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.
- f(x)=sinhx,a=0.
- f(x)=√1+x,a=0.
- f(x)=ln1+x1−x,a=0.
- f(x)=xex,a=1.
- f(x)=1√x,a=1.
- f(x)=x4+x−2,a=2.
- f(x)=x−12+x,a=1.
Exercise 11.9.15
Consider Gregory’s expansion tan−1x=x−x33+x55−⋯=∞∑k=0(−1)k2k+1x2k+1.
- Derive Gregory’s expansion by using the definition tan−1x=∫x0dt1+t2, expanding the integrand in a Maclaurin series, and integrating the resulting series term by term.
- From this result, derive Gregory’s series for π by inserting an appropriate value for x in the series expansion for tan−1x.
Exercise 11.9.16
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.
- Differentiate the series representation for f(x)=11−x to sum the series ∑∞n=1nxn,|x|<1.
- Use the result from part a to sum the series ∑∞n=1n5n.
- Sum the series ∑∞n=2n(n−1)xn,|x|<1.
- Use the result from part c to sum the series ∑∞n=2n2−n5n.
- Use the results from this problem to sum the series ∑∞n=4n25n.
Exercise 11.9.17
Evaluate the integral ∫π/60sin2xdx by doing the following:
- Compute the integral exactly.
- Integrate the first three terms of the Maclaurin series expansion of the integrand and compare with the exact result.
Exercise 11.9.18
Determine the next term in the time dilation example, 11.8.2. That is, find the v4c2 term and determine a better approximation to the time difference of 1 ns.
Exercise 11.9.19
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√1+x3−cosx2 at x=0.015.
- ln√1+x1−x−tanx at x=0.0015.
- f(x)=1√1+2x2−1+x2 at x=5.00×10−3.
- f(R,h)=R−√R2+h2 for R=1.374×103 km and h=1.00 m.
- f(x)=1−1√1−x for x=2.5×10−13.