# Let S_{N}(x)=frac{4}{pi} sum_{n=1}^{N} frac{1 - (-1)^{n}}{n^{3}} sin(nx). Construct graphs of S_{N}(x) and x(pi - x), for 0 leq x leq pi, for N=2 and then N=10. This will give some sense of the correctness of Fourier’s claim that this polynomial could be exactly represented by the infinite series frac{4}{pi} sum_{n=1}^{infty} frac{1 - (-1)^{n}}{n^{3}} sin(nx) on [0, pi].

Question
Polynomial graphs
Let $$\displaystyle{S}_{{{N}}}{\left({x}\right)}={\frac{{{4}}}{{\pi}}}\ {\sum_{{{n}={1}}}^{{{N}}}}\ {\frac{{{1}\ -\ {\left(-{1}\right)}^{{{n}}}}}{{{n}^{{{3}}}}}}\ {\sin{{\left({n}{x}\right)}}}.$$
Construct graphs of $$\displaystyle{S}_{{{N}}}{\left({x}\right)}\ {\quad\text{and}\quad}\ {x}{\left(\pi\ -\ {x}\right)},\ {f}{\quad\text{or}\quad}\ {0}\ \leq\ {x}\ \leq\ \pi,\ {f}{\quad\text{or}\quad}\ {N}={2}\ {\quad\text{and}\quad}\ {t}{h}{e}{n}\ {N}={10}.$$
This will give some sense of the correctness of Fourier’s claim that this polynomial could be exactly represented by the infinite series
$$\displaystyle{\frac{{{4}}}{{\pi}}}\ {\sum_{{{n}={1}}}^{{\infty}}}\ {\frac{{{1}\ -\ {\left(-{1}\right)}^{{{n}}}}}{{{n}^{{{3}}}}}}\ {\sin{{\left({n}{x}\right)}}}\ {o}{n}\ {\left[{0},\ \pi\right]}.$$

2021-01-06
Step 1
Note that for even n the coefficient in the sum is zero, so some terms are absent.

Step 2
$$\displaystyle{S}_{{{10}}}{\left({x}\right)}={\frac{{{8}}}{{\pi}}}{\left({\sin{\ }}{x}\ +\ {\frac{{{1}}}{{{27}}}}\ {\sin{\ }}{3}{x}\ +\ {\frac{{{1}}}{{{125}}}}\ {\sin{\ }}{5}{x}\ +\ {\frac{{{1}}}{{{273}}}}\ {\sin{\ }}{7}{x}\ +\ {\frac{{{1}}}{{{729}}}}\ {\sin{\ }}{9}{x}\right)}$$

Step 3
We see that $$\displaystyle{S}_{{{10}}}{\left({x}\right)}$$ is a very good a approximation of the function
$$\displaystyle{f{{\left({x}\right)}}}={x}{\left(\pi\ -\ {x}\right)}.$$
The absolute value of their difference is given below for the comparison:

### Relevant Questions

A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of $$25^{\circ}F$$. However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to $$25^{\circ}F$$. One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 5.1. Another similar frozen food case was equipped with the old thermostat, and a random sample of 19 temperature readings gave a sample variance of 12.8. Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a $$5\%$$ level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings? (Let population 1 refer to data from the old thermostat.)
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}>?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}\neq?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}?_{2}^{2},H1:?_{1}^{2}=?_{2}^{2}$$
(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)
What are the degrees of freedom?
$$df_{N} = ?$$
$$df_{D} = ?$$
What assumptions are you making about the original distribution?
The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.
(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings. Fail to reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.Reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.
Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of $$\alpha = 0.05$$. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.) Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities? $$\begin{matrix} \text{Lemon Imports} & 230 & 265 & 358 & 480 & 530\\ \text{Crashe Fatality Rate} & 15.9 & 15.7 & 15.4 & 15.3 & 14.9\\ \end{matrix}$$
Determine the area of the region below the parametric curve given by the set of parametric equations. For each problem you may assume that each curve traces out exactly once from right to left for the given range of t. For these problems you should only use the given parametric equations to determine the answer. 1.$$x = t^2 + 5t - 1 y = 40 - t^2 -2 \leq t \leq 5$$ 2.$$x = 3cos^2 (t) — sin^2 (t) y = 6 + cos(t) -\frac{\pi}{2} \neq t \leq 0$$ 3.$$x = e^{\frac{1}{4} t} —2 y = 4 + e^{\frac{1}{4 t}} — e^{\frac{1}{4} t} - 6 \leq t \leq 1$$
Finance bonds/dividends/loans exercises, need help or formulas
Some of the exercises, calculating the Ri is clear, but then i got stuck:
A security pays a yearly dividend of 7€ during 5 years, and on the 5th year we could sell it at a price of 75€, market rate is 19%, risk free rate 2%, beta 1,8. What would be its price today? 2.1 And if its dividend growths 1,7% each year along these 5 years-what would be its price?
A security pays a constant dividend of 0,90€ during 5 years and thereafter will be sold at 10 €, market rate 18%, risk free rate 2,5%, beta 1,55, what would be its price today?
At what price have i purchased a security if i already made a 5€ profit, and this security pays dividends as follows: first year 1,50 €, second year 2,25€, third year 3,10€ and on the 3d year i will sell it for 18€. Market rate is 8%, risk free rate 0,90%, beta=2,3.
What is the original maturity (in months) for a ZCB, face value 2500€, required rate of return 16% EAR if we paid 700€ and we bought it 6 month after the issuance, and actually we made an instant profit of 58,97€
You'll need 10 Vespas for your Parcel Delivery Business. Each Vespa has a price of 2850€ fully equipped. Your bank is going to fund this operation with a 5 year loan, 12% nominal rate at the beginning, and after increasing 1% every year. You'll have 5 years to fully amortize this loan. You want tot make monthly installments. At what price should you sell it after 3 1/2 years to lose only 10% of the remaining debt.
a) Identify the parameters a, k, d, and c in the polynomial function $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{4}}}$$. b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function. c) Describe two possible orders in which the transformations can be applied to the graph of $$\displaystyle{y}={x}^{{{4}}}$$ to produce the graph of $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. d) Sketch graphs of the base function and the transformed function on the same set of axes.
Find the Maclaurin series for $$f(x)=x^4\cos x$$. Also, find the third term for this series.
The Maclaurin series for $$\cos(x)$$ is given by:
$$\cos(x)=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+...$$
$$\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}$$
(a) find the Maclaurin polynomial $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ for f(x), (b) complete the following $$\displaystyle{x}:-{0.75},-{0.50},-{0.25},{0},{0.25},{0.50},{0.75}{f}{\quad\text{or}\quad}{f{{\left({x}\right)}}}$$ and $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$, and (c) sketch the graphs of f(x) and $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ on the same set of coordinate axes. $$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}$$
Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\displaystyle{\sec{{x}}}\approx{1}+{\frac{{{x}^{{{2}}}}}{{{2}!}}}+{\frac{{{5}{x}^{{{4}}}}}{{{4}!}}}$$ where x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?
Find the quadratic polynomial $$\displaystyle{g{{\left({x}\right)}}}-{a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c}\ \text{which best fits the function}\ {f{{\left({x}\right)}}}={e}^{{{x}}}\ \text{at}\ {x}={0},\ \text{in the sense that}\ {g{{\left({0}\right)}}}={f{{\left({0}\right)}}},\ \text{and}\ {g}'{\left({0}\right)}={f}'{\left({0}\right)},\ \text{and}\ {g}{''}{\left({0}\right)}={f}{''}{\left({0}\right)}.$$ Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?