# (a) find the Maclaurin polynomial P_{3}(x) for f(x), (b) complete the following x: -0.75, -0.50, -0.25, 0, 0.25, 0.50, 0.75 for f(x) and P_{3}(x), and (c) sketch the graphs of f(x) and P_{3}(x) on the same set of coordinate axes. f(x) = arcsin x

Question
Polynomial graphs
(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. $$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}$$

2020-12-16
a) $$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}$$
$$\displaystyle{P}_{{{3}}}{\left({x}\right)}={\sum_{{{k}={0}}}^{{{3}}}}{\frac{{{{f}^{{{\left({k}\right)}}}{\left({0}\right)}}}}{{{k}!}}}{x}^{{{k}}}$$
$$\displaystyle{P}_{{{3}}}{\left({x}\right)}={x}+{\frac{{{x}^{{{3}}}}}{{{6}}}}$$ $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&-{0.75}&-{0.50}&-{0.25}&-{0.0}&{0.25}&{0.50}&{0.75}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&-{0.848}&-{0.524}&-{0.253}&{0}&{0.253}&{0.524}&{0.848}\backslash{h}{l}\in{e}{P}_{{{3}}}{\left({x}\right)}&-{0.820}&-{0.521}&-{0.253}&{0}&{0.253}&{0.521}&{0.820}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ On the graph below: Purple curve = $$\displaystyle{f{{\left({x}\right)}}}$$ Orange curve = $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ Step 3

### Relevant Questions

(a) find the Maclaurin polynomial $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ for PSKf(x), (b) complete the following PSKx: -0.75, -0.50, -0.25, 0, 0.25, 0.50, 0.75 for f(x) and P_{3}(x) and (c) sketch the graphs of f(x) and $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ on the same set of coordinate axes. $$\displaystyle{f{{\left({x}\right)}}}={\arctan{{x}}}$$
a) Identify the parameters a, b, h, and k in the polynomial $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left({x}+{3}\right)}^{{{3}}}-{2}$$ Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{3}}}$$.
b) State the domain and range of the transformed function.
c) Sketch graphs of the base function and the transformed function on the same set of axes.
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 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?
Investigate the change in the graph of a sinusoidal function of the form $$\displaystyle{\quad\text{and}\quad}={\sin{{x}}}{\quad\text{or}\quad}{\quad\text{and}\quad}={\cos{{x}}}$$ when multiplied by a polynomial function. Use a graphing calculator to sketch the graphs of and $$\displaystyle={2}{x},{\quad\text{and}\quad}=-{2}{x},{\quad\text{and}\quad}={2}{x}{\cos{{x}}}$$ on the same coordinate plane, on the interval $$\displaystyle{\left[-{20},{20}\right]}.$$
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
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 $$\displaystyle{S}_{{{N}}}{\left({x}\right)}\ {\quad\text{and}\quad}\ {x}{\left(\pi\ -\ {x}\right)},\ {f}{\quad\text{or}\quad}\ {0}\ \leq\ {x}\ \leq\ \pi,\ {f}{\quad\text{or}\quad}\ {N}={2}\ {\quad\text{and}\quad}\ {t}{h}{e}{n}\ {N}={10}.$$
$$\displaystyle{\frac{{{4}}}{{\pi}}}\ {\sum_{{{n}={1}}}^{{\infty}}}\ {\frac{{{1}\ -\ {\left(-{1}\right)}^{{{n}}}}}{{{n}^{{{3}}}}}}\ {\sin{{\left({n}{x}\right)}}}\ {o}{n}\ {\left[{0},\ \pi\right]}.$$
Make rough sketches of the graphs of each of the following polynomial functions. Be sure to label the x- and y- intercepts. $$\displaystyle{a}{)}{y}={x}{\left({2}{x}+{5}\right)}{\left({2}{x}-{7}\right)}$$ $$\displaystyle{b}{)}{y}={\left({15}-{2}{x}\right)}^{{{2}}}{\left({x}+{3}\right)}$$
Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$
Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of $$\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}$$