Find the quadratic polynomial g(x)-ax^{2} + bx + c text{which best fits the function} f(x)=e^{x} text{at} x=0, text{in the sense that} g(0)=f(0), text{and} g'(0)=f'(0), text{and} g''(0)=f''(0). Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?

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asked 2021-02-02

(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. \(\displaystyle{f{{\left({x}\right)}}}={\arctan{{x}}}\)

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asked 2020-11-08

The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.
Source:
Centers for Disease Control and Prevention.
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{amp, Cases}\backslash{h}{l}\in{e}{1995}&{a}\mp,\ {295}\backslash{h}{l}\in{e}{1997}&{a}\mp,\ {166}\backslash{h}{l}\in{e}{1999}&{a}\mp,\ {109}\backslash{h}{l}\in{e}{2001}&{a}\mp,\ {115}\backslash{h}{l}\in{e}{2003}&{a}\mp,\ {94}\backslash{h}{l}\in{e}{2005}&{a}\mp,\ {107}\backslash{h}{l}\in{e}{2007}&{a}\mp,\ {79}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
a) Plot the data on a graphing calculator, letting \(\displaystyle{t}={0}\) correspond to the year 1995.
b) Using the regression feature on your calculator, find a quadratic, a cubic, and an exponential function that models this data.
c) Plot the three functions with the data on the same coordinate axes. Which function or functions best capture the behavior of the data over the years plotted?
d) Find the number of cases predicted by all three functions for 20152015. Which of these are realistic? Explain.

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asked 2021-02-26

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.

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asked 2020-12-15

(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. \(\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}\)

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asked 2021-02-25

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.

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asked 2021-02-22

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 \(\displaystyle{\left[-{20},{20}\right]}.\)

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asked 2021-02-15

Using calculus, it can be shown that the tangent function can be approximated by the polynomial \(\displaystyle{\tan{\ }}{x}\ \approx\ {x}\ +\ {\frac{{{2}{x}^{{{3}}}}}{{{3}!}}}\ +\ {\frac{{{16}{x}^{{{5}}}}}{{{5}!}}}\) where x is in radians. Use a graphing utility to graph the tangent function and its polynomial approximation in the same viewing window. How do the graphs.

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asked 2020-10-23

A random sample of \(\displaystyle{n}_{{1}}={16}\) communities in western Kansas gave the following information for people under 25 years of age.
\(\displaystyle{X}_{{1}}:\) Rate of hay fever per 1000 population for people under 25
\(\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}\)
A random sample of \(\displaystyle{n}_{{2}}={14}\) regions in western Kansas gave the following information for people over 50 years old.
\(\displaystyle{X}_{{2}}:\) Rate of hay fever per 1000 population for people over 50
\(\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}\)
(i) Use a calculator to calculate \(\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.\) (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use \(\displaystyle\alpha={0.05}.\)
(a) What is the level of significance?
State the null and alternate hypotheses.
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}\)
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference \(\displaystyle\mu_{{1}}-\mu_{{2}}\). Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference \(\displaystyle\mu_{{1}}-\mu_{{2}}\). Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value \(\displaystyle>{0.250}\)
\(\displaystyle{0.125}<{P}-\text{value}<{0},{250}\)
\(\displaystyle{0},{050}<{P}-\text{value}<{0},{125}\)
\(\displaystyle{0},{025}<{P}-\text{value}<{0},{050}\)
\(\displaystyle{0},{005}<{P}-\text{value}<{0},{025}\)
P-value \(\displaystyle<{0.005}\)
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value

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asked 2020-12-03

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. \(\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}\)

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asked 2020-11-01

For the following exercise, for each polynomial \(\displaystyle{f{{\left({x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}\): a) find the degree, b) find the zeros, if any, c) find the y-intercept(s), if any, d) use the leading coefficient to determine the graph’s end behavior, e) determine algebraically whether the polynomial is even, odd, or neither.

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Answer 1

Step 1 The functions f(x) and g(x) are: \(\displaystyle{f{{\left({x}\right)}}}={e}^{{{x}}}\)
\(\displaystyle{g{{\left({x}\right)}}}={a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c}\)
\(\displaystyle{f}'{\left({x}\right)}=\ {\frac{{{d}{\left({f{{\left({x}\right)}}}\right)}}}{{{\left.{d}{x}\right.}}}}=\ {\frac{{{d}{\left({e}^{{{x}}}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using formula for derivative of exponential function: \(\displaystyle{f}'{\left({x}\right)}={e}^{{{x}}}\)
\(\displaystyle{f}{''}{\left({x}\right)}=\ {\frac{{{d}{\left({f},{\left({x}\right)}\right)}}}{{{\left.{d}{x}\right.}}}}=\ {\frac{{{d}{\left({e}^{{{x}}}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using formula for derivative of exponential function: \(\displaystyle{f}{''}{\left({x}\right)}={e}^{{{x}}}\)
\(\displaystyle{g}'{\left({x}\right)}=\ {\frac{{{d}{\left({g{{\left({x}\right)}}}\right)}}}{{{\left.{d}{x}\right.}}}}=\ {\frac{{{d}{\left({a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using Theorem 3.2: \(\displaystyle{g}'{\left({x}\right)}=\ {\frac{{{d}{\left({a}{x}^{{{2}}}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {\frac{{{d}{\left({b}{x}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {\frac{{{d}{\left({c}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using Theorem 3.1: \(\displaystyle{g}'{\left({x}\right)}={a}\ \times\ {\frac{{{d}{\left({x}^{{{2}}}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {b}\ \times\ {\frac{{{d}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {0}\) Step 2 Using Power Law: \(\displaystyle{g}'{\left({x}\right)}={a}\ \times\ {2}\ \times\ {x}^{{{2}\ -\ {1}}}\ +\ {b}\ \times\ {1}\ +\ {0}\)
\(\displaystyle{g}'{\left({x}\right)}={2}{a}{x}\ +\ {b}\)
\(\displaystyle{g}{''}{\left({x}\right)}=\ {\frac{{{d}{\left({g}'{\left({x}\right)}\right)}}}{{{\left.{d}{x}\right.}}}}=\ {\frac{{{d}{\left({2}{a}{x}\ +\ {b}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using Theorem 3.2: \(\displaystyle{g}{''}{\left({x}\right)}=\ {\frac{{{d}{\left({2}{a}{x}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {\frac{{{d}{\left({b}\right)}}}{{{\left.{d}{x}\right.}}}}\) Using Theorem 3.1: \(\displaystyle{g}{''}{\left({x}\right)}={2}{a}\ \times\ {\frac{{{d}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}}\ +\ {0}\)
\(\displaystyle{g}{''}{\left({x}\right)}={2}{a}\ \times\ {1}\)
\(\displaystyle{g}{''}{\left({x}\right)}={2}{a}\) Substituting \(\displaystyle{x}={0}\ \text{we get}:\)
\(\displaystyle{f{{\left({0}\right)}}}={f}'{\left({0}\right)}={f}{''}{\left({0}\right)}={1}\)
\(\displaystyle{g}{''}{\left({0}\right)}={2}{a}\)
\(\displaystyle{g}'{\left({0}\right)}={b}\)
\(\displaystyle{g{{\left({0}\right)}}}={c}\) Using \(\displaystyle{g{{\left({0}\right)}}}={f{{\left({0}\right)}}},\ {g}'{\left({0}\right)}={f}'{\left({0}\right)}\ \text{and}\ {g}{''}{\left({0}\right)}={f}{''}{\left({0}\right)},\ \text{we get}:\)
\(\displaystyle{2}{a}={1}\)
\(\displaystyle{a}=\ {\frac{{{1}}}{{{2}}}}\)
\(\displaystyle{b}={1}\)
\(\displaystyle{c}={1}\) Thus the quadratic polynomial g(x) that fits f(x) is: \(\displaystyle{g{{\left({x}\right)}}}={\left({\frac{{{x}^{{{2}}}}}{{{2}}}}\ +\ {x}\ +\ {1}\right)}\) Step 3 The two functions look very much similar near \(\displaystyle{x}={0}.\) Both functions increase for large positive values of x, but \(\displaystyle{e}^{{{x}}}\) increases quickly compared to quadratic function. For negative values of x, the quadratic goes to infinity while the exponential goes to 0. The exponential function is and increasing function while quadratic function is initially decreasing till \(\displaystyle{x}=\ -{1}\) and then increasing function.

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Find the quadratic polynomial g(x)-ax^{2} + bx + c text{which best fits the function} f(x)=e^{x} text{at} x=0, text{in the sense that} g(0)=f(0), text{and} g'(0)=f'(0), text{and} g''(0)=f''(0). Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?

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