# Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. y=x^{3}-3x^{2},[-2,5] by [-10,10]

Question
Polynomial graphs
Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. $$\displaystyle{y}={x}^{{{3}}}-{3}{x}^{{{2}}},{\left[-{2},{5}\right]}{b}{y}{\left[-{10},{10}\right]}$$

2021-02-21
Step 1
$$\displaystyle{y}={x}^{{{3}}}-{3}{x}^{{{2}}},{\left[-{2},{5}\right]}\times{\left[-{10},{10}\right]}$$
The graph of this function in the indicated viewing window is shown in the following picture.
$$\displaystyle{h}{\mathtt{{p}}}{s}:{/}{q}{2}{a}.{s}{3}-{u}{s}-{w}{e}{s}{t}-{1}.{a}{m}{a}{z}{o}{n}{a}{w}{s}.{c}{o}\frac{{m}}{{d}}{e}\frac{{v}}{{19610300971}}.{j}{p}{g}$$
Step 2
As we can see in this picture this function has:
A local maximum that occurs at the point (0, 0).
A local minimum that occurs at the point (2,-4).
For the domain:
This is a polynomial function so it is defined for all values of x in the set of real numbers.
The domain is: $$\displaystyle{D}={\left(-\infty,+\infty\right)}$$
For the range:
This function can get all values in the set of real numbers (vertical axis) so:
The range is: $$\displaystyle{R}={\left(-\infty,+\infty\right)}$$

### Relevant Questions

Find all zeros of p(x), real or imaginary. $$p(x) = x^{4} + 6x^{3} + 6x^{2} -18x -27$$ List all of the possible rational zeros according to the rational zero theorem and state the values for C, A, B and D in the following partial factorization of $$p(x) = (x-c)(x^{3}+Ax^{2}+Bx+D)$$ State the exact answer and a decimal approximation of each zero to the tenths place

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. 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 known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Express irrational solutions in exact form and as a decimal rounded to three decimal places.
$$\log_{2}(3x+2)-\log_{4}x=3$$
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.
For each polynomial function, one zero is given. Find all rational zeros and factor the polynomial. Then graph the function. f(x)=3x^{3}+x^{2}-10x-8ZSK, zero:2
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 boy is to sell lemonade to make some money to afford some holiday shopping. The capacity of the lemonade bucket is 1. At the start of each day, the amount of lemonade in the bucket is a random variable X, from which a random variable Y is sold during the day. The two random variables X and Y are jointly uniform.
Find and sketch the CDF and the pdf of 'Z' which is the amount of lemonade remaining at the end of the day. Clearly indicate the range of Z
Using calculus, it can be shown that the arctangent function can be approximated by the polynomial
$$\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}$$
a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
b) Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part (a). How does the accuracy of the approximation change when an additional term is added?
In calculus, it can be shown that the arctangent function can be approximated by the polynomial
$$\displaystyle{\arctan{{x}}}\approx{x}-\frac{{x}^{{{3}}}}{{3}}+\frac{{x}^{{{5}}}}{{5}}-\frac{{x}^{{{7}}}}{{7}}$$
where x is in radians. Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
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.