# Answer this question and show the steps. sqrt{x}+sqrt{y}=3, find the value of frac{dy}{dx} at the point (4,1)

Question
Decimals
Answer this question and show the steps. $$\displaystyle\sqrt{{{x}}}+\sqrt{{{y}}}={3}$$, find the value of $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}$$ at the point (4,1)

2020-12-25
Step 1 Consider the given function $$\displaystyle\sqrt{{{x}}}+\sqrt{{{y}}}={3}$$ Step 2 Differentiate implicitly with respect to x. $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(\sqrt{{{x}}}+\sqrt{{{y}}}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({3}\right)}$$
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{\frac{{{1}}}{{{2}}}}}}\right)}+{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({y}^{{{\frac{{{1}}}{{{2}}}}}}\right)}={0}$$
$$\displaystyle{\frac{{{1}}}{{{2}}}}{x}^{{{\frac{{{1}}}{{{2}}}}-{1}}}+{\left({\frac{{{1}}}{{{2}}}}{y}^{{{\frac{{{1}}}{{{2}}}}-{1}}}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0}$$
$$\displaystyle{\frac{{{1}}}{{{2}}}}{x}^{{-{\frac{{{1}}}{{{2}}}}}}+{\left({\frac{{{1}}}{{{2}}}}{y}^{{-{\frac{{{1}}}{{{2}}}}}}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0}$$
$$\displaystyle{\frac{{{1}}}{{{2}\sqrt{{{x}}}}}}+{\frac{{{1}}}{{{2}\sqrt{{{y}}}}}}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0}$$
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}=-{\frac{{\sqrt{{{y}}}}}{{\sqrt{{{x}}}}}}=-\sqrt{{{\frac{{{y}}}{{{x}}}}}}$$ Step 3 Now, find the value of $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}$$ at the point (4, 1) $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}{\left({x},{y}\right)}={\left({4},{1}\right)}=-\sqrt{{{\frac{{{1}}}{{{4}}}}}}=-{\frac{{{1}}}{{{2}}}}$$ The answer is $$\displaystyle-{\frac{{{1}}}{{{2}}}}$$ or -0.5(in decimals).

### Relevant Questions

Marks will be awarded for accuracy in the rounding of final answers where you are asked to round. To ensure that you receive these marks, take care in keeping more decimals in your intermediate steps than what the question is asking you to round your final answer to.
A fair 7 -sided die with the numbers 1 trough 7 is rolled five times. Express each of your answers as a decimal rounded to 3 decimal places.
(a) What is the probability that exactly one 3 is rolled?
(b) What is the probability that at least one 3 is rolled?
(c) What is the probability that exactly four of the rolls show an even number?
Consider the quantity$$a^{2}\ -\ b^{2}$$ where a and b are real numbers.
(a) Under what conditions should one expect an unusually large relative error in the computed value of $$a^{2}\ -\ b^{2}$$ when this expression is evaluated in finite precision arithmetic?
(b)cWs 4-digit (decimal) rounding arithmetic to evaluate both $$a^{2}\ -\ b^{2}\ and\ (a\ +\ b)(a\ -\ b)\ with\ a\ = 995.1\ and\ b = 995.0.$$ Calculate th relative error in each result.
(c) The expression $$(a\ +\ b)(a\ -\ b)\ is\ algebraically\ equivalent\ to\ a^{2}\ -\ b^{2},$$ but it is a more accurate way to calculate this quantity if both a and b have exact floating point representations. Why?
factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 70 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H_{0}:\sigma=60,\ H_{1}:\sigma\ <\ 60H_{0}:\sigma\ >\ 60,\ H_{1}:\sigma=60H_{0}:\sigma=60,\ H_{1}:\sigma\ >\ 60H_{0}:\sigma=60,\ H_{1}:\sigma\ \neq\ 60$$
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original distribution?
We assume a binomial population distribution.We assume a exponential population distribution. We assume a normal population distribution.We assume a uniform population distribution.
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
To check: Whether the set of numbers $$\displaystyle{\left\lbrace\sqrt{{{3}}},\pi,{\frac{{\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}}}{{{4}}}},\sqrt{{{5}}}\right\rbrace}$$ contains integers, rational numbers, and (or) irrational numbers.
Solve $$2t\ -\ 3(t\ +\ 8) = -(1\ -\ 4t)\ + 10$$ for t. Simplfy all fractions or round decimals to 2 places. Show all steps.
For the given fraction and decimals we have to write its equivalent percent. Given fractions are $$\displaystyle{a}{)}{\frac{{{3}}}{{{25}}}}{b}{)}{\frac{{{1}}}{{{5}}}}{c}{)}{\frac{{{2}}}{{{5}}}}$$ And the decimals are, $$\displaystyle{d}{)}{0.01},{e}{)}{4.06},{f}{)}{0.6}$$ We have to find its equivalent percent.
Find the solution of the equation rounded to two decimals. 1) $$\displaystyle{3.02}{x}+{1.48}={10.92}$$ 2) $$\displaystyle{8.36}-{0.95}{x}={9.97}$$ 3) $$\displaystyle{2.15}{x}-{4.63}={x}+{1.19}$$