/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q54E In Example \(7\) we saw that \(y... [FREE SOLUTION] | 91影视

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Chapter 8: Q54E (page 356)

In Example \(7\) we saw that \(y = {\phi _1}(x) = \sqrt {25 - {x^2}} \) and \(y = {\phi _2}(x) = - \sqrt {25 - {x^2}} \) are solutions of \(dy/dx = - x/y\) on the interval \(( - 5,5)\). Explain why the piecewise-defined function \(y = \left\{ {\begin{array}{*{20}{c}}{\sqrt {25 - {x^2}} }&{ - 5 < x < 0}\\{ - \sqrt {25 - {x^2}} ,}&{0 \le x < 5}\end{array}} \right.\) is not a solution of the differential equation on the interval \(( - 5,5)\).

Short Answer

Expert verified

The piecewise-defined function is not a solution of the differential equation because of the discontinuity.

Step by step solution

01

Determine the left-hand limit of the function.

To check the continuity of the function at\(x = 0\), find the limits of the function.

The left-hand limit of the function is given by,

\(\begin{array}{c}\mathop {lim}\limits_{x \to {0^ - }} y = \mathop {lim}\limits_{x \to {0^ - }} \sqrt {25 - {0^2}} \\ = \sqrt {25} \\ = 5\end{array}\)

02

Determine the right-limit of the function.

The right-hand limit of the function is given by,

\(\begin{array}{c}\mathop {lim}\limits_{x \to {0^ + }} y = \mathop {lim}\limits_{x \to {0^ + }} - \sqrt {25 - {0^2}} \\ = - \sqrt {25} \\ = - 5\end{array}\)

Thus, the left-hand limit is not equal to the right-hand limit, so the function is not continuous at \(x = 0\). Then, the solution is not differential at \(x = 0\).

Hence, the piecewise-defined function is not a solution of the differential equation on the interval \(( - 5,5)\).

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Most popular questions from this chapter

For which of the given P-values would the null hypothesis be rejected when performing a level .05 test?

a. .001 b. .021 c. .078

d..047 e. .148

In Problems \(19\) and \(20\) verify that the indicated expression is an implicit solution of the given first-order differential equation. Find atleast one explicit solution \(y = \phi (x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi \).

Pairs of P-values and significance levels, 伪, are given.

For each pair, state whether the observed P-value would lead to rejection of H0 at the given significance level.

a.P颅-value = .084, 伪= .05

b.P颅-value = .003, 伪= .001

c.P-颅value = .498, 伪= .05

d.P-颅value = .084, 伪= .10

e.P-颅value = .039, 伪= .01

f.P-颅value = .218, 伪 = .10

Reconsider the paint-drying situation of Example 8.5, in which drying time for a test specimen is normally distributed with 蟽 = 9. The hypotheses H0: 碌 =75 versus Ha: 碌 <75 are to be tested using a random sample of n= 25 observations.

a.How many standard deviations (of X) below the null value is \(\overline x = 72.3\)?

b.If \(\overline x = 72.3\), what is the conclusion using 伪 =.002?

c.For the test procedure with 伪 =.002, what is 尾(70)?

d.If the test procedure with 伪 =.002 is used, what n is necessary to ensure that 尾(70) = .01?

e.If a level .01 test is used with n5 100, what is the probability of a type I error when m5 76?Answer the following questions for the tire problem in Example 8.7.

a.If \(\overline x = 30,960\) 30,960 and a level 伪=.01 test is used, what is the decision?

b.If a level .01 test is used, what is 尾(30,500)?

c.If a level .01 test is used and it is also required that 尾(30,500) = .05, what sample size n is necessary?

d.If \(\overline x = 30,960\), what is the smallest 伪 at which H0 can be rejected (based on n = 16)?

The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. The article 鈥淣utrient Intakes and Dietary Patterns of Older Americans: A National Study鈥� (J. of Gerontology, 1992: M145鈥�150) reports the following summary data on intake for a sample of males age 65鈥�74 years: n = 115, \(\overline x = 11.3\), and s = 6.43. Does this data indicate that average daily zinc intake in the population of all males ages 65鈥�74 falls below the recommended allowance?
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