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timurjin [86]
3 years ago
7

How do you know where to put the constant when finding general solutions for differential equations?

Mathematics
1 answer:
Norma-Jean [14]3 years ago
5 0
Let's suppose we want to solve y'=y with y(0)=2. Separating variables and integrating, we get

\displaystyle\int\dfrac{\mathrm dy}y=\int\mathrm dx\implies\ln|y|=x+C\implies y=e^{x+C}

Leaving the solution in this form, the initial condition gives

2=e^{0+C}=e^C\implies C=\ln2

This means the solution is y=e^{x+\ln2}.

Now if we were to write y=e^{x+C}=e^xe^C=Ce^x, then we would have found

2=Ce^0\implies C=2

so that the solution would have been y=2e^x.

But these two solutions are the same, since y=e^{x+\ln2}=e^xe^{\ln2}=2e^x. So we get the same solution regardless of where we place C, despite getting different values for C.
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Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body b
kramer

Answer:

(a)

\bar x = 0.0075

s = 0.0049

(b)

B. The sample is too small to make judgments about skewness or symmetry.

Step-by-step explanation:

Given:

n = 8

\begin{array}{ccccccccc}{} & {S} & {u} & {b} &{j} & {e} & {c} & {t} & {s} &{Operator} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {1} & 1.326 & 1.337 & 1.079 & 1.229 & 0.936 & 1.009 & 1.179 & 1.289 & 2 & 1.323 & 1.322 & 1.073 & 1.233 & 0.934 & 1.019 & 1.184 & 1.304 \ \end{array}

Solving (a):

First, calculate the difference between the recorded TBBMC for both operators:

\begin{array}{ccccccccc}{} & {S} & {u} & {b} &{j} & {e} & {c} & {t} & {s} &{Operator} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {1} & 1.326 & 1.337 & 1.079 & 1.229 & 0.936 & 1.009 & 1.179 & 1.289 & 2 & 1.323 & 1.322 & 1.073 & 1.233 & 0.934 & 1.019 & 1.184 & 1.304 &{|1 - 2|} &0.003 & 0.015 & 0.006 & 0.004 & 0.002 & 0.010 & 0.005 & 0.015 \ \end{array}

The last row which represents the difference between 1 and 2 is calculated using absolute values. So, no negative entry is recorded.

The mean is then calculated as:

\bar x = \frac{\sum x}{n}

\bar x = \frac{0.003 + 0.015 + 0.006 + 0.004 + 0.002 + 0.010 + 0.005 + 0.015}{8}

\bar x = \frac{0.06}{8}

\bar x = 0.0075

Next, calculate the standard deviation (s).

This is calculated using:

s = \sqrt{\frac{\sum (x - \bar x)^2}{n}}

So, we have

s = \sqrt\frac{(0.003 - 0.0075)^2 + (0.015 - 0.0075)^2+ (0.006- 0.0075)^2+ ....... + (0.005- 0.0075)^2+ (0.015- 0.0075)^2 }{8}s = \sqrt\frac{0.00019}{8}

s = \sqrt{0.00002375

s = 0.0049

Solving (b):

Of the given options (A - E), option B is correct because the sample is actually too small

6 0
3 years ago
What is the answer please.... need help
mina [271]

Answer:

f\left(x\right)=x^3-6x^2+3x+10 is the function of the least degree has the real coefficients and the leading coefficients of 1 and with the zeros -1, 5, and 2.

Step-by-step explanation:

Given the function

f\left(x\right)=x^3-6x^2+3x+10

As the highest power of the x-variable is 3 with the leading coefficients of 1.

  • So, it is clear that the polynomial function of the least degree has the real coefficients and the leading coefficients of 1.

solving to get the zeros

f\left(x\right)=x^3-6x^2+3x+10

0=x^3-6x^2+3x+10              ∵  f(x)=0

as

Factor\:x^3-6x^2+3x+10\::\:\left(x+1\right)\left(x-2\right)\left(x-5\right)=0

so

\left(x+1\right)\left(x-2\right)\left(x-5\right)=0    

Using the zero factor principle

if  ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)

x+1=0\quad \mathrm{or}\quad \:x-2=0\quad \mathrm{or}\quad \:x-5=0

x=-1,\:x=2,\:x=5

Therefore, the zeros of the function are:

x=-1,\:x=2,\:x=5

f\left(x\right)=x^3-6x^2+3x+10 is the function of the least degree has the real coefficients and the leading coefficients of 1 and with the zeros -1, 5, and 2.

Therefore, the last option is true.    

8 0
3 years ago
The test statistic of z= -2.39 is obtained when testing the claim that p<0.52.
krek1111 [17]

Considering the situation described, we have that:

a) The critical value is of z = -1.645.

b) Since the test statistic is less than the critical value, we should reject the null hypothesis H0.

<h3>What is the critical value?</h3>

We have a left-tailed test, as we are testing if the proportion is less than a value. Hence the critical value is z with a p-value equals to the significance level, hence z with a p-value of 0.05, hence z = -1.645.

<h3>What is the decision?</h3>

Considering the test statistic, for a left-tailed test, we have that:

  • Less than the critical value: Reject H0.
  • Equal or greater: Do not reject.

In this problem, z = -2.39 is less than -1.645, hence we should reject the null hypothesis H0.

More can be learned about the test of an hypothesis at brainly.com/question/13873630

#SPJ1

6 0
2 years ago
Line parallel to y=1/2x-5
Sveta_85 [38]

Answer:

1/2

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
This one I can’t figure out! It has fractions too..
viva [34]

Answer:

3x+ 5 1/3

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
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