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Leni [432]
2 years ago
10

prove that if f is integrable on [a,b] and c is an element of [a,b], then changing the value of f at c does not change the fact

that f is integrable or the value of its integral on [a,b]
Mathematics
1 answer:
Neko [114]2 years ago
6 0

Answer with Step-by-step explanation:

We are given that if f is integrable  on [a,b].

c is an element which lie in the interval [a,b]

We have to prove that when we change the value of f at c then the value of f does not change on interval [a,b].

We know that  limit property of an  integral

\int_{a}^{b}f dt=\int_{a}^{c}fdt+\int_{c}^{b} fdt

\int_{a}^{b} fdt=f(b)-f(a)....(Equation I)

Using above property of integral then we get

\int_{a}^{b}fdt=\int_{a}^{c}fdt+\int_{c}^{b} fdt......(Equation II)

Substitute equation I and equation II are equal

Then we get

\int_{a}^{b}fdt= f(c)-f(a)+{f(b)-f(c)}

\int_{a}^{b}fdt=f(c)-f(a)+f(b)-f(c)=f(b)-f(a)

\int_{a}^{c}fdt+\int_{c}^{b}fdt=f(b)-f(a)

Therefore, \int_{a}^{b}fdt=\int_{a}^{c}fdt+\int_{c}^{b}fdt.

Hence, the value of function does not change after changing the value of function at c.

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In the June 2007 issue, Consumer Reports also examined the relative merits of top-loading and front-loading washing machines, te
saveliy_v [14]

Answer:

Yes

Step-by-step explanation:

Given that in the June 2007 issue, Consumer Reports also examined the relative merits of top-loading and front-loading washing machines, testing samples of several different brands of each type.

The difference in mean values test gave a p value of 0.32

Confidence level = 95%

Alpha = 1-0.95 = 0.05

Compare p with alpha, here p >alpha

Hence we accept null hypothesis that there is no difference in the means.

Confidence interval method also will yield the same result.  i.e. confidence interval for difference of means would definitely contain 0 at 95% conf level.

So answer is yes

3 0
2 years ago
For the following pairs of equations explain why the equations are NOT equivalent.
Serga [27]
If you were to make both x's equivalent, you'd have to multiply the first equation by 2 to get rid of the coefficient, 1/2. However, you'd have to multiply 2 onto -8 as well, therefore the equation would turn into x - 16 = 18. That is not equivalent to x - 8 = 18.
7 0
2 years ago
Which equation represents the standard form of the equation y= (x + 3)² - 4?
lilavasa [31]

Answer:

The answer is B. y = x^2 + 6x + 5

Step-by-step explanation:

Because y = (x + 3)^2 - 4 = x^2 + 6x + 9 - 4 = x^2 + 6x + 5

So the answer in this question is B.

4 0
2 years ago
X^2-21.75x=-15.75 what is one solution
Jlenok [28]
Well, there are two solutions.

Add both sides by 15.75 then from here, you can use quadratic formula.

So your answer should be x = -2.44949 or 2.44949.
5 0
3 years ago
Please help: A box contains 5 yellow balls, 3 blue, and 1 red ball. Two balls are drawn at random. Find the probability that:
Y_Kistochka [10]

Answer:

a) 0.2778

b) 0.3611

c) 0.1389

d) 0.0833

Step-by-step explanation:

We have a total of 5 + 3 + 1 = 9 balls

a) First ball being yellow: we have 5 yellow balls, so P1 = 5/9

Second ball being yellow after one yellow was drawn: we have 4 yellows and 8 balls, so P2 = 4/8 = 1/2

Both yellows: P = P1 * P2 = 5/18 = 0.2778

b) Both blues:

P1 = 3/9 = 1/3

P2 = 2/8 = 1/4

P = P1 * P2 = 1/12 = 0.0833

Both yellows or both blues: 5/18 + 1/12 = 0.2778 + 0.0833 = 0.3611

c) First yellow: P1 = 5/9

Second red: P2 = 1/8

Pa = P1 * P2 = 5/72

or

First red: P3 = 1/9

Second yellow: P4 = 5/8

Pb = P3 * P4 = 5/72

P = Pa + Pb = 10/72 = 5/36 = 0.1389

d) First blue: P1 = 3/9 = 1/3

Second red: P2 = 1/8

Pa = P1 * P2 = 1/24

or

First red: P3 = 1/9

Second blue: P4 = 3/8

Pb = P3 * P4 = 1/24

P = Pa + Pb = 2/24 = 1/12 = 0.0833

8 0
3 years ago
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