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scZoUnD [109]
2 years ago
6

Use the definition of continuity to determine whether f is continuous at a.

Mathematics
1 answer:
dmitriy555 [2]2 years ago
5 0
f(x) will be continuous at x=a=7 if
(i) \displaystyle\lim_{x\to7}f(x) exists,
(ii) f(7) exists, and
(iii) \displaystyle\lim_{x\to7}f(x)=f(7).

The second condition is immediate, since f(7)=8918 has a finite value. The other two conditions can be established by proving that the limit of the function as x\to7 is indeed the value of f(7). That is, we must prove that for any \varepsilon>0, we can find \delta>0 such that

|x-7|

Now,


|f(x)-f(7)|=|5x^4-9x^3+x-8925|

Notice that when x=7, we have 5x^4-9x^3+x-8925=0. By the polynomial remainder theorem, we know that x-7 is then a factor of this polynomial. Indeed, we can write

|5x^4-9x^3+x-8925|=|(x-7)(5x^3+26x^2+182x+1275)|=|x-7||5x^3+26x^2+182x+1275|

This is the quantity that we do not want exceeding \varepsilon. Suppose we focus our attention on small values \delta. For instance, say we restrict \delta to be no larger than 1, i.e. \delta\le1. Under this condition, we have

|x-7|

Now, by the triangle inequality,


|5x^3+26x^2+182x+1275|\le|5x^3|+|26x^2|+|182x|+|1275|=5|x|^3+26|x|^2+182|x|+1275

If |x|, then this quantity is moreover bounded such that

|5x^3+26x^2+182x+1275|\le5\cdot8^3+26\cdot8^2+182\cdot8+1275=6955

To recap, fixing \delta\le1 would force |x|, which makes


|x-7||5x^3+26x^2+182x+1275|

and we want this quantity to be smaller than \varepsilon, so


6955|x-7|

which suggests that we could set \delta=\dfrac{\varepsilon}{6955}. But if \varepsilon is given such that the above inequality fails for \delta=\dfrac{\varepsilon}{6955}, then we can always fall back on \delta=1, for which we know the inequality will hold. Therefore, we should ultimately choose the smaller of the two, i.e. set \delta=\min\left\{1,\dfrac{\varepsilon}{6955}\right\}.

You would just need to formalize this proof to complete it, but you have all the groundwork laid out above. At any rate, you would end up proving the limit above, and ultimately establish that f(x) is indeed continuous at x=7.
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Find the size of the angle <br> 3<br> x<br> .
AlexFokin [52]

Answer:

Given - Two angles measures 2x and 3x

To find - size of 3x

Solution -

2x + 3x = 360° ( Make reflex angle )

5x = 360°

x = 360/5

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3x = 3 * 72 = 216°

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2 years ago
A bale of fabric is 189 metres long. If lengths of 15 metres, 36 metres and 29 metres are cut off, how much fabric is left?.
Xelga [282]

Answer:

109 metres

Step-by-step explanation:

1. Do 15+36+29 which equals 80

2. Do 189-80 and u get 109

Hope this helps :3

5 0
2 years ago
g natasha is in a class of 30 students that selects 4 leaders. How many ways are there to select the 4 leaders so that natasha i
Papessa [141]

Answer:

<h2>3,654 different ways.</h2>

Step-by-step explanation:

If there are 30 students in a class with natasha in the class and natasha is to select four leaders in the class of which she is already part of the selection, this means there are 3 more leaders needed to be selected among the remaining 29 students (natasha being an exception).

Using the combination formula since we are selecting and combination has to do with selection, If r object are to selected from n pool of objects, this can be done in nCr number of ways.

nCr = n!/(n-r)!r!

Sinca natasha is to select 3 more leaders from the remaining 29students, this can be done in 29C3 number of ways.

29C3 = 29!/(29-3)!3!

29C3 = 29!/(26!)!3!

29C3 = 29*28*27*26!/26!3*2

29C3 = 29*28*27/6

29C3 = 3,654 different ways.

This means that there are 3,654 different ways to select the 4 leaders so that natasha is one of the leaders

5 0
2 years ago
What are the real zeros of x^3+4x^2-9x-36
Katyanochek1 [597]

Answer:

x = -4\\x = -3\\x = 3

Step-by-step explanation:

To find the real zeros you must match the function to zero and factor the expression.

We have a polynomial of degree 3.

We try to group the terms to perform the factorization

x ^ 3 + 4x ^ 2-9x-36 = 0\\\\x ^ 2(x + 4) - 9(x + 4) = 0

Now we take (x + 4) as a common factor

(x + 4)(x ^ 2 -9) = 0

If we have an expression of the form (a ^ 2-b ^ 2) we know that this expression is equivalent to:

(a ^ 2-b ^ 2) = (a + b)(a-b)

In this case

a = x\\b = 3

So:

(x ^ 2 -9) = (x + 3)(x-3)

Finally:

x ^ 3 + 4x ^ 2-9x-36 = (x + 4)(x + 3)(x-3) = 0

The solutions are:

x = -4\\x = -3\\x = 3

6 0
3 years ago
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