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s2008m [1.1K]
3 years ago
12

Find the linear approximation of the function g(x) = 3 1 + x at a = 0. g(x) ≈ Correct: Your answer is correct. Use it to approxi

mate the numbers 3 0.95 and 3 1.1 . (Round your answers to three decimal places.)
Mathematics
1 answer:
EleoNora [17]3 years ago
7 0

Answer:

3.296x+3

2.835

3.330

Step-by-step explanation:

g(x) = 3^{1+x}

Let the linear approximation be L(x) at a = 0. This is given by

L(x) = g(a) + g'(a)(x-a)

g'(x) is the derivative of g(x). To find this, we use the form

If f(x) = a^x, then f'(x) =a^x\ln a

By doing this and applying chain rule for the power (which is a function of x), we have

g'(x) = 3^{1+x}\ln 3

Then

g'(0) = 3^{1+0}\ln 3 = 3.296

Also g(0) = 3^{1+0} = 3

Hence L(x) = 3+(x-0)\times3.296 = 3.296x + 3

For 3^{0.95}, 1+x = 0.95 and x=-0.05

Using this in L(x),

3^{0.95} = 3.296(-0.05) + 3 = 2.835

For 3^{1.1}, 1+x = 1.1 and x=0.1

Using this in L(x),

3^{1.1} = 3.296(0.1) + 3 = 3.330

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3 years ago
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ElenaW [278]

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2 years ago
Suppose that you need to create a list of n values that have a specific known mean. Some of the n values can be freely selected.
Leno4ka [110]

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Assume that the mean sought is x and consider that the list is composed of a single element a, in that case no random number can be selected, since the mean x must correspond to that number.

If the list were composed of two elements a and b, one of the two values ​​could be chosen randomly, and according to the chosen value the second should be the one whose sum with the previous one results in 2x, this given that the formula of the average \sum\limits_{i=1}^n \frac{ x_{i}}{n}.

With three values ​​a, b and c, it is possible to select two freely, since the thirteen must be the one that balances the sum of a+b, that is (a + b) + c = 3x.

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Answer

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3 years ago
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