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mojhsa [17]
3 years ago
6

Evaluate the upper and lower sums for f(x) = 1 + x2, −1 ≤ x ≤ 1, with n = 3 and 4.

Mathematics
1 answer:
Elenna [48]3 years ago
7 0
I'm going to assume that your function is f(x) = 1 + x^2    (NOT x2).

I suspect you're trying to estimate the "area under the curve of f(x) = 1 + x^2.  You need to use this or a similar description to explain what you're doing.

Also, you need to specify whether you want "left end points" or "right end points" or "midpoints."  Again I must assume you want one or the other (and will assume that you meant "left end points").  

First, let's address the case n=3.  You must graph f(x) = 1 + x^2 between -1 and +1.     We will find the "lower sum," using "left end points."  The 3 x-values are {-1, -1/3, 1/3}.  Evaluate the function f(x) = 1 + x^2 at these 3 x-values.  Keep in mind that the interval width is 2/3.
The function (y) values are {0, 2/3, 4/3}.

Sorry, Michael, but I must stop here and await clarification from you regarding what you've been told to do in this problem.  Otherwise too much guessing (regarding what you meant) is necessary.  Please review the original problem and ensure that you have copied it exactly as presented, and also please verify whether this problem does indeed involve estimating areas under curves between starting and ending x-values.
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Use the "rule of 72" to estimate the doubling time (in years) for the interest rate, and then calculate it exactly. (Round your
Law Incorporation [45]

Answer:

According to the rule of 72, the doubling time for this interest rate is 8 years.

The exact doubling time of this amount is 8.04 years.

Step-by-step explanation:

Sometimes, the compound interest formula is quite complex to be solved, so the result can be estimated by the rule of 72.

By the rule of 72, we have that the doubling time D is given by:

D = \frac{72}{Interest Rate}

The interest rate is in %.

In our exercise, the interest rate is 9%. So, by the rule of 72:

D = \frac{72}{9} = 8.

According to the rule of 72, the doubling time for this interest rate is 8 years.

Exact answer:

The exact answer is going to be found using the compound interest formula.

A = P(1 + \frac{r}{n})^{nt}

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this exercise, we have:

We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.

is double the initial amount, double the principal.

A = 2P

r = 0.09

The interest is compounded anually, so n = 1

A = P(1 + \frac{r}{n})^{nt}

2P = P(1 + \frac{0.09}{1})^{t}

2 = (1.09)^{t}

Now, we apply the following log propriety:

\log_{a} a^{n} = n

So:

\log_{1.09}(1.09)^{t} = \log_{1.09} 2

t = 8.04

The exact doubling time of this amount is 8.04 years.

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5 0
2 years ago
Write the number that is 10 less than 526
Leona [35]
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So, 526 - 10 = 516

Hope this helped :)
6 0
3 years ago
Read 2 more answers
Find the perimeter of the quarter-circle. Give your answer to the nearest whole number.
Bezzdna [24]

Answer:

The perimeter of the quarter-circle is 200 cm.

Step-by-step explanation:

Given that the angle of quarter-circle is 90°. So using Length of Arc formula, you are able to find the length of curve :

arc =  \frac{θ}{360}  \times 2 \: \pi \: r

Let r be radius = 56 cm,

Let θ = 90°

arc =  \frac{90}{360}  \times 2 \times\pi \times 56

=  \frac{1}{4}  \times 2 \times \pi \times 56

=  \frac{1}{2}  \times \pi \times 56

= 28\pi \: cm

We have already found the arc. Next we have to add up together to find the perimeter :

p = 56 + 56 + 28\pi

p = 112 + 28\pi

p = 200 \: cm \: (3s.f)

6 0
3 years ago
Read 2 more answers
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