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3 years ago

Use your calculator to work out this equation

Mathematics

2 answers:

alisha [4.7K]3 years ago

4 0

Answer:

2.123842852 is the first answer

for part b it'll be 2.12

hope that helps

Step-by-step explanation:

Charra [1.4K]3 years ago

4 0

Answer:

a) 2.123842862

b) 2.12

Step-by-step explanation:

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Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n.

Vera_Pavlovna [14]

Split up the integration interval into 4 subintervals:

$\left[0,\dfrac\pi8\right],\left[\dfrac\pi8,\dfrac\pi4\right],\left[\dfrac\pi4,\dfrac{3\pi}8\right],\left[\dfrac{3\pi}8,\dfrac\pi2\right]$

The left and right endpoints of the $i$ -th subinterval, respectively, are

$\ell_i=\dfrac{i-1}4\left(\dfrac\pi2-0\right)=\dfrac{(i-1)\pi}8$

$r_i=\dfrac i4\left(\dfrac\pi2-0\right)=\dfrac{i\pi}8$

for $1\le i\le4$ , and the respective midpoints are

$m_i=\dfrac{\ell_i+r_i}2=\dfrac{(2i-1)\pi}8$

Trapezoidal rule

We approximate the (signed) area under the curve over each subinterval by

$T_i=\dfrac{f(\ell_i)+f(r_i)}2(\ell_i-r_i)$

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4T_i\approx\boxed{3.038078}$

Midpoint rule

We approximate the area for each subinterval by

$M_i=f(m_i)(\ell_i-r_i)$

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4M_i\approx\boxed{2.981137}$

Simpson's rule

We first interpolate the integrand over each subinterval by a quadratic polynomial $p_i(x)$ , where

$p_i(x)=f(\ell_i)\dfrac{(x-m_i)(x-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+f(m)\dfrac{(x-\ell_i)(x-r_i)}{(m_i-\ell_i)(m_i-r_i)}+f(r_i)\dfrac{(x-\ell_i)(x-m_i)}{(r_i-\ell_i)(r_i-m_i)}$

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx$

It so happens that the integral of $p_i(x)$ reduces nicely to the form you're probably more familiar with,

$S_i=\displaystyle\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx=\frac{r_i-\ell_i}6(f(\ell_i)+4f(m_i)+f(r_i))$

Then the integral is approximately

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4S_i\approx\boxed{3.000117}$

Compare these to the actual value of the integral, 3. I've included plots of the approximations below.

3 0

3 years ago

What are these < > =???????????

madam [21]

These Are Less Than(<) and Greater Than(>) Symbols!!

Hope This Helps!!!

7 0

3 years ago

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Is Happyville linear or exponential

nordsb [41]

Answer:

expoential

Step-by-step explanation:

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4 years ago

Which expressions are equivalent to the one below? Check all that apply.

Shalnov [3]

Answer:

The answer would be A. 3.3x

Step-by-step explanation:

3.3x can be broken down as 3.3.x

3.3 equals 9 and the 'x' is multiplied to 9.

Hence, it equals to 9x.

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3 years ago

A. Is this mapping diagram a function? Why or why not?

nikitadnepr [17]

Answer:

40--C A ---is not[figure it out in your own words,your teacher might suspect something] and B---times 40

Step-by-step explanation:

b--40+40=80 80+40=120 and 120+40=160 A=subtract 40 from all the other answers such as 40-160=120 40-120=80 and 40-80=40

5 0

3 years ago