Answer:
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.
Step-by-step explanation:
Answer:
81
Step-by-step explanation:
go hgv thx bgg TV vvt g h.c g ccx d
Answer: y=2x+5
Slope is 2
Step-by-step explanation: y=2(x+1)+3
y=2x+2+3
y=2x+5
For the first question, the region is a bit ambiguous.
![x](https://tex.z-dn.net/?f=x)
and
![x^3](https://tex.z-dn.net/?f=x%5E3)
intersect three times, and there are two regions between them. So either you're approximating
![\displaystyle\int_{-1}^1|x^3-x|\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B-1%7D%5E1%7Cx%5E3-x%7C%5C%2C%5Cmathrm%20dx)
or
![\displaystyle\int_0^1(x-x^3)\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E1%28x-x%5E3%29%5C%2C%5Cmathrm%20dx)
I'll assume the second case. Split the interval into 4 smaller ones, taking
![(0,1)=\left(0,\dfrac14\right)\cup\left(\dfrac14,\dfrac12\right)\cup\left(\dfrac12,\dfrac34\right)\cup\left(\dfrac34,1\right)](https://tex.z-dn.net/?f=%280%2C1%29%3D%5Cleft%280%2C%5Cdfrac14%5Cright%29%5Ccup%5Cleft%28%5Cdfrac14%2C%5Cdfrac12%5Cright%29%5Ccup%5Cleft%28%5Cdfrac12%2C%5Cdfrac34%5Cright%29%5Ccup%5Cleft%28%5Cdfrac34%2C1%5Cright%29)
with respective midpoints of
![\dfrac18,\dfrac38,\dfrac58,\dfrac78](https://tex.z-dn.net/?f=%5Cdfrac18%2C%5Cdfrac38%2C%5Cdfrac58%2C%5Cdfrac78)
. The length of each interval is
![\dfrac14](https://tex.z-dn.net/?f=%5Cdfrac14)
. Note that I'm also assuming you are supposed to use equally spaced intervals.
![\displaystyle\int_0^1(x-x^3)\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E1%28x-x%5E3%29%5C%2C%5Cmathrm%20dx)
![\approx\dfrac{\frac18-\left(\frac18\left)^3}4+\dfrac{\frac38-\left(\frac38\left)^3}4+\dfrac{\frac58-\left(\frac58\left)^3}4+\dfrac{\frac78-\left(\frac78\left)^3}4=\dfrac{33}{128}](https://tex.z-dn.net/?f=%5Capprox%5Cdfrac%7B%5Cfrac18-%5Cleft%28%5Cfrac18%5Cleft%29%5E3%7D4%2B%5Cdfrac%7B%5Cfrac38-%5Cleft%28%5Cfrac38%5Cleft%29%5E3%7D4%2B%5Cdfrac%7B%5Cfrac58-%5Cleft%28%5Cfrac58%5Cleft%29%5E3%7D4%2B%5Cdfrac%7B%5Cfrac78-%5Cleft%28%5Cfrac78%5Cleft%29%5E3%7D4%3D%5Cdfrac%7B33%7D%7B128%7D)
Skipping the second one since I already answered it.
For the third, split up the region of integration at some arbitrary constant
![c](https://tex.z-dn.net/?f=c)
between
![2x](https://tex.z-dn.net/?f=2x)
and
![5x](https://tex.z-dn.net/?f=5x)
, then differentiate and apply the fundamental theorem of calculus.
![F(x)=\displaystyle\int_{2x}^{5x}\frac{\mathrm dt}t](https://tex.z-dn.net/?f=F%28x%29%3D%5Cdisplaystyle%5Cint_%7B2x%7D%5E%7B5x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt)
![F(x)=\displaystyle\int_c^{5x}\frac{\mathrm dt}t+\int_{2x}^c\frac{\mathrm dt}t](https://tex.z-dn.net/?f=F%28x%29%3D%5Cdisplaystyle%5Cint_c%5E%7B5x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt%2B%5Cint_%7B2x%7D%5Ec%5Cfrac%7B%5Cmathrm%20dt%7Dt)
![F(x)=\displaystyle\int_c^{5x}\frac{\mathrm dt}t-\int_c^{2x}\frac{\mathrm dt}t](https://tex.z-dn.net/?f=F%28x%29%3D%5Cdisplaystyle%5Cint_c%5E%7B5x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt-%5Cint_c%5E%7B2x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt)
![F'(x)=\dfrac1{5x}\cdot\dfrac{\mathrm d(5x)}{\mathrm dx}-\dfrac1{2x}\cdot\dfrac{\mathrm d(2x)}{\mathrm dx}](https://tex.z-dn.net/?f=F%27%28x%29%3D%5Cdfrac1%7B5x%7D%5Ccdot%5Cdfrac%7B%5Cmathrm%20d%285x%29%7D%7B%5Cmathrm%20dx%7D-%5Cdfrac1%7B2x%7D%5Ccdot%5Cdfrac%7B%5Cmathrm%20d%282x%29%7D%7B%5Cmathrm%20dx%7D)
![F'(x)=\dfrac5{5x}-\dfrac2{2x}](https://tex.z-dn.net/?f=F%27%28x%29%3D%5Cdfrac5%7B5x%7D-%5Cdfrac2%7B2x%7D)
![F'(x)=\dfrac1x-\dfrac1x](https://tex.z-dn.net/?f=F%27%28x%29%3D%5Cdfrac1x-%5Cdfrac1x)
![F'(x)=0](https://tex.z-dn.net/?f=F%27%28x%29%3D0)
Since
![F'(x)=0](https://tex.z-dn.net/?f=F%27%28x%29%3D0)
, it follows that
![F(x)](https://tex.z-dn.net/?f=F%28x%29)
is constant.
32m = 8m · 4
56mp = 8m · 7p
32m + 56mp = 8m(4 + 7p)