Answer: D 2/3
Step-by-step explanation:
Answer:
a) So, this integral is convergent.
b) So, this integral is divergent.
c) So, this integral is divergent.
Step-by-step explanation:
We calculate the next integrals:
a)
![\int_1^{\infty} e^{-2x} dx=\left[-\frac{e^{-2x}}{2}\right]_1^{\infty}\\\\\int_1^{\infty} e^{-2x} dx=-\frac{e^{-\infty}}{2}+\frac{e^{-2}}{2}\\\\\int_1^{\infty} e^{-2x} dx=\frac{e^{-2}}{2}\\](https://tex.z-dn.net/?f=%5Cint_1%5E%7B%5Cinfty%7D%20e%5E%7B-2x%7D%20dx%3D%5Cleft%5B-%5Cfrac%7Be%5E%7B-2x%7D%7D%7B2%7D%5Cright%5D_1%5E%7B%5Cinfty%7D%5C%5C%5C%5C%5Cint_1%5E%7B%5Cinfty%7D%20e%5E%7B-2x%7D%20dx%3D-%5Cfrac%7Be%5E%7B-%5Cinfty%7D%7D%7B2%7D%2B%5Cfrac%7Be%5E%7B-2%7D%7D%7B2%7D%5C%5C%5C%5C%5Cint_1%5E%7B%5Cinfty%7D%20e%5E%7B-2x%7D%20dx%3D%5Cfrac%7Be%5E%7B-2%7D%7D%7B2%7D%5C%5C)
So, this integral is convergent.
b)
![\int_1^{2}\frac{dz}{(z-1)^2}=\left[-\frac{1}{z-1}\right]_1^2\\\\\int_1^{2}\frac{dz}{(z-1)^2}=-\frac{1}{1-1}+\frac{1}{2-1}\\\\\int_1^{2}\frac{dz}{(z-1)^2}=-\infty\\](https://tex.z-dn.net/?f=%5Cint_1%5E%7B2%7D%5Cfrac%7Bdz%7D%7B%28z-1%29%5E2%7D%3D%5Cleft%5B-%5Cfrac%7B1%7D%7Bz-1%7D%5Cright%5D_1%5E2%5C%5C%5C%5C%5Cint_1%5E%7B2%7D%5Cfrac%7Bdz%7D%7B%28z-1%29%5E2%7D%3D-%5Cfrac%7B1%7D%7B1-1%7D%2B%5Cfrac%7B1%7D%7B2-1%7D%5C%5C%5C%5C%5Cint_1%5E%7B2%7D%5Cfrac%7Bdz%7D%7B%28z-1%29%5E2%7D%3D-%5Cinfty%5C%5C)
So, this integral is divergent.
c)
![\int_1^{\infty} \frac{dx}{\sqrt{x}}=\left[2\sqrt{x}\right]_1^{\infty}\\\\\int_1^{\infty} \frac{dx}{\sqrt{x}}=2\sqrt{\infty}-2\sqrt{1}\\\\\int_1^{\infty} \frac{dx}{\sqrt{x}}=\infty\\](https://tex.z-dn.net/?f=%5Cint_1%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7B%5Csqrt%7Bx%7D%7D%3D%5Cleft%5B2%5Csqrt%7Bx%7D%5Cright%5D_1%5E%7B%5Cinfty%7D%5C%5C%5C%5C%5Cint_1%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7B%5Csqrt%7Bx%7D%7D%3D2%5Csqrt%7B%5Cinfty%7D-2%5Csqrt%7B1%7D%5C%5C%5C%5C%5Cint_1%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7B%5Csqrt%7Bx%7D%7D%3D%5Cinfty%5C%5C)
So, this integral is divergent.
Answer:
Step-by-step explanation:
Parameterize the ellipse as (acos∙,bsin∙). Take points P:=(acosp,bsinp) and Q:=(acosq,bsinq) on the ellipse, with midpoint M:=(P+Q)/2.
If |PQ|=2k, then
a2(cosp−cosq)2+b2(sinp−sinq)2=4k2
The coordinates of M are
xy==a2(cosp+cosq)b2(sinp+sinq)
Answer:
The first blank: not equal
The 2nd blank: not right triangle (it's acute triangle)
The 3rd blank: equal
The 4th blank: it's right triangle
Step-by-step explanation:
4^2 = 3^2 + 3^2
<=> 16 = 9 + 9
<=> 16 = 18 ( nonsense)
so triangle LKJ is not right triangle
5^2 = 4^2 + 3^2
<=> 25 = 16 + 9
<=> 25 = 25 ( make sense)
so triangle ABC is right triangle
Done!! :))
