Answer:
1) The temperature of Bryan is 101.39°
Step-by-step explanation:
Here, the Temperature of Isabell is given = 103.78 degrees
Now, the temperature of Bryan = Iaabell's Tempt. - 2.36
So, temperature of Bryan = (103.78 - 2.36) degrees
= 101.4 degree
Hence, the temperature of Bryan = 101.4 degree
So, the most appropriate option is 101.39°
Answer:
<h2>y = 5x + 11</h2>
Step-by-step explanation:
![\text{The slope-intercept form of an equation of a line:}\\\\y=mx+b\\\\m-\text{slope}\\b-\text{y-intercept}\\\\\text{Let}\\\\k:y=m_1x+b_1\\l:y=m_2x+b_2\\\\k\ \perp\ l\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}\\\\k\ ||\ l\iff m_1=m_2\\=========================](https://tex.z-dn.net/?f=%5Ctext%7BThe%20slope-intercept%20form%20of%20an%20equation%20of%20a%20line%3A%7D%5C%5C%5C%5Cy%3Dmx%2Bb%5C%5C%5C%5Cm-%5Ctext%7Bslope%7D%5C%5Cb-%5Ctext%7By-intercept%7D%5C%5C%5C%5C%5Ctext%7BLet%7D%5C%5C%5C%5Ck%3Ay%3Dm_1x%2Bb_1%5C%5Cl%3Ay%3Dm_2x%2Bb_2%5C%5C%5C%5Ck%5C%20%5Cperp%5C%20l%5Ciff%20m_1m_2%3D-1%5Cto%20m_2%3D-%5Cdfrac%7B1%7D%7Bm_1%7D%5C%5C%5C%5Ck%5C%20%7C%7C%5C%20l%5Ciff%20m_1%3Dm_2%5C%5C%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D)
![\text{We have}\ y=5x+2\to m_1=5\\\\\text{therefore}\ m_2=5.\\\\\text{Put the value of the slope and the coordinates of the given point (-2, 1)}\\\text{to the equation of a line:}\\\\1=5(-2)+b\\1=-10+b\qquad\text{add 10 to both sides}\\1+10=-10+10+b\\11=b\to b=11\\\\\text{Finally:}\\\\y=5x+11](https://tex.z-dn.net/?f=%5Ctext%7BWe%20have%7D%5C%20y%3D5x%2B2%5Cto%20m_1%3D5%5C%5C%5C%5C%5Ctext%7Btherefore%7D%5C%20m_2%3D5.%5C%5C%5C%5C%5Ctext%7BPut%20the%20value%20of%20the%20slope%20and%20the%20coordinates%20of%20the%20given%20point%20%28-2%2C%201%29%7D%5C%5C%5Ctext%7Bto%20the%20equation%20of%20a%20line%3A%7D%5C%5C%5C%5C1%3D5%28-2%29%2Bb%5C%5C1%3D-10%2Bb%5Cqquad%5Ctext%7Badd%2010%20to%20both%20sides%7D%5C%5C1%2B10%3D-10%2B10%2Bb%5C%5C11%3Db%5Cto%20b%3D11%5C%5C%5C%5C%5Ctext%7BFinally%3A%7D%5C%5C%5C%5Cy%3D5x%2B11)
Answer:
See below for proof.
Step-by-step explanation:
<u>Given</u>:
![y=\left(x+\sqrt{1+x^2}\right)^m](https://tex.z-dn.net/?f=y%3D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em)
<u>First derivative</u>
![\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If $f(g(x))$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=f'(g(x))\:g'(x)$\\\end{minipage}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Bminipage%7D%7B5.4%20cm%7D%5Cunderline%7BChain%20Rule%20for%20Differentiation%7D%5C%5C%5C%5CIf%20%20%24f%28g%28x%29%29%24%20then%3A%5C%5C%5C%5C%24%5Cdfrac%7B%5Ctext%7Bd%7Dy%7D%7B%5Ctext%7Bd%7Dx%7D%3Df%27%28g%28x%29%29%5C%3Ag%27%28x%29%24%5C%5C%5Cend%7Bminipage%7D%7D)
<u />
<u />![\boxed{\begin{minipage}{5 cm}\underline{Differentiating $x^n$}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=xn^{n-1}$\\\end{minipage}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Bminipage%7D%7B5%20cm%7D%5Cunderline%7BDifferentiating%20%24x%5En%24%7D%5C%5C%5C%5CIf%20%20%24y%3Dx%5En%24%2C%20then%20%24%5Cdfrac%7B%5Ctext%7Bd%7Dy%7D%7B%5Ctext%7Bd%7Dx%7D%3Dxn%5E%7Bn-1%7D%24%5C%5C%5Cend%7Bminipage%7D%7D)
<u />
![\begin{aligned} y_1=\dfrac{\text{d}y}{\text{d}x} & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{2x}{2\sqrt{1+x^2}} \right)\\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{x}{\sqrt{1+x^2}} \right) \\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(\dfrac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}} \right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(x+\sqrt{1+x^2}\right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20y_1%3D%5Cdfrac%7B%5Ctext%7Bd%7Dy%7D%7B%5Ctext%7Bd%7Dx%7D%20%26%20%3Dm%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5E%7Bm-1%7D%20%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7B2x%7D%7B2%5Csqrt%7B1%2Bx%5E2%7D%7D%20%5Cright%29%5C%5C%5C%5C%20%26%20%3Dm%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5E%7Bm-1%7D%20%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7Bx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%20%5Cright%29%20%5C%5C%5C%5C%20%26%20%3Dm%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5E%7Bm-1%7D%20%5Ccdot%20%5Cleft%28%5Cdfrac%7Bx%2B%5Csqrt%7B1%2Bx%5E2%7D%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%20%5Cright%29%5C%5C%5C%5C%20%26%20%3D%20%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%20%5Ccdot%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5E%7Bm-1%7D%20%20%5Ccdot%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5C%5C%5C%5C%20%26%20%3D%20%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Cend%7Baligned%7D)
<u>Second derivative</u>
<u />
![\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Bminipage%7D%7B5.5%20cm%7D%5Cunderline%7BProduct%20Rule%20for%20Differentiation%7D%5C%5C%5C%5CIf%20%20%24y%3Duv%24%20%20then%3A%5C%5C%5C%5C%24%5Cdfrac%7B%5Ctext%7Bd%7Dy%7D%7B%5Ctext%7Bd%7Dx%7D%3Du%5Cdfrac%7B%5Ctext%7Bd%7Dv%7D%7B%5Ctext%7Bd%7Dx%7D%2Bv%5Cdfrac%7B%5Ctext%7Bd%7Du%7D%7B%5Ctext%7Bd%7Dx%7D%24%5C%5C%5Cend%7Bminipage%7D%7D)
![\textsf{Let }u=\dfrac{m}{\sqrt{1+x^2}}](https://tex.z-dn.net/?f=%5Ctextsf%7BLet%20%7Du%3D%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D)
![\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7B%5Ctext%7Bd%7Du%7D%7B%5Ctext%7Bd%7Dx%7D%3D-%5Cdfrac%7Bmx%7D%7B%5Cleft%281%2Bx%5E2%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%7D)
![\textsf{Let }v=\left(x+\sqrt{1+x^2}\right)^m](https://tex.z-dn.net/?f=%5Ctextsf%7BLet%20%7Dv%3D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em)
![\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^m](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7B%5Ctext%7Bd%7Dv%7D%7B%5Ctext%7Bd%7Dx%7D%3D%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%20%5Ccdot%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em)
![\begin{aligned}y_2=\dfrac{\text{d}^2y}{\text{d}x^2}&=\dfrac{m}{\sqrt{1+x^2}}\cdot\dfrac{m}{\sqrt{1+x^2}}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}\\\\&=\dfrac{m^2}{1+x^2}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\\\\ &=\left(x+\sqrt{1+x^2}\right)^m\left(\dfrac{m^2}{1+x^2}-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\right)\\\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dy_2%3D%5Cdfrac%7B%5Ctext%7Bd%7D%5E2y%7D%7B%5Ctext%7Bd%7Dx%5E2%7D%26%3D%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Ccdot%5Cdfrac%7Bm%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Ccdot%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%2B%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Ccdot-%5Cdfrac%7Bmx%7D%7B%5Cleft%281%2Bx%5E2%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%7D%5C%5C%5C%5C%26%3D%5Cdfrac%7Bm%5E2%7D%7B1%2Bx%5E2%7D%5Ccdot%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%2B%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Ccdot-%5Cdfrac%7Bmx%7D%7B%5Cleft%281%2Bx%5E2%5Cright%29%5Csqrt%7B1%2Bx%5E2%7D%7D%5C%5C%5C%5C%20%26%3D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Cleft%28%5Cdfrac%7Bm%5E2%7D%7B1%2Bx%5E2%7D-%5Cdfrac%7Bmx%7D%7B%5Cleft%281%2Bx%5E2%5Cright%29%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cright%29%5C%5C%5C%5C%5Cend%7Baligned%7D)
![= \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\right)\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%7D%7B1%2Bx%5E2%7D%5Cright%29%5Cleft%28m%5E2-%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cright%29)
<u>Proof</u>
![(x^2+1)y_2+xy_1-m^2y](https://tex.z-dn.net/?f=%28x%5E2%2B1%29y_2%2Bxy_1-m%5E2y)
![= (x^2+1) \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m](https://tex.z-dn.net/?f=%3D%20%28x%5E2%2B1%29%20%5Cdfrac%7B%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%7D%7B1%2Bx%5E2%7D%5Cleft%28m%5E2-%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cright%29%2B%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em-m%5E2%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%5Cright%29%5Em)
![= \left(x+\sqrt{1+x^2}\right)^m\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m](https://tex.z-dn.net/?f=%3D%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Cleft%28m%5E2-%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cright%29%2B%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em-m%5E2%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%5Cright%29%5Em)
![= \left(x+\sqrt{1+x^2}\right)^m\left[m^2-\dfrac{mx}{\sqrt{1+x^2}}+\dfrac{mx}{\sqrt{1+x^2}}-m^2\right]](https://tex.z-dn.net/?f=%3D%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Cleft%5Bm%5E2-%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D%2B%5Cdfrac%7Bmx%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D-m%5E2%5Cright%5D)
![= \left(x+\sqrt{1+x^2}\right)^m\left[0]](https://tex.z-dn.net/?f=%3D%20%5Cleft%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%5Cright%29%5Em%5Cleft%5B0%5D)
![= 0](https://tex.z-dn.net/?f=%3D%200)
Answer:
30/52
Step-by-step explanation:
its a 26/52% chance to get a black card because 13 spain cards and 13 clovers add it together you get 26 and theirs 4 kings in a dack so add them both together 26/52+4/52=30/52
<h2>Answer: 30/52</h2>