Answer:
B. whole numbers, integers, rational numbers, natural numbers
Answer:
w=-7
Step-by-step explanation:
Soooo-
W(x)=-3x-4
divide both sides by x
W=-3-4
combine like terms
W=-7
Median is where if you put them in ascending order and cross them off, one from each side at a time, you get to the middle number.
56789 is your answer but so would any group of 5 numbers where two are bigger and two smaller than seven.
Answer:
Negative.
Step-by-step explanation:
Hope this helps.
Question:
Find the point (,) on the curve
that is closest to the point (3,0).
[To do this, first find the distance function between (,) and (3,0) and minimize it.]
Answer:
![(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})](https://tex.z-dn.net/?f=%28x%2Cy%29%20%3D%20%28%5Cfrac%7B5%7D%7B2%7D%2C%5Cfrac%7B%5Csqrt%7B10%7D%7D%7B2%7D%7D%29)
Step-by-step explanation:
can be represented as: ![(x,y)](https://tex.z-dn.net/?f=%28x%2Cy%29)
Substitute
for ![y](https://tex.z-dn.net/?f=y)
![(x,y) = (x,\sqrt x)](https://tex.z-dn.net/?f=%28x%2Cy%29%20%3D%20%28x%2C%5Csqrt%20x%29)
So, next:
Calculate the distance between
and ![(3,0)](https://tex.z-dn.net/?f=%283%2C0%29)
Distance is calculated as:
![d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x_1-x_2%29%5E2%20%2B%20%28y_1%20-%20y_2%29%5E2%7D)
So:
![d = \sqrt{(x-3)^2 + (\sqrt x - 0)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x-3%29%5E2%20%2B%20%28%5Csqrt%20x%20-%200%29%5E2%7D)
![d = \sqrt{(x-3)^2 + (\sqrt x)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x-3%29%5E2%20%2B%20%28%5Csqrt%20x%29%5E2%7D)
Evaluate all exponents
![d = \sqrt{x^2 - 6x +9 + x}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7Bx%5E2%20-%206x%20%2B9%20%2B%20x%7D)
Rewrite as:
![d = \sqrt{x^2 + x- 6x +9 }](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7Bx%5E2%20%2B%20x-%206x%20%2B9%20%7D)
![d = \sqrt{x^2 - 5x +9 }](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7Bx%5E2%20-%205x%20%2B9%20%7D)
Differentiate using chain rule:
Let
![u = x^2 - 5x +9](https://tex.z-dn.net/?f=u%20%3D%20x%5E2%20-%205x%20%2B9)
![\frac{du}{dx} = 2x - 5](https://tex.z-dn.net/?f=%5Cfrac%7Bdu%7D%7Bdx%7D%20%3D%202x%20-%205)
So:
![d = \sqrt u](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%20u)
![d = u^\frac{1}{2}](https://tex.z-dn.net/?f=d%20%3D%20u%5E%5Cfrac%7B1%7D%7B2%7D)
![\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}](https://tex.z-dn.net/?f=%5Cfrac%7Bdd%7D%7Bdu%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7Du%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D)
Chain Rule:
![d' = \frac{du}{dx} * \frac{dd}{du}](https://tex.z-dn.net/?f=d%27%20%3D%20%5Cfrac%7Bdu%7D%7Bdx%7D%20%2A%20%5Cfrac%7Bdd%7D%7Bdu%7D)
![d' = (2x-5) * \frac{1}{2}u^{-\frac{1}{2}}](https://tex.z-dn.net/?f=d%27%20%3D%20%282x-5%29%20%2A%20%5Cfrac%7B1%7D%7B2%7Du%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D)
![d' = (2x - 5) * \frac{1}{2u^{\frac{1}{2}}}](https://tex.z-dn.net/?f=d%27%20%3D%20%282x%20-%205%29%20%2A%20%5Cfrac%7B1%7D%7B2u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
![d' = \frac{2x - 5}{2\sqrt u}](https://tex.z-dn.net/?f=d%27%20%3D%20%5Cfrac%7B2x%20-%205%7D%7B2%5Csqrt%20u%7D)
Substitute: ![u = x^2 - 5x +9](https://tex.z-dn.net/?f=u%20%3D%20x%5E2%20-%205x%20%2B9)
![d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}](https://tex.z-dn.net/?f=d%27%20%3D%20%5Cfrac%7B2x%20-%205%7D%7B2%5Csqrt%7Bx%5E2%20-%205x%20%2B%209%7D%7D)
Next, is to minimize (by equating d' to 0)
![\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0](https://tex.z-dn.net/?f=%5Cfrac%7B2x%20-%205%7D%7B2%5Csqrt%7Bx%5E2%20-%205x%20%2B%209%7D%7D%20%3D%200)
Cross Multiply
![2x - 5 = 0](https://tex.z-dn.net/?f=2x%20-%205%20%3D%200)
Solve for x
![2x =5](https://tex.z-dn.net/?f=2x%20%20%3D5)
![x = \frac{5}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B5%7D%7B2%7D)
Substitute
in ![y = \sqrt x](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%20x)
![y = \sqrt{\frac{5}{2}}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%7B%5Cfrac%7B5%7D%7B2%7D%7D)
Split
![y = \frac{\sqrt 5}{\sqrt 2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Csqrt%205%7D%7B%5Csqrt%202%7D)
Rationalize
![y = \frac{\sqrt 5}{\sqrt 2} * \frac{\sqrt 2}{\sqrt 2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Csqrt%205%7D%7B%5Csqrt%202%7D%20%2A%20%20%5Cfrac%7B%5Csqrt%202%7D%7B%5Csqrt%202%7D)
![y = \frac{\sqrt {10}}{\sqrt 4}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Csqrt%20%7B10%7D%7D%7B%5Csqrt%204%7D)
![y = \frac{\sqrt {10}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Csqrt%20%7B10%7D%7D%7B2%7D)
Hence:
![(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})](https://tex.z-dn.net/?f=%28x%2Cy%29%20%3D%20%28%5Cfrac%7B5%7D%7B2%7D%2C%5Cfrac%7B%5Csqrt%7B10%7D%7D%7B2%7D%7D%29)