Answer:
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%203x%5E2%2B%203x%20%5Ccdot%20%5Ctriangle%20x%20%2B%20%28%5Ctriangle%20x%29%5E2)
Step-by-step explanation:
Given
![f(x) = x^3](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E3)
Required
Evaluate
![\frac{f(x + \triangle x) - f(x)}{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D)
becomes
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = \frac{(x + \triangle x)^3 - x^3}{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%20%5Cfrac%7B%28x%20%2B%20%5Ctriangle%20x%29%5E3%20-%20x%5E3%7D%7B%5Ctriangle%20x%7D)
Expand
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = \frac{x^3 + 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 - x^3}{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%20%5Cfrac%7Bx%5E3%20%2B%203x%5E2%20%5Ccdot%20%5Ctriangle%20x%2B%203x%20%5Ccdot%20%28%5Ctriangle%20x%29%5E2%20%2B%20%28%5Ctriangle%20x%29%5E3%20%20-%20x%5E3%7D%7B%5Ctriangle%20x%7D)
Collect like terms
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = \frac{x^3 - x^3+ 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 }{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%20%5Cfrac%7Bx%5E3%20-%20x%5E3%2B%203x%5E2%20%5Ccdot%20%5Ctriangle%20x%2B%203x%20%5Ccdot%20%28%5Ctriangle%20x%29%5E2%20%2B%20%28%5Ctriangle%20x%29%5E3%20%20%7D%7B%5Ctriangle%20x%7D)
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = \frac{3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 }{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%20%5Cfrac%7B3x%5E2%20%5Ccdot%20%5Ctriangle%20x%2B%203x%20%5Ccdot%20%28%5Ctriangle%20x%29%5E2%20%2B%20%28%5Ctriangle%20x%29%5E3%20%20%7D%7B%5Ctriangle%20x%7D)
Factorize
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = \frac{\triangle x(3x^2+ 3x \cdot \triangle x + (\triangle x)^2) }{\triangle x}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%20%5Cfrac%7B%5Ctriangle%20x%283x%5E2%2B%203x%20%5Ccdot%20%5Ctriangle%20x%20%2B%20%28%5Ctriangle%20x%29%5E2%29%20%20%7D%7B%5Ctriangle%20x%7D)
![\frac{f(x + \triangle x) - f(x)}{\triangle x} = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%20%2B%20%5Ctriangle%20x%29%20-%20f%28x%29%7D%7B%5Ctriangle%20x%7D%20%3D%203x%5E2%2B%203x%20%5Ccdot%20%5Ctriangle%20x%20%2B%20%28%5Ctriangle%20x%29%5E2)
Answer:
-.857
Step-by-step explanation:
Answer:
48
Step-by-step explanation:
First you would subtract 9 from 393 which is 384 then you would divide the 384 remaining student among the 8 busses which is 48.
Hope this helps.
The main formula is:
Distance traveled = (average)Speed*Time.
i) when Ted Climbs up:
let D be the distance, T the time it takes Ted to climb, S_u=3mph is the speed as Ted climbs up.
then: D=3T
ii) when Ted goes down:
D is the distance, T-40 the time it takes Ted go down, S_d=5mph is the speed as Ted climbs up.
then: D=5(T-40)
equalizing the equations in i and ii:
(D=) 3T=5(T-40)
3T=5T-200
200=2T
T=100 (minutes)
the total length of the hike is T+T-40=100+100-40=160 (minutes)
Answer: 160 minutes
The largest number of groups Emma can make is 6