Question

Help Please!(:
Consider the function f(x) = 6sqrt {x}+7 on the interval [ 1 , 10 ]. Find the average or mean slope of the function on this interval.

Answer 1

Well if you want to find the average, what do you do normally? Don't even think calculus yet.

Answer 2

solve for x?

Answer 3

@Kainui

Answer 4

Nope, the average is like when you have 3 people and each person has their own height, like 5 feet tall, 6 feet tall, and 4 feet tall.

So you add up all the heights and divide by the total number,

So what's the average height of the people?

Answer 5

5 feet @Kainui

Answer 6

Alright, cool. So now you need to find the formula for slope of that function.

Then this is what you're taking the average of.

You need to add up all the individual amounts of slope over that function from the boundary points (1 to 10) by integration and divide by the total interval which is (10-1). It's basically the same as the average.

Answer 7

I don't know how to go about starting this problem..
add of the individual amounts of the slope over the function?
how do i do that?
@kainui

Answer 8

the integral definition of avg slope is:
\[\frac{1}{b-a}\int\limits_{a}^{b}f'(x) dx\]
using fund thm of calculus, this reduces to
\[\frac{f(b)-f(a)}{b-a}\]
which is the same as simple slope formula for the 2 endpoints of interval

Answer 9

so by drawing a line from the endpoints, the slope of that line is no different than if you averaged all the slopes at every point on the curve

Answer 10

So how do I go about doing this problem. I understand i need to find the slope. but how?

Answer 11

from algebra 1 you learned slope is change in y over change in x

Answer 12

That's what the derivative is all about. Remember y=mx+b, where m is the slope?

That's rise over run, which is (y2-y1)/(x2-x1) which is essentially what the definition of a derivative is, but it's [f(x+h)-f(x)]/[(x+h)-(x)] as h approaches 0, so it's instantaneous.

Answer 13

yes but there's only one point. (1,10)

Answer 14

no thats the interval...x varies from 1 to 10
plug those into function to get y values

Answer 15

wow you prob didn't get anything i wrote huh

Answer 16

yeahh... I'm so confused int his class... But i'm trying.

so 13 and 25.97? haha

Answer 17

yes..i would leave it in exact form though

Answer 18

Well pdd what you have to understand is that an integral is the reverse operation of a derivative, so you are kind of not calculating anything much since you're basically doing something, undoing it, and then plugging in those values... lol

Answer 19

so... 6sqrt{1}7 and 6sqrt{10}+7 ?

Answer 20

no need for sqrt1 ... 13 and 6sqrt10 +7

now plug those into slope formula and simplify

Answer 21

\[\rightarrow \frac{(6\sqrt{10}+7) -13}{10-1} = \frac{6\sqrt{10}-6}{9} = \frac{2(\sqrt{10}-1)}{3}\]

Answer 22

I was just about to type that out.... but for {1,10}

Answer 23

By the Mean Value Theorem, we know there exists a c in the open interval ( 1 , 10 ) such that f'( c) is equal to this mean slope. For this problem, there is only one c that works. how do I find it?

Answer 24

i find the derivative of the function first right?

Answer 25

right set f'(x) equal to avg slope found above ... solve for "x" and that is your c value

Answer 26

so f'(x)=3/sqrt(x)

then x becomes c.
so 3/sqrt(c)=2(sqrt(10)-1)?
and c=0.230886157?

Answer 27

derivative is right .... the "c" value seems way too small

wait it has to be bigger than 1

Answer 28

im lazy

http://www.wolframalpha.com/input/?i=3%2Fsqrt%28c%29+%3D+2%28sqrt%2810%29-1%29%2F3