If f and g are the functions whose graphs are shown below, let u(x)=f(g(x)) and v(x)=g(f(x)) .
find u'(3) and v'(3)

- anonymous

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- anonymous

##### 1 Attachment

- TuringTest

start with finding expressions for u'(x) and v'(x)

- TuringTest

u'(x)=f'(g(x))g'(x)
v'(x)=g'(f(x))f'(x)

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## More answers

- anonymous

yeah i dont understand this. am i just looking at 3 on the graph?

- TuringTest

say you want u'(3), you first need g(3) because g(x) is part of the formula you need
what is g(3) according to the graph?

- anonymous

looks like 4

- TuringTest

looks like g(3)=10 to me
each square is 2 vertically, and 1 horizontally as far as I can tell

- TuringTest

g(3)=8 I mean, sorry

- TuringTest

I'm not really sure about the increments on the graph though, it's a bit vague...

- anonymous

8, how so?

- TuringTest

It looks to me like each square vertically is 2 (by the marking), but I guess it must be 4 since 8 is not on the graph horizontally
so ok, g(3)=4
what is the value of f'(4) ?
[remember we need f'(g(3))g'(3)]

- TuringTest

also remember that f'(4) represents the slope of f(x) at x=4

- anonymous

f(4) is 2

- TuringTest

we want f'(4)
that is the slope of f(x) at x=4
what is the slope of the line in the graph f(x) at x=4 ?

- anonymous

when x is 4, f is 2 right?

- TuringTest

yes, but that is the value of f(4) not f'(4)
f'(4) is the SLOPE, not the value of the function
do you remember how to find the slope of a straight line from algebra?

- anonymous

ohh

- anonymous

y2-y1/x2-x1?

- TuringTest

actually sorry, they are the same in this case, but that is pure luck! lol
notice that the slope (rise over run) is also 2 !

- TuringTest

that is what f' means, the slope of f
so similarly what is the last piece of the formula we need ?
u'(3)=f'(g(3))g'(3)
and we still need g'(3)
what is it?

- anonymous

rise over run right? 4/3? no?

- TuringTest

at x=3 it looks to me like g has a slope of -1

- anonymous

how did you figure that out?

- TuringTest

look at the tail last portion of g(x), the far right portion of the top graph.
it dips downward
it seems to go down one unit for every unit it goes to the right
rise/run=-1/1=-1

- TuringTest

here's a sketch of g|dw:1329536578080:dw|

- anonymous

ok i see now i see

- TuringTest

here's the part from x=2 to x=4

- TuringTest

|dw:1329536693848:dw|in your graph you can see it goes one over and one down on that region

- TuringTest

(I'm not gonna draw the little squares in...)

- anonymous

i see that thanks

- TuringTest

so you have all the pieces now
u'(3)=f'(g(3))g'(3)
you know f'(g(3)) and g'(3), so multiply them to find u'(3)

- TuringTest

what is g(3) ?

- anonymous

4
slope is -1

- TuringTest

right, g(3)=4
and what is f'(g(3)) ?

- anonymous

f is also 4 right?
so that makes 4 x 4?

- TuringTest

no, we want f'(g(3))=f'(4)
because g(3)=3, right ?
what is f'(4) ?

- anonymous

the slope of at 4 is 3

- TuringTest

you are thinking about the idea correctly, but it looks like 2 to me

- anonymous

ok the gradients on this graph is crap

- TuringTest

yeah, I agree with that, but the problem only seems to make sense if we count each square as 1...|dw:1329537340258:dw|so we want the slope of the upward portion of f

- anonymous

4/2

- anonymous

2

- TuringTest

right, looks that way
so what is u'(3)=f'(g(3))g'(3) ?

- anonymous

(2)(4) (4)

- TuringTest

there should be no 4 in the answer
what is f'(g(3)) ?

- TuringTest

(we just figured it out above)

- anonymous

2

- TuringTest

yes, and what is g'(3) ?

- anonymous

-1

- TuringTest

so f'(g(3))g'(3) is what?

- anonymous

(2) (-1)

- TuringTest

u'(3)=f'(g(3))g'(3)=-2
yep :D

- anonymous

thanks for the help

- TuringTest

so now you get to try for v'(3) !
good luck :)

- anonymous

I got zero

- TuringTest

yep!

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