Question

please help me! photo attached

Answer 1

I don't see a photo? @Dodo1

Answer 2

I attached two photos, its 2 questions"

Answer 3

@genius12

Answer 4

\(\large 65)\)
\(\large u=f(g(x))\)

\(\large a)\) \(\large u'(1)\). Ok so we'll want to find the derivative of u by applying the chain rule.
Remember how to do that?

Answer 5

yes i do
f'(g(x)*g(x)'

Answer 6

Ok good. From there they want us to find \(\large u'(1)\)
Which will be given by \(\large f'(\color{royalblue}{g(1)})g'(1)\).

So let's try to read this piece by piece.
Start with \(\large \color{royalblue}{g(1)}\). What is the value?

Answer 7

Yes, g(1)=3?

Answer 8

Ok good, that changes our expression to this, \(\large f'(\color{royalblue}{3})g'(1)\)

Answer 9

Luckily these are straight lines, so even without a function we can easily determine the slope of these points.

What is the slope of that line segment of \(\large f\) at \(\large x=3\) ?

Answer 10

Just count the boxes if you're not sure.

It looks like we're moving DOWN1 for every 4RIGHT yes?

We can think of this as -1UP and 4RIGHT.

Rise/Run = -1/4
This is our slope. Our \(\large f'(3)\). Understand how I got that value? <:o

Answer 11

Mmmm im confsued.

Answer 12

g(1)=1?
f(1)=3.

Answer 13

i got this part so far

Answer 14

No, g(1)=3, as you said earlier.

Answer 15

ops, yes.

Answer 16

i was going to say f(1)=2

Answer 17

First thing we notice is that each function is piece-wise and has corners so will not be differentiable at certain points. We know that u(x) = f(g(x)). Since a) asks us for u'(1), we look at functions f(x) and g(x) from (0, 2). We notice that f(x), in the interval (0, 2), is the same as saying f(x) = 2x. In the same interval (0, 2), we notice that g(x) = -3x + 6. Since we are asked for u'(1), we must first find f(g(x)) in the interval (0, 2).

f(g(x)) = u(x) = -6x + 12
Therefore, u'(x) = -6
u'(1) = -6

For v'(1), we do the exact same as above. We are still using the interval (0, 2) except this time it's g(f(x)).

g(f(x)) = v(x) = -3(2x) + 6 = -6x + 6
Therefore, v'(x) = -6
v'(1) = -6

For w'(1), we are doing the same thing in the interval (0, 2). This time only, we are finding g(g(x)).

g(g(x)) = w(x) = -3(-3x + 6) + 6 = (9x - 18) + 6 = 9x - 12
Therefore, w'(x) = 9
w'(1) = 9

Hope that helps.
@Dodo1

Answer 18

hold on,
for the first graph the answer is 3/4

Answer 19

What graph are you talking about? I am talking about question 65.

Answer 20

not 65, the graph of u. u=3/4 or something

Answer 21

Which graph? The plotted in red? I was answering the other question 65. where it asks for derivatives.

Answer 22

the pix of 1917?

Answer 23

What's 1917 lol -.-

Answer 24

the drivative askins u' question right?
the answer is 3/4

Answer 25

sorry asking

Answer 26

@zepdrix any ideas?

Answer 27

for 65, the answer for u is 3/4 but i dont know how i get to this answer

Answer 28

The question asking u'(1) v'(1) w'(1), that's what I answered.

Answer 29

yes,
but my answer book says thst u'(1)= 3/4

Answer 30

:/

Answer 31

Lol.

Answer 32

thanks for your time tho.

Answer 33

What does it say for w'(1) and v'(1)?

Answer 34

for v says DNE. and for w, -2

Answer 35

this is tricky question is it?

Answer 36

This isn't tricky. The answers just don't make any sense to me. 0,o What I got should be correct.

Answer 37

\(\large f'(\color{royalblue}{g(1)})g'(1) \qquad = \qquad f'(\color{royalblue}{3})g'(1)\qquad = \qquad -\dfrac{1}{4}(2) \qquad = \qquad -\dfrac{1}{2}\)

The book says u'(1) is 3/4??? Hmm...

Answer 38

Oh oh oh i see, i made a stupid mistake :)

Answer 39

yes! it does, it says use slope the slope is (2,4)(6,3) so the slope is 3-4/6-2 g is linear from (0,6)(2,0)

Answer 40

you got it?? :)

Answer 41

g'(1)=-3.
Which gives us,\[\large -\frac{1}{4}(-3)\]

Answer 42

I'm not really sure how to explain this to you though :(
Do you remember how to find the slope of a line?

Answer 43

Given two points*

Answer 44

ok i will write what book says.. i dont get it at all tho

Answer 45

to find f'(3) note that f is linear from (2,4)(6,3) so its lope is 3-4/6-2 =-1/4

Answer 46

i have no idea why the slope is nesscary to solve this question

Answer 47

The derivative represents the slope of the line at a given point.

Answer 48

given point?

Answer 49

For example,

g(x)=2x from the interval [0,2] as genius mentioned.
We know that the derivative will give us,
g'(x)=2.

This is telling us that the slope of that line segment is 2.
If I were to ask you what is the value of g'(1)...
I'm asking, what is the slope of the line segment at x=1.

Well we determined that the entire line segment has a slope of 2 in the given interval. So at the point x=1, the slope will also be 2.

A way we could verify that is by plugging 1 into our derivative.
g'(1)=2.

Answer 50

oh i see, thank you so much uguys. I have test tomorrow. wish me luck :)