Question

The graphs of the function f (given in blue) and g (given in red) are plotted above. Suppose that u(x)=f(x)g(x) and v(x)=f(x)/g(x). Find each of the following:
u'(1) =
v'(1) =

Answer 1

Since these are piece-wise linear functions, the slopes of the functions are constant over certain intervals.

Answer 2

And the slope of the function at a point is its derivative at that point.

Answer 3

First you need to find the equation, u(x). All we care about are the equations of the red and blue lines when x = 1.

Answer 4

If you have u(x) = f(x)g(x), and f and g are both continuous at a point, then u(x) should be differentiable at that point. And yes, we can find the equations of f and g, since they are piece-wise linear. :)

Answer 5

What's the equation of that upward sloping part of the blue line...f(x)?

Answer 6

Yes, f is blue and g is red.

Answer 7

what's the equation of that line?

Answer 8

I'm assuming f(x)=x in this case

Answer 9

yes.
the other line...the red one...is slightly trickier...can you do that one?

Answer 10

y-intercept is 3.5 or 7/2
slope appears to be -3/2...right?

Answer 11

Seems right

Answer 12

Looks like it to me too.

Answer 13

slope-intercept form....all we need is the slope and y intercept.
so, what is g(x)?

Answer 14

g(x) = -3/2x + 7/2
I'm just using y = mx + b there

Answer 15

So..now, what's f(x)g(x).....just multiply them together and expand.

Answer 16

May I make a suggestion?

Answer 17

Go ahead

Answer 18

BangkokGarrett is right, you can multiply the functions together then take the derivative of the product. Or, you can use the product rule:

Answer 19

for derivatives.

Answer 20

Since you know f(1), g(1), f'(1) and g'(1)

Answer 21

f(x)g(x) = x(-3/2x + 7/2) = -3/2x^2 + 7/2x
That's your u(x). Now find u'(x)...and then plug in 1 to get u'(1).
You can probably figure out now how to get f(x)/g(x)
Good luck!

Answer 22

ahhhh....yes. your way is probably quicker.

Answer 23

I don't know for sure whether it is. :) I was just trying to avoid messing with fractions because I'm lazy ;)

Answer 24

i'm sure your teacher would prefer anteater's approach.

Answer 25

But yes, you could use f(x)g'(x) + f'(x)g(x)

Answer 26

u(x) = f(x)g(x)
so...u'(x) = f(x)g'(x) + f'(x)g(x)

Answer 27

yeah. right.

Answer 28

Well, it depends on whether they have discussed those rules yet. It just occurred to me that they might not have

Answer 29

and that this may be an exercise to help them "appreciate" that next lesson :D

Answer 30

But I think your idea is good as well. I think it is good to verify that the derivative of the product will give the same result.

Answer 31

good luck. i'm outta here!

Answer 32

One last thing. So to find v'(1), I use the quotient rule for derivatives instead?

Answer 33

Yes :D

Answer 34

Thank you! :)

Answer 35

You're welcome! Have a good evening! :)