Question

Suppose g (x) is increasing and concave up everywhere and g(A)=7, gprime(A)=13, h=.01
Estimate g(A-h) and g(A+h).
Can someone help me with this its the last problem in my homework and I have no idea how to do it.

Answer 1

Abe: Hello!
To start off: are you familiar with linear approximations? The problem you've posted strikes me as one in such approximations.

Answer 2

Sounds familiar

Answer 3

Mind looking that up? See whether you can find a formula, or an explanation, in your book or online. I know this material well, but would like for you to be able to find this sort of info yourself when you need it. What can you tell me about linear approximations?

Answer 4

f(x)=f(a)+fprime(a)(x-a)

Answer 5

Perfect. Exactly what I was hoping you'd type.

Answer 6

Now we're going to tweak that a bit. Where you typed " a " please type in A instead. We don't know what the value of A is, but our linear approx will be centered at x = A. Where you've typed "f" replace that with " g ". Go ahead.

Answer 7

g(x)=g(A)+g'(A)(x-A)

Answer 8

and where you've typed fprime, type in " g ' (A) instead.

Answer 9

Nice work!

Answer 10

Now, Abe, what's the value of g(A)? of g '(A)?

Answer 11

7 and 13

Answer 12

Right. Would you plese substitute those values into your current equation in the appropriate places.

Answer 13

7+13(x-7)

Answer 14

Now would you please label that with L(x) =
Note that we use L because what we're looking for here is a Linear Approximation.

Answer 15

L(x)=7+13(x-7)

Answer 16

Beatiful. Right on target!

Answer 17

Now, Abe, we want to estimate the value of your function g(x) at the x value A+h, or A+0.01. The original problem statement tells us that h=0.01.

What I'd like for you to do is to go back to the equation you've just typed in and replace x on both sides with [A+0.01] (be certain to include the brackets, [ ] ).

Answer 18

L[A+.01]=7+13([A+.01]+7)

Answer 19

That's great, with one exception: That " 7 " should be " A ". I've missed that.
the point here, Abe, is that g(A) = 7. So, simply rewrite what you've just typed in but replace that 7 by A. Then look for ways to simplify what you've written.

Answer 20

so L[A+.01]=7+13([A+.01]+A)

Answer 21

A-1!

Now, what about that ([A+0.01]+A]? could this be reduced? Upon closer inspection I've found that we both missed the fact that the 2nd A should be negative, not positive. I will review this with you later. But for now, fix it and then simplify ([A+0.01]-A).

Answer 22

type out the whole thing, starting with L[ ...
and simplifying ([A=0.01]-A) as much as possible.

Answer 23

just .01

Answer 24

Right, and so your approximation L[A+0.01] equals what? either write out the formula or go ahead and evaluate it.

Answer 25

Please use the label L[A+0.01]= as before.

Answer 26

7.01?

Answer 27

What I see is L[A+0.01]=7 + 13(0.01). Check this out and say wehther y ou agree or disagree.

Answer 28

h=0.01, not 0.1.

Answer 29

so L[A+.01]=7.13

Answer 30

Perfect. Congrats. That's your estimation of g(x) when x is 0.01 greater than A.

Answer 31

Your last step is to estimate g(x) for the input A-h, or A-0.01.

You are now to write

L[A-0.01] = blah , blah.

Want to give that a try? The work is practically identical to what you've just done.

Answer 32

L[A-0.01]=7 + 13(-0.01)?

Answer 33

Exactly. Nice labelling, too. Simplify your result as much as possible, please/

Answer 34

Okay 6.87 is final answer

Answer 35

Right on. You've done very, very well here.

Answer 36

All right great thanks really helped me understand how to do it

Answer 37

My pleasure. Hope y ou'll reveiw our work and take some notes on it so that you'll be that much better prepared next time to do similar problems. Hope to work with you again! Good night, Abe!