h of x Problem

[Jason76]

Interpret below as a chart:

\(\displaystyle x = 10 | f(x) = 36 | f'(x) = 16 | g(x) = 7 | g'(x) = 5\)

\(\displaystyle x = 21 | f(x) = 9 | f'(x) = 6 | g(x) = 13 | g'(x) = 11\)

\(\displaystyle x = 31 | f(x) = 25 | f'(x) = 26 | g(x) = 21 | g'(x) = 11\)

End of graph

if \(\displaystyle h(x) = f(g(x))\), then what is the value of \(\displaystyle h'(30)\)

Any starting hints?

 

[HallsofIvy]

The "chain rule" in calculus tells you that (f(g(x))'= f'(g(x))g'(x)

(the first f'(g(x)) means the derivative of f (ignoring "g") then evaluated at g(x). For example, if \(\displaystyle f(x)= x^2\) and g(x)= 3x- 1, then \(\displaystyle f'(x)= 2x\) so that \(\displaystyle f'(g(x))= 2(3x- 1)\).)

Here, you are told that h(x)= f(g(x)) so that h'= f'(g(x))g'(x). You are asked to find h'(30).

The first thing I think you should do is reread the problem! h'(30)= f'(g(30))g'(30) but you are NOT told anything about the values of f, g, f', or g' at x= 30. Using the information given it is impossible to answer the question. In order that this problem be possible, the last line must give values at x= 30, not 31, or the problem must ask for h'(31) not 30.

A third possibility is that you are only asked to approximate h'(30). There are many different ways to do that, giving many different answers since there is NO one answer to an "approximation". The simplest thing to do is probably to find h'(21) and h'(31), then do a linear "interpolation", that is, find the line that goes through (21, h'(21)) and (31, h'(31)) and evaluate that at x= 30.

 

[Jason76]

How about with these numbers???

Interpret below as a chart:

\(\displaystyle x = 10 | f(x) = 35 | f'(x) = 15 | g(x) = 6 | g'(x) = 4\)

\(\displaystyle x = 20 | f(x) = 8 | f'(x) = 5 | g(x) = 12 | g'(x) = 10\)

\(\displaystyle x = 30 | f(x) = 24 | f'(x) = 25 | g(x) = 20 | g'(x) = 10\)

End of graph

if \(\displaystyle h(x) = f(g(x))\), then what is the value of \(\displaystyle h'(30)\)

Any starting hints?

 

[DrPhil]

How about with these numbers???

Interpret below as a chart:

\(\displaystyle x = 10 | f(x) = 35 | f'(x) = 15 | g(x) = 6 | g'(x) = 4\)

\(\displaystyle x = 20 | f(x) = 8 | f'(x) = 5 | g(x) = 12 | g'(x) = 10\)

\(\displaystyle x = 30 | f(x) = 24 | f'(x) = 25 | g(x) = 20 | g'(x) = 10\)

End of graph

if \(\displaystyle h(x) = f(g(x))\), then what is the value of \(\displaystyle h'(30)\)

Any starting hints?

MUCH BETTER!

1) you have differentiated f[g(x)] using the chain rule, f' = ..

2) When x=30, what is g(x)? what is g'(x)?

3) What is f'[g(30)]?

Thus, what is the answer?

 

[Jason76]

MUCH BETTER!

1) you have differentiated f[g(x)] using the chain rule, f' = ..

2) When x=30, what is g(x)? what is g'(x)?

3) What is f'[g(30)]?

Thus, what is the answer?

The answer should be 10 according to the graph, but the actual answer is 50.

 

[HallsofIvy]

The answer should be 10 according to the graph, but the actual answer is 50.

Yes, doing what we suggested you get "50". You have NOT shown any graph so we have no idea what you mean by "according to the graph".

 

[Jason76]

Yes, doing what we suggested you get "50". You have NOT shown any graph so we have no idea what you mean by "according to the graph".

I mean according to the chart, sorry. If you look under \(\displaystyle g'(x)\) for the line with \(\displaystyle x = 30\) then you see \(\displaystyle 10\)

 

[HallsofIvy]

Yes, "10" is the answer to Dr. Phil's question "what is g'(30)". But your question was "What is h'(30)" where h(x)= f(g(x)). The answer to that is "50".

 

[Jason76]

Yes, "10" is the answer to Dr. Phil's question "what is g'(30)". But your question was "What is h'(30)" where h(x)= f(g(x)). The answer to that is "50".

\(\displaystyle g(x)\) is \(\displaystyle 25\) at \(\displaystyle x = 30\). So what is \(\displaystyle f(g(x))\)? Is there some multiplication involved?

 

[HallsofIvy]

No, what's involved is learning the definitions! It is annoying that, after all of this, you finally tell us that you do not know what "f(g(x))" means!
Given a value of x, to find f(g(x)) you calculate a= g(x) and then find f(a).

You are told that "when x= 30, g(x)= 20" so f(g(30))= f(20)= 8.

(It was bad enough that you wrote "x= 21" and "x= 31" when it was supposed to be "x= 20" and "x= 30". Now you have written "g is 25 when x= 30" which is NOT what was given in your chart. One might begin to suspect that you do not care enough to pay attention!

 

[soroban]

Hello, Jason76!

I believe there is a typo . . .

\(\displaystyle \text{Given:}\)

\(\displaystyle \begin{Bmatrix} f(10) =36 & g(10) =7 \\ f'(10) =16 & g'(10) =5 \end{Bmatrix} \quad \begin{Bmatrix} f(21) = 9 & g(21) = 13 \\ f'(21) = 6 & g'(21) = 11 \end{Bmatrix} \quad \begin{Bmatrix}f(\color{blue}{30}) = 25 & g(\color{blue}{30}) = 21 \\ f'(\color{blue}{30}) = 26 & g'(\color{blue}{30}) = 11 \end{Bmatrix}\)


\(\displaystyle \text{If }h(x) \,=\, f(g(x)),\,\text{ find the value of }\,h'(30)\)

First, find \(\displaystyle h'(x).\)

. . \(\displaystyle h'(x) \:=\:f'(g(x))\cdot g'(x)\)


Let \(\displaystyle x = 30.\)

. . \(\displaystyle h'(30) \;=\;f'(g(30))\cdot g'(30)\)

. . . . . . . \(\displaystyle =\; f'(21)\cdot 11\)

. . . . . . . \(\displaystyle =\;6\cdot 11\)

. . . . . . . \(\displaystyle =\;66\)

 

[DrPhil]

\(\displaystyle x = 10 | f(x) = 35 | f'(x) = 15 | g(x) = 6 | g'(x) = 4\)

\(\displaystyle x = 20 | f(x) = 8 | f'(x) = 5 | g(x) = 12 | g'(x) = 10\)

\(\displaystyle x = 30 | f(x) = 24 | f'(x) = 25 | g(x) = 20 | g'(x) = 10\)

if \(\displaystyle h(x) = f(g(x))\), then what is the value of \(\displaystyle h'(30)\)

Here is what I told you to do:

1) you have differentiated f[g(x)] using the chain rule, f' = ..

2) When x=30, what is g(x)? what is g'(x)?

3) What is f'[g(30)]?

I get frustrated when you ignore my advice.

Show me the (symbolic) derivative of h(x) = f[g(x)]

Then you will see what numbers you need from the chart.

 

[Jason76]

I would understand this stuff if we were dealing with two equations Ex. \(\displaystyle f(x) = 4x^{2} + 2\) and \(\displaystyle g(x) = 9x^{3} + 3\) Given \(\displaystyle h(x) = f(g(x))\) What is \(\displaystyle h'(3)\)
But in this thread, there are no two equations, so what to do?

 

[DrPhil]

Do you ever check if someone has posted to your thread while you were posting?