Inverse functions... Need help with this problem...

[casseyakens]

I'm in AP Calculus AB, and my instructor has assigned me the problem:
f(x)=(1/27)(x^5+2x^3), a=-11
I have to on find (f^-1)'(a) for this function, but I have difficulty finding the inverse first. Any help would be greatly appreciated.

 

[stapel]

f(x)=(1/27)(x^5+2x^3), a=-11
I have to on find (f^-1)'(a) for this function, but I have difficulty finding the inverse first. Any help would be greatly appreciated.

Have you been given the formula \(\displaystyle \frac{df^{-1}(a)}{dx}\, =\, \frac{1}{f'(f^{-1}(a))}\)?

 

[Quaid]

f(x) = (1/27)(x^5 + 2x^3)

a = -11

find (f^-1)'(a)

I have difficulty finding the inverse first.

Hi Cassey:

It's too difficult, to find f^-1(x). In this case, we may use the Inverse Function Theorem.


(click thumbnail for larger image)



Part (d) states the derivative of f-inverse in terms of the derivative of f

Remember, y=f(x)

Questions still? If not, post your answer. I'll compare it against mine, to see whether I made a mistake.

Ciao

 

[Quaid]

Hi Stapel, just saw your post.

I may have already derailed; I assumed a = -11 to be a given value of x instead of y.

Is that a goof? I haven't seen this type of exercise in a while. :cool:

 

[ahorn]

I'm in AP Calculus AB, and my instructor has assigned me the problem:
f(x)=(1/27)(x^5+2x^3), a=-11
I have to on find (f^-1)'(a) for this function, but I have difficulty finding the inverse first. Any help would be greatly appreciated.

\(\displaystyle f(f^{-1}(-11))=-11\)

Let \(\displaystyle f^{-1}(-11)=y\)

\(\displaystyle \Rightarrow f(y)=-11\\\
\\
\frac{1}{27}(y^5+2y^3)=-11\\\
\\
y^5+2y^3+297=0\\\
\\
y=-3 \text{ (I used wolframalpha to find this solution)}\\\
\\
f^{-1}(-11)=-3\)

 

[casseyakens]

Thank you for the tips on finding the derivative... I really appreciate it...However, im still stuck on how exactly to find the inverse function, in order to place it in the other equation... I might be overlooking or not looking enough, but I'm still stumped. How do I find the inverse function first? :/

 

[Quaid]

im still stuck on how exactly to find the inverse function, in order to place it in the other equation.

You do not need the rule for the inverse function.

The theorem requires only the output of the inverse function.

f^-1(y) = x when y = f(x)

f(x)=(1/27)(x^5+2x^3), a=-11

Can you define symbol a? I know its value is -11, but what does symbol a represent?

Is it an input to function f (x-value), or is it an output from function f (y-value)?


Cheers

 

[casseyakens]

thank you

Ohhhhhhhhhhhh......okay... I think I understand it now.... thank you all so much... it just clicked that I didn't NEED to find the actual inverse equation... I just needed the right equation to get me started, so thank you all...

 

[casseyakens]

What a is...

My book only states that a=-11and is a given real number for finding: (f^-1)'(a)...

 

[Quaid]

Let \(\displaystyle f^{-1}(-11) = y\)

Hi ahorn:

That's one interpretation.

y = f(x) is another.

If there's a standard, I've forgotten it.

Cheers

 

[Quaid]

My book only states that a=-11 and [it] is a given real number for finding: (f^-1)'(a).

Okay, then I probably interpreted the notation backwards.

I hope your class allows technology, to solve equations like f(x) = -11