Problem from 2003 AP Calc AB Exam! Help!

Test Preparation
1

<p>Once again, this was on the 2003 AP Calculus AB Exam multiple choice without a calculator:</p>

<p>Let f be the function defined by f(x) = x^3 + x. If g(x) = f^(-1)(x) and g(2) = 1, what is the value of g'(2)?</p>

2

<p>so we know f^(-1) = g(x)</p>

<p>based on that, we can use the rule of inverse functions to say that</p>

<p>F'(x) = 1/G'(f(x))</p>

<p>therefore, if g(2) = 1
f(1) = 2</p>

<p>find the derivitive of the first function
f(x) = x^3 + x
F'(x) = 3x^2 + 1</p>

<p>G'(2) = 1/F'(g(2))
= 1/F'(1)
=1/(3(1) + 1)
= 1/4</p>

<p>:)</p>

3

<p>is it just me, or is this calc exam gonna b easy? i mean the actuall frq from collegeboard (like the one above) just seem like plugging in straight formulas, and if u can pull down a 50 on the frq, then u only need like around 25 mc right for a 5 right?</p>

4

<p>it looks rather complicated, but here's my wording-explanation</p>

<p>"a derivative of an inverse function is
an inverse of derivative of an original function."</p>

<p>of course, x in an original function is y in an inverse function and vicec verse,
therefore you need f'(g(x)) rather than just f(x)
(however pay attention to and notice that g(x) is, in fact, x in an original function.)</p>