Question

if f is the function defined by f(x) = (x^2+4x)^(1/3) and g is the antiderivative of f such that g(5) = 7, then g(1) =?

Answer 1

I'm tutoring an AP calc student, and this one sort of stumped me, honestly. The method that seemed to make sense to me would be something like:
1) take antiderivative of f.
2) Use g(5) = 7 to solve for the constant of integration
3) Use the particular solution to solve for g(1)

However, I got stuck at taking the antiderivative of f(x). Am I missing some obvious way of taking that antiderivative? How do you think this problem is meant to be solved? This was a calculator problem, so my student seemed to think the he could use his ti89 to take the antiderivative, although we didn't have one with us, so I wasn't able to try that.

Answer 2

\[g(x)=\int\limits _0^x (t^2+4t)^\frac{1}{3} dt +C \\ g(5)=\int\limits_0^5(t^2+4t)^\frac{1}{3}+C=7 \\ \text{ use a calculator to solve for the definite integral } \]
So say we get some number N.
\[g(5)=N+C=7 \\ N+C=7 \\ C=7-N\]
whatever N is you got from the calculator
\[g(1)=\int\limits_0^1(t^2+4t)^\frac{1}{3} dt+7-N\]
use a calculator again for the definite integral

Answer 3

That makes a lot of sense, myininaya. I wasn't aware the ti89 would do definite integrals like that. That's helpful.

Answer 4

I assumed it could because ti-83 could and ti-89>ti-83. :p

Answer 5

http://www.batesville.k12.in.us/physics/calcnet/calculator/ti_89/definite_integral.htm I think this explains how to use it if the student doesn't know how to use it.