00:01
So with this question, they say that a cup of coffee at 95 degrees celsius is put into a 20 degree celsius room when t is equal to zero.
00:11
And the coffee's temperature, f of t, is changing at a rate given by f prime of t equals negative 5 times 0 .7 being raised to the t power degrees celsius per minute, where t is in minutes.
00:29
And i want to estimate the coffee's temperature when t is equal to 9.
00:35
So what are we doing here? we are using the fundamental theorem of calculus.
00:41
So you may remember that the fundamental theorem of calculus says, if you want to compute the definite integral from a to b of little f of x, d x, this is equal to capital f of b minus capital f of a.
00:59
In other words, to evaluate a definite interval, the first thing i do is i find an anti -derivative.
01:06
The antiderivative of little f is denoted big f, and then i plug in my top limit of integration to that antiderivative, and from that, i subtract what i get when i plug in my bottom limit of integration.
01:22
So here, i have been given f prime of t, and i'm on the t interval from t equals 0 to t equals 9.
01:32
So what do i know? i know that the integral from 0 to 9 of f prime of t d t, this would be equal to little f of 9 minus little f of 0.
01:49
To evaluate this definite interval, we start by finding our anti -derivative.
01:54
My antiderivative of f prime is just little f...