WEBVTT
1
00:00:01.439 --> 00:00:05.459 A:middle L:90%
yeah. Were given a function and were asked to
2
00:00:05.459 --> 00:00:08.269 A:middle L:90%
use part one of the fundamental theorem of calculus to
3
00:00:08.269 --> 00:00:12.519 A:middle L:90%
find the derivative of this function, the function is
4
00:00:12.539 --> 00:00:18.260 A:middle L:90%
Y equals the integral from the sine of X.
5
00:00:18.739 --> 00:00:23.579 A:middle L:90%
Yes, 21 of square root of one plus T
6
00:00:23.579 --> 00:00:30.989 A:middle L:90%
squared. Mhm. GT Mhm. Yeah, recall
7
00:00:30.989 --> 00:00:34.159 A:middle L:90%
what part one of the fundamental theorem of calculus says
8
00:00:34.840 --> 00:00:38.100 A:middle L:90%
this is that if a function little F is continuous
9
00:00:38.539 --> 00:00:44.409 A:middle L:90%
on a closed interval, how old and the function
10
00:00:44.409 --> 00:00:47.990 A:middle L:90%
big F. Which is the integral from one end
11
00:00:47.990 --> 00:00:50.859 A:middle L:90%
of that closed interval two X. Of little F
12
00:00:52.439 --> 00:00:55.179 A:middle L:90%
. Then the derivative of big F F. Prime
13
00:00:55.179 --> 00:01:00.060 A:middle L:90%
of X is little F of X. So,
14
00:01:00.539 --> 00:01:03.250 A:middle L:90%
first of all thanks. It is true that the
15
00:01:03.250 --> 00:01:10.489 A:middle L:90%
square root of one plus T square is continuous for
16
00:01:10.500 --> 00:01:18.430 A:middle L:90%
all values of T. We can also Use integral
17
00:01:18.430 --> 00:01:22.709 A:middle L:90%
properties to rewrite this as the opposite of the integral
18
00:01:22.709 --> 00:01:26.760 A:middle L:90%
from 1 to the sine of X. Of the
19
00:01:26.760 --> 00:01:32.359 A:middle L:90%
square root of one plus T squared D. T
20
00:01:33.040 --> 00:01:34.650 A:middle L:90%
. And now using part one of the fundamental theorem
21
00:01:36.239 --> 00:01:42.150 A:middle L:90%
, it follows that G. Prime not prime that
22
00:01:42.930 --> 00:01:47.250 A:middle L:90%
I guess I would say why prime of X.
23
00:01:48.140 --> 00:01:56.150 A:middle L:90%
Black? Well this is the opposite of the derivative
24
00:01:56.150 --> 00:02:00.659 A:middle L:90%
of the integral which using the chain rule this is
25
00:02:00.659 --> 00:02:07.260 A:middle L:90%
going to be yeah um the derivative of the integral
26
00:02:07.260 --> 00:02:09.250 A:middle L:90%
from 12 X. Of the square one plus T
27
00:02:09.250 --> 00:02:13.560 A:middle L:90%
squared, which is simply the square root of one
28
00:02:13.560 --> 00:02:22.349 A:middle L:90%
plus X squared times the derivative of the inside sign
29
00:02:22.349 --> 00:02:29.669 A:middle L:90%
index, which is co sign of X. Which
30
00:02:29.669 --> 00:02:32.490 A:middle L:90%
we could also rewrite as get a couple of cents
31
00:02:32.490 --> 00:02:39.430 A:middle L:90%
for this. Right? I'm sorry. Instead of
32
00:02:39.430 --> 00:02:44.629 A:middle L:90%
one plus X squared, we substitute Synnex for X
33
00:02:44.629 --> 00:02:46.909 A:middle L:90%
here. So this is squared of one plus sine
34
00:02:46.909 --> 00:02:54.240 A:middle L:90%
squared of X. And so this is negative co
35
00:02:54.240 --> 00:02:58.060 A:middle L:90%
sign of X times the square root of one plus
36
00:02:58.889 --> 00:03:02.960 A:middle L:90%
sine squared of X. That's what more people do
37
--> A:middle L:90%
.