00:01
For this, probably we want to find the seventh degree mclaurian polynomial for f of x, which is equal to the integral from 0 to x of sine of 5t squared d t.
00:12
Now let's recall, mclaurin polynomial for a function is given by f of x that's equal to f of 0 plus f prime of 0 times x, plus f double prime of 0 times x squared over 2 factorial and so on.
00:30
Same as summation from n equals 0 to infinity of the nth derivative evaluated 0 over n factorial times x raised a power of n now let's look at our function and in here we can see that it's the integral of sine of 5t squared if we start with the integrand sign of 5t squared it will be easier to get the derivative more so if we start with a basic function sine of x.
01:02
So let's say f of x is equal to sine of x.
01:08
Note that in the integran the inner function here is 5t squared or it's in quadratic form.
01:15
So we think of x here as quadratic and if it's the case then we need only take the derivative of f until the third derivative.
01:25
Now f of 0 is sine of 0 which is equal to 0.
01:32
F prime of x is cosine of x.
01:35
So evaluating f prime at 0, this gives us cosine of 0, which is 1...