00:01
Let f be defined as f of x is equal to the score root of 9 minus x square for x greater than or equal to negative 3 and less than or equal to 0.
00:16
And the function is negative x plus 3 cosine of pi x half for x greater than 0 and less than or equal to 4.
00:27
So we want to write an equation for the line tangent to the graph of f at x equal 3.
00:38
Okay, so the first thing we got to note is that this function is piecewise function.
00:45
It has two parts or pieces.
00:47
And the tangent line we want to write down the equation.
00:54
It corresponds to x equal 3.
00:56
So we are using this piece of the function.
01:00
That is the formula negative x plus 3 cosine of pi x half because number 3 is just inside it's in the interior of the interval 0 4 where this formula is applying.
01:17
So we get to find two things.
01:19
The first thing we got to find is the image of 3 because the point through which the tangent line passes is the point 3f of 3.
01:29
So let's say the tangent line passes through the point 3 f at 3.
02:02
So we got to calculate that image of 3.
02:06
And for that we use again this piece of the function because x equals 3 belongs to the interval 04.
02:17
So we have that f at 3 is negative 3 plus negative 3 plus negative 3 plus 3 % of 3 by 1.
02:44
And that is negative 3 plus 3 times cosine of 3 by half is 0.
03:00
So we get 3 times 0 so we get negative 3 okay so the tangent line passes through the point 3 negative 3 now we need that's the first thing we we need the second thing is the slope of the tangent line we know the slope of the tangent line at x equal 3 is just a derivative of f at 3 so we can say that this slope of the tangent line to the graph of f at x equal 3 is f derivative of 3...