00:01
Okay, so we have this function, g of x, which is the cube root of 1 plus x.
00:05
We want to find the linear approximation at a equals 0.
00:09
So in general, the linear approximation of a function is the following, g of a plus g prime of a times x minus a.
00:19
In our case, a is equal to 0, so we have g of 0 plus g prime of 0 times x.
00:27
So we need to find g of 0 and g prime of 0.
00:31
So firstly, g of 0 is the cube root of 1 plus 0, which is the cube root of 1, which is 1.
00:39
Now we need to find a g prime.
00:42
So if i rewrite g of x as 1 plus x to the power of a third, the cube root, g prime of x, taking the derivative is a third.
00:52
So bring the power down 1 plus x and then subtract 1 from the power, so minus 2 thirds.
00:59
So g prime of 0.
01:02
It's going to be a third times 1 plus 0 to the minus 2 thirds...