Let f(x) = 3(x-1)^3 + 4(x-1)^2 + 2(x-1) -7
(a) Determine f'''(1) without taking any derivatives. Explain your method.
(b) Determine the general Taylor polynomial of f aobut c =2 by computing any for each k, using the formula above for f(x).
(a) To determine f'''(1) without taking any derivatives, we can use the concept of the derivative as the rate of change of a function.
First, recall that the first derivative f'(x) represents the rate of change of a function at a given point. In this case, f(x) is defined as 3(x-1)^3 + 4(x-1)^2 + 2(x-1) -7, so f'(x) represents the rate of change of f(x) at any point x.
Next, the second derivative f''(x) represents the rate of change of the first derivative. In other words, it tells us how the rate of change of the function's slope is changing. If f'(x) is a linear function, f''(x) would be a constant.
Finally, the third derivative f'''(x) represents the rate of change of the second derivative. So f'''(1) would tell us how the rate of change of the second derivative is changing at x = 1.
Using this information, we can observe that f''(x) is a constant since f'(x) is a linear function. Therefore, the rate of change of the slope of f(x) is not changing. Hence, f'''(x) would be equal to 0 at any point.
So, f'''(1) is equal to 0.
(b) To determine the general Taylor polynomial of f about c = 2, we need to compute the coefficients for each term using the formula for f(x).
The formula for the Taylor polynomial of f(x) about c is given by:
P(x) = f(c) + f'(c)(x - c) + f''(c)(x - c)^2 / 2! + f'''(c)(x - c)^3 / 3! + ...
In our case, c = 2, and we need to find the coefficients for each term in the expansion.
First, we find the value of f(2):
f(2) = 3(2-1)^3 + 4(2-1)^2 + 2(2-1) - 7
= 3(1)^3 + 4(1)^2 + 2(1) - 7
= 3 + 4 + 2 - 7
= 2
Next, we find the value of f'(2):
f'(x) = 9(x-1)^2 + 8(x-1) + 2
f'(2) = 9(2-1)^2 + 8(2-1) + 2
= 9(1)^2 + 8(1) + 2
= 9 + 8 + 2
= 19
We can repeat this process to find the values of higher derivatives up to f'''(2) if needed.
Using these values, we can now construct the general Taylor polynomial:
P(x) = f(2) + f'(2)(x - 2) + f''(2)(x - 2)^2 / 2! + f'''(2)(x - 2)^3 / 3! + ...
P(x) = 2 + 19(x - 2) + f''(2)(x - 2)^2 / 2! + f'''(2)(x - 2)^3 / 3! + ...
The specific terms involving f''(2) and f'''(2) can be expanded further if needed, but this is the general form of the Taylor polynomial of f about c = 2.