Evaluate each of the following. Show all your calculations.
f' ���(−2)� if ���f(x) = 2x^3�� + 4��x^2 − 5x� + 8
To evaluate f' at x = -2, we need to find the derivative of f(x) and then substitute x = -2.
First, let's find the derivative of f(x):
f(x) = 2x^3 + 4x^2 - 5x + 8
To find the derivative, we apply the power rule for each term:
d/dx (x^n) = n*x^(n-1)
For the first term, 2x^3, the derivative will be:
d/dx (2x^3) = 2 * 3x^(3-1) = 6x^2
For the second term, 4x^2, the derivative will be:
d/dx (4x^2) = 4 * 2x^(2-1) = 8x
For the third term, -5x, the derivative will be:
d/dx (-5x) = -5 * 1x^(1-1) = -5
The fourth term, 8, is a constant, so its derivative is zero:
d/dx (8) = 0
Now let's put all the derivatives together to get f'(x):
f'(x) = 6x^2 + 8x - 5
To evaluate f'(-2), substitute x = -2 into the derivative:
f'(-2) = 6(-2)^2 + 8(-2) - 5
Now let's calculate this expression step by step:
f'(-2) = 6 * 4 + (-16) - 5
= 24 - 16 - 5
= 8 - 5
= 3
Therefore, f'(-2) = 3.