The total cost C(x) (in dollars) incurred by Aloha Company in manufacturing x surfboards a day is given by the following function. C(x) = −8x2 + 500x + 130 (0 ≤ x ≤ 15) (a) Find C'(x).

Answer 1

When applying a derivative operator to a polynomial, you apply it to each term independently. Let's first apply it to the first term, which is of the form ab, where a is a constant and b is a function of x. We therefore have (ab)' (where ' means we take the derivative), which becomes a(b') because of the constant multiple rule. b is x^2, so its derivative is 2x because of the power rule. Therefore, the first term becomes 2ax.

Now let's apply the derivative operator to the second term, which is also of the form ab, but to avoid confusion let's use the letters c and d. Then (cd)' becomes c(d'). d is x, so its derivative is 1. Therefore, the second term becomes c.

Now let's apply the derivative operator to the third term, which is merely a constant. The derivative of a constant is 0, so that term disappears. We are then left with C' = 2ax + c. I'll let you substitute the appropriate values for a and c.

Answer 2

The function given is a standard polynomial function. The standard rules for taking the derivative of a polynomial function apply

For a term ax^n, the derivative of the term is anx^(n-1). The derivative of any constant c = 0

Applying these rules gives

C'(x) = -16x+500