Question

Given the function f(x) = x² - 2x, which of the following is the correct limit definition of f'(2)?

a. Limit as h approaches 0 of [(2 + h)² - 2(2 + h) - (2² - 2 * 2)] / h
b. Limit as h approaches 2 of [(2 + h)² - 2(2 + h) - (2² - 2 * 2)] / h
c. Limit as h approaches 0 of [2² - 2 * 2 - ((2 + h)² - 2(2 + h))] / h
d. Limit as h approaches 2 of [2² - 2 * 2 - ((2 + h)² - 2(2 + h))] / h

Answer 1

The correct limit definition of f'(2) is option a, which applies the limit as h approaches 0 to the difference quotient [(2 + h)² - 2(2 + h) - (2² - 2 × 2)] / h. This expression accurately represents the definition of the derivative for the function f(x) = x² - 2x at x=2.

Step-by-step explanation:

The correct limit definition of f'(2) for the function f(x) = x² - 2x is obtained using the definition of the derivative. Applying the limit as h approaches 0 to the difference quotient of the function we get:

Option a. Limit as h approaches 0 of [((2 + h)² - 2(2 + h)) - (2² - 2 × 2)] / h

This expression is derived from the difference quotient which is:

1. f(x + h) - f(x)

2. Divided by h

3. And lastly taking the limit as h approaches 0.

In this case, f(x + h) becomes (2 + h)² - 2(2 + h) and f(x) is 2² - 2 × 2. When these terms are plugged into the difference quotient and we take the limit as h approaches 0, we get the derivative of the function at x=2. Therefore, the correct option from the given choices is a.