Question

Find the derivative from the left at x = 3. a) f' (3) b) lim h→0 f(3) - f(3-h) / h c) lim x→3⁻ f'(x) d) lim x→3⁺ f'(x)

Answer 1

Option b, y = x², is the correct answer as the slope of this function, given by 2x, is positive and decreasing in magnitude as x approaches 3 from the left.

Step-by-step explanation:

The question involves finding the one-sided derivative of a function from the left at x = 3. Since we are given that at x = 3, the function f(x) has a positive value and a positive slope that is decreasing, we need to examine the options for the function f(x) to determine which matches these conditions.

• For option a, y = 13x, the slope is constant and does not vary with x.
• For option b, y = x², the slope is given by the derivative f'(x) = 2x, which is positive and decreasing in magnitude as x approaches 3 from the left.

Therefore, option b, y = x², correctly corresponds to the function f(x).