Question

example 1: the figure shows the graph of a function ( f ) in the ( xy )-plane with three labeled points. order the rates of change of ( f ) at the three labeled points from least to greatest.

Explanation:

Step1: Recall rate of change definition

The rate of change of a function \( f \) at a point is the slope of the tangent line to the graph of \( f \) at that point. A positive slope means the function is increasing, a negative slope means it's decreasing, and the steeper the slope (in absolute value), the larger the magnitude of the rate of change.

Step2: Analyze slope at each point

• Point A: The graph is increasing at point A (since moving from left to right, the function goes up). The tangent line here has a positive slope.
• Point B: The graph is decreasing at point B (moving from left to right, the function goes down). The tangent line here has a negative slope.
• Point C: The graph is increasing at point C (moving from left to right, the function goes up from the minimum). The tangent line here has a positive slope, and it's steeper than at point A (since the graph is rising more steeply at C than at A).

Step3: Compare slopes

Negative slope (at B) is less than positive slopes (at A and C). Between A and C, the slope at C is steeper (more positive) than at A. So the order from least to greatest rate of change is the order of their slopes: \( \text{Rate of change at B} < \text{Rate of change at A} < \text{Rate of change at C} \).

Answer:

The rates of change of \( f \) at the three labeled points from least to greatest are \( \boldsymbol{f'(B) < f'(A) < f'(C)} \) (or in terms of the points: Rate of change at B, Rate of change at A, Rate of change at C).