Question

1. the graph of y = f(x) is given. estimate the instantaneous rate of change of f at x = 1.

Explanation:

Step1: Recall the concept

The instantaneous rate of change at a point is the slope of the tangent line at that point.

Step2: Estimate the tangent - line slope

Draw the tangent line at \(x = 1\) on the graph. Observe two points on the tangent - line. Let's say we can estimate two points \((0,4)\) and \((2,0)\) on the tangent line (approximate points based on the graph).
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute \(x_1 = 0,y_1 = 4,x_2 = 2,y_2 = 0\) into the formula: \(m=\frac{0 - 4}{2-0}=\frac{-4}{2}=- 2\).

Answer:

\(-2\)