Suppose that f(x) is a function with f(120)=40 and f'(120)=4. Estimate f(118.5).
f(120) -1.5*f'(120)
= 40 - 1.5*4 = 34
You are approximating the function by a straight line tangent at x = 120.
To estimate f(118.5), we can use the information given: f(120)=40 and f'(120)=4. The derivative of a function represents its rate of change or slope at a specific point. In this case, f'(120) = 4 tells us that the function is increasing at a rate of 4 units for every 1 unit increase in x at x = 120.
To estimate f(118.5), which is a point just before x = 120, we can make use of this rate of change. Since there is a 1.5 unit decrease in x from 120 to 118.5, we can estimate a corresponding decrease in the function value.
We'll start with the given information: f(120) = 40. Since the function is increasing at a rate of 4 units for every 1 unit increase in x at x = 120, we can estimate that for every 1 unit decrease in x, the function decreases by 4 units.
So, for the 1.5 unit decrease from 120 to 118.5, we can estimate a decrease of 6 units in the function value. Therefore, we can estimate f(118.5) to be:
f(118.5) ≈ f(120) - 6
≈ 40 - 6
≈ 34
Therefore, f(118.5) is estimated to be 34.