Rate of Change for the value of y

Question

If y'(8)=1/12, how much would you think y would change between x=8 and x=8.12? (Hint: y' measures the rate of change of y per unit of change of x)

I tried to use y2-y1/x2-x1 and set it equal to 1/12 but I was wasn't sure which values to use for y2 and y1. Please help!

Answer 1

Given that y'(8) = 1/12, we will use this value as the slope (since a derivative is a tangent which is the generalization of a slope -- linear approximation -- at a point).

So m = 1/12

Our line equation then becomes...

y = mx + b

Since we want to know how much y changes between x = 8 and x = 8.12, we have Δx = 0.12

y₁ = mx₁ + b

y₂ = mx₂ + b

y₂ - y₁ = m * (x₂ - x₁)

But this is nothing more than the "change in y" related to the "change in x"

Δy = m Δx

Δy = 1/12 * 0.12 = 1/12 * 12/100 = 1/100 = 0.01