Finding rate of change formulas with the given information

[Dustobusto]

Homework Statement

Let Q(t)=t². Find a formula for the average rate of change (ROC) of Q over the interval [1, t] and use it to estimate the instantaneous ROC at t = 1.

Homework Equations

For x₁ ≠ x₀, the average rate of change of y with respect to x over [x₀, x₁] is the ratio

Average ROC = Δf/Δx = f(x₁) - f(x₀) / x₁ - x₂

Finding the instantaneous rate of change is basically the same as finding the avg ROC, except instead of using the intervals that are given (say [7, 10] for example) they give you one number, and you create your own intervals by choosing numbers extremely close to that given number (say [7, 7.01] or [7, 6.99] etc.)

The Attempt at a Solution

All I can think of is plugging in the intervals as with previous problems. Plugging t into t gives you t.

Plugging 1 into t gives you one squared which is one. Then you subtract those two to get

t² - 1. The bottom portion which is x₁ - x₂ would look just like t - 1.

So maybe t squared minus one over t minus one is the formula? I suppose I could factor it out to

(t +1)(t - 1) / (t - 1) cancel out the expressions and get t + 1. So t + 1 would be the requested formula for the average rate of change? To get instantaneous ROC at t = 1, wouldn't I just plug one into "t" and get 2?

That can't be right

 

[Dustobusto]

Anyone have some ideas?

 

[ehild]

It looks right.

ehild