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Dustobusto
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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
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