"How do you find the instantaneous rate of..." solved and explained

Annie Rice

Answered question

2022-04-17

How do you find the instantaneous rate of change for

f (x) = 3 \sqrt{x}

for [4,25]?

Answer & Explanation

davelucasnp0j

Beginner2022-04-18Added 16 answers

The instantaneous rate of change is defined for each point, not in an interval. We say
"What is the instantaneous rate of change of f(x) at x=4?"
not "What is the instantaneous rate of change of f(x) for

4 \leq x \leq 25

?"
unless you are asking for all the values in that interval. Usually people ask for average rate of change in an interval.
To find the instantaneous rate of change, differentiate.

f^{'} (x) = \frac{3}{2 \sqrt{x}}

That's it. If you want to know the instantaneous rate of change at say at x=16, just plug it in.

f^{'} (16) = \frac{3}{2 \sqrt{16}} = \frac{3}{8}

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