Not sure if I'm solving this right

[CoreyyV]

To the question
You are driving 20 miles. If f(x) is the number of minutes your trip takes when you drive x miles per hour, then f′(x) =−1200/x2. If you drive 20 miles per hour, your trip takes 60 minutes. Use the formula: f(a+h)≈f(a) +h·f′(a) to estimate how long your trip would take if you drive 23 miles per hour.

When I used the formula, I got:
f(23)≈ 60 + 3 - (1200/20^2) = 51

I was wondering if I was correct in using 20 for x in f'(x) and have the correct answer to the problem or not?

 

[Dr.Peterson]

To the question
You are driving 20 miles. If f(x) is the number of minutes your trip takes when you drive x miles per hour, then f′(x) =−1200/x2. If you drive 20 miles per hour, your trip takes 60 minutes. Use the formula: f(a+h)≈f(a) +h·f′(a) to estimate how long your trip would take if you drive 23 miles per hour.

When I used the formula, I got:
f(23)≈ 60 + 3 - (1200/20^2) = 51

I was wondering if I was correct in using 20 for x in f'(x) and have the correct answer to the problem or not?

You appear to have changed the multiplication (dot) to subtraction, by putting the negative sign outside the parentheses; but I think that's a typo, since you actually did the multiplication.

And that is correct. You are making an approximation taking a = 20 and h = 3. (Writing that down first helps in being sure of yourself.)

f(a+h) ≈ f(a) +h·f′(a), so f(20+3) ≈ f(20) + 3·f′(20) = 60 + 3(-1200/20^2) =60 + 3 (-3) = 51​

Of course, now you can calculate the exact value (using the actual formula for f(x)) and compare. It's fairly close.

 

[CoreyyV]

Thank you so much!