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A differentiable function has the property that f(5) = 4 and f'(5) = 3. What is the estimate for f(4.8) using the local linear approximation for f at x=5?

Answer 1

Ezra Adams

#f(4.8)~~3.4#

We know that, #f(x+deltax)~~f(x)+f'(x)*deltax.........(star)#

Letting, #x=5, and, x+deltax=4.8 rArr deltax=-0.2" in "(star)," we get,"#

#f(5-0.2)=f(4.8)~~f(5)+f'(5)(-0.2),#

#=4-3(-0.2)=4-0.6#

#"Therefore, "f(4.8)~~3.4#

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Answer 2

Luke Alvarez

Using the local linear approximation formula:

[ f(x) \approx f(a) + f'(a)(x - a) ]

Given (a = 5), (f(a) = f(5) = 4), and (f'(a) = f'(5) = 3):

[ f(4.8) \approx f(5) + f'(5)(4.8 - 5) ] [ f(4.8) \approx 4 + 3(4.8 - 5) ] [ f(4.8) \approx 4 + 3(-0.2) ] [ f(4.8) \approx 4 - 0.6 ] [ f(4.8) \approx 3.4 ]

So, the estimate for (f(4.8)) using the local linear approximation for (f) at (x = 5) is (3.4).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.