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How do you use part 1 of the fundamental theorem of calculus to find f ' (4) for #f(x) = int sqrt ((t^2)+3) dt# from 1 to x?

Answer 1

Evelyn Agard

FTC 1 tells us that #f'(x) = sqrt(x^2+2)#

So #f'(4) = sqrt(4^2+2) = sqrt18 = 3sqrt2#

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Answer 2

Evelyn Agard

To find ( f'(4) ) for ( f(x) = \int_{1}^{x} \sqrt{t^2 + 3} , dt ) using Part 1 of the Fundamental Theorem of Calculus, first evaluate the integral at the upper limit ( x ) to get ( f(x) = F(x) - F(1) ). Then differentiate ( F(x) ) with respect to ( x ) to find ( f'(x) ), and finally substitute ( x = 4 ) to get ( f'(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.