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(6 points) Use part 1 of tbe Flundamental Theorem of Calculus to fnd 9 (2):g(2) =8in %? &...

Question

(6 points) Use part 1 of tbe Flundamental Theorem of Calculus to fnd 9 (2):g(2) =8in %? &

(6 points) Use part 1 of tbe Flundamental Theorem of Calculus to fnd 9 (2): g(2) = 8in %? &



Answers

Use the Fundamental Theorem of Calculus given in Theorem 5.5 .2 to find the indicated derivative. $$ \frac{d}{d x} \int_{3}^{6 x-1} \sqrt{4 t+9} d t $$

We have G f R is equal to the integral from zero to our of the square root of X squared plus four D X. Now we're going to go ahead and use the fundamental theorem of calculus. Take the derivative with respect to our that's going to be just the in a grand evaluated at our

For this problem, we are asked to use the fundamental theorem of calculus to find the derivative of the integral from X 29 of the cubed root of U squared plus two D. U. So the first thing that we can note is that the expression above, based on the fundamental theorem of calculus, we know that it's going to equal capital F of nine minus capital F of X. Where capital F is the anti derivative of our into grand. Now taking the derivative of this abstract version, we can see that we will just be left with negative F prime of X because capital F of nine will be a constant. And by definition F prime of X is going to be just our original function with an X instead of A you. So our final result is going to be the negative of the cube root of X squared plus two.

To use Part two of the fundamental theorem of calculus, we first need to find the anti derivative and then evaluate both the upper and lower limit and find the difference. In this case the anti derivative of one is X, evaluating from negative 2 to 3 gives us three minus negative two for a value of five, find a financial.

Function g of X equal to into you. Of course, T square, DT 12 X. So in this kitchen, we have to use part first for fundamental Europe of calculus to find the derivative of the function. So here we know that mhm the fundamental theorem of calculus part first we have to use here. So we know that the fundamental theorem of calculus Spot force which is D in triangle fft DT upon p X here 82 x equal to a full facts. So here we use that term and we have to differentiate g of X to the both sides with respect to X. Then we get derivative of cheap X equal to B and Regal. Of course, the Squire DT 12 eggs upon D x. Okay, now here you can see that to the Tehran we get f of X. So here we're plugging equal to x. So here, report X then we get calls. Mm x is quiet so we can say the everybody of G of X equal to, of course, X squared. It is all final answer


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