4. Let P and Q be the points in the graph of with x-coordinates a and...

 

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4. Let P and Q be the points in the graph of with x-coordinates a and b respectively. Write an equation for the distance between P and Q. Then use the Mean Value Theorem to prove there exists CE (a, b) such that this distance is dist(P,Q) = 1+[f'(c)] (ba). 5. The result you obtained in Question 4 is an approximation for the length of the graph of f. (It is the exact value when the graph of happens to be a straight line, but otherwise only an approximation.) Now use a partition of the interval [a, b] to break the graph into many smaller pieces, and use the same approximation for each piece. The result you obtain (a sum) is a better approximation for the length of the graph of f. a 2 6. Now "break the interval [a, b] into infinitely many pieces which are infinitesimally small". Obtain a formula for the length of the graph of as an integral. Hint: use the definition of the Riemann sum. 4. Let P and Q be the points in the graph of with x-coordinates a and b respectively. Write an equation for the distance between P and Q. Then use the Mean Value Theorem to prove there exists CE (a, b) such that this distance is dist(P,Q) = 1+[f'(c)] (ba). 5. The result you obtained in Question 4 is an approximation for the length of the graph of f. (It is the exact value when the graph of happens to be a straight line, but otherwise only an approximation.) Now use a partition of the interval [a, b] to break the graph into many smaller pieces, and use the same approximation for each piece. The result you obtain (a sum) is a better approximation for the length of the graph of f. a 2 6. Now "break the interval [a, b] into infinitely many pieces which are infinitesimally small". Obtain a formula for the length of the graph of as an integral. Hint: use the definition of the Riemann sum.