Workshop on the definite integral and the Fundamental Theorem of Calculus part 1.

1. Math everywhere...

2. Let for 0 ≤ x ≤ 1.
(a) Using n = 4 subintervals, Find L(4) and U(4), lower and upper estimates for
(b) Find the difference between the upper and lower estimates, U(4) - L(4).
(c) Represent U(4) - L(4) geometrically on a graph of ƒ.
(d) How large would n need to be to be sure that U(n) - L(n) ≤ 0.005?

3. Let for 0 ≤ x ≤ 1.
(a) Using n = 4 subintervals, Find L(4) and U(4), lower and upper estimates for
(b) Find the difference between the upper and lower estimates, U(4) - L(4).
(c) Represent U(4) - L(4) geometrically on a graph of ƒ.
(d) How large would n need to be to be sure that U(n) - L(n) ≤ 0.005?

4. The table below gives values of a continuous function g at several inputs x.
Estimate using a midpoint sum with three equal subintervals.
Then draw a sketch that illustrates this sum geometrically.

5. Real cars don't adhere to explicit speed formulas. What can be observed,
practically speaking, are numerical speed data, like those below. (Notice
the irregular time intervals; real data often have gaps.)
How far did the car travel in one hour? Find the best approximation you can.

6. Real cars don't adhere to explicit speed formulas. What can be observed,
practically speaking, are numerical speed data, like those below. (Notice
the irregular time intervals; real data often have gaps.)
How far did the car travel in one hour? Find the best approximation you can.