Spring+Week+04

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Monday 1/25
The Mean Value Theorem was the topic of the day and we found out, through an exploration, that this theorem had a lot to do with the existence of parallel tangent line and secant lines. Apparently we are going to need this theorem to prove the Fundamental Theorem of Calculus which sounds pretty important! The exploration made us think about the existence of these parallel lines depending upon the continuity and differentiability of the function and interval of x values under consideration. The exploration can be seen in the slide-show below. media type="custom" key="5225873" Unfortunately we ran out time so we didn't have chance to discuss Rolle's Theorem. Rolle's Theorem, named after Michel Rolle (1652-1719) states that : If f is continuous on [a,b] and differentiable on (a,b) AND if f(a) = f(b) = 0, then there is at least one number x = c in (a,b) such that f '(c) = 0. This is a special case of the Mean Value Theorem guaranteeing, if f is continuous and differentiable on the appropriate intervals, the existence of a local extrema for an x value between a and b if f(a) and f (b) are both zero.

**Tuesday 1/26**
Problems of the day(see below) followed by a discussion of Rolle's theorem (see Monday)



Wednesday 1/27
The best day ever! We proved the Fundamental Theorem of Calculus using Riemann Sums and the Mean Value Theorem. Tomorrow we have a test on anti-derivatives (reversing the chain rule) and definite integrals so happy studying!

Thursday1/28
Test today.

Friday1/29
We know how to find a definite integral by definition and by using the Fundamental Theorem so today we looked at some properties of definite integrals and continued to look at theri connection with the area between the graph and the x axis. The first example demonstrates that a coefficient inside the integral can be extracted outside the integral.

a graphical example of this would be to compare the area under the curve of f(x) = 3sinx on the interval [o,pi] to three time the area under the curve g(x) = sinx on the same interval. The property indicates that they would be equal.

This property suggests that reversing the bounds of a definite integral results in an opposite numerical value.

Referencing the limit definition of the definite integral, based on the limit of a Riemann sum as n approaches infinity, this is caused by the sign of the change in x or dx. If we start our rectangles form the right bound of the region instead of the left bound, as the reversed bounds indicate, dx would be a negative quantity.

Here we show that a definite integral of a function with negative and positive function values does not always equate to the area between the function and the x axis.

Also, when the function values are positive and negative on a given interval, we need to use the property in the previous graphic to calculate integrals over such bounds.