AP Calculus

*disclaimer: these notes are unfortunately belated due to login issues and personal computer problems. my apologies.

In class we went over the previous quiz and homework. We then focused on three main questions:

1) When do you use the trapezoidal rule?

2) What is a definite integral?

3) What exactly is the trapezoidal rule?

The trapezoidal rule is used to find an approximate definite integral. The definite integral, geometrically speaking, is the area beneath the curve of a given function. Algebraically, it is the product of x and f(x). For instance, if one were to to use calculate the definite integral of velocity (f(x)) over time (x), the units of the answer would be velocity (eg. meters/sec) * time (eg. seconds) = distance (seconds cancel, left with meters). A definite integral could also represent volume (eg. distance (x=m) * area (f(x)=m^2) = volume (x*f(x)=m^3).

Let’s take a look at fig 1.1:

fig 1.1

We use this rule to calculate total area, or the integral:

A = ((b1 + b2)/2) * h

or, the area is the total area of all the trapezoids, each trapezoid being the average of the bases times the height.

We can also use a Traprule program, which, if you have a function on y1, you can easily find the total area of the trapezoids. You just plug in L (lower bound, the value of x at the beginning of the curve), U (upper bound), and N, the number of trapezoids. As you will note, the more trapezoids there are, the more accurate the area. Looking at fig 1.1, there is a margin of error where the line is deviated from the curve.

We also were introduced to limits. We noticed that if a function is simplified to

0/0 = indeterminate, but still has a limit

0/5 = 0

and 5/0 = undefined (asymptote)

Limits were to later be explored.

(HW (from ages ago): Quiz on Thursday, 45 points on Chapter 1.1 – 1.4, know trig values and unit circle. Pp. 28 – 31, all quickies. 1 – 19 odd.)

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