Fall+Week+02

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Monday 8/24
Our problem of the day was an application of the trapezoidal rule. We were presented with a table of values representing the velocity of a train at various times within a ten-second interval. The question asked us to estimate the definite integral of the function,to estimate the instantaneous rate of change of the function, and to interpret the meaning of each of these answers.

Following the discussion, students requested another problem of the day; an instantaneous rate of change. The one selected is shown in the graphic below. Business of the day was to demonstrate that not only are limits critical to calculating the derivative of a function but they are also fundamental to the calculation of a definite integral of a function. The example taken was a specific function over a specific interval so that the algebra involved wouldn't be too tough to handle! The graphic below shows the steps taken to find the definite integral of f(x) = x² on the interval [0,2]. The set up required us to think about sketching an infinite number of trapezoids in the region under the curve and between the x axis; to do this we started out with n trapezoids and then thought about the limit of our expression for the sum of the area of n trapezoids as n approaches infinity.

"Normal" set- up for the trapezoidal rule.

x values of the sub-intervals are related to the width of the trapezoids by a factor of 1,2, 3 .....(n-1) respectively.

Applying the fiunction f(x) = x²

Factoring out (4/n²) fro each term

Using the formula for the first n-1 square numbers

Distributing the 2/n

Algebra

Evaluating the limit!

HOW COOL IS THAT!!!!!!!

Tuesday 8/25
Problem of the day: Following this we went over investigation 2.1 and worked on section 2.2 which are introductions to the formal definition of limits. The explorations are shown below. media type="custom" key="4279015"

Wednesday 8/26
Review for our test tomorrow!

Thursday 8/27
Test followed by NO HOMEWORK !!!!!!

Friday 8/28
Lots of limits was the title of the smartnotebook which we worked through in class, learning some algebraic techniques for evaluating limits of the form 0/0 infinity/infinity and k/0 for non-zero values of k. media type="custom" key="4279019"

Homework Worksheet