Leonhard Euler 1707 – 1783 Leonhard Euler 1707 – 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics.

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Presentation transcript:

Leonhard Euler 1707 – 1783 Leonhard Euler 1707 – 1783 Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory.

Leonhard Euler

Euler’s collected works, the Opera Omnia, run to over 25,000 pages. After his death, Euler published … … 228 papers. In 1910 Gustav Eneström catalogued Euler’s 866 books and papers in a document that itself ran to 388 pages. Quantity: Leonhard Euler’s Work

Mathematical terms (from MATHWORLD Wolfram): Euler – 96 entries Gauss – 70 entries Cauchy – 33 entries Bach – 0 entries Quality:

The number e (1748)

Euler’s Identity (1748)

Another look at FTC Part 1

t f (t) A(x)A(x) x

Use the First Fundamental Theorem of Calculus to find the derivative of the function:

1. Derivative of an integral. First Fundamental Theorem:

2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral.

2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. First Fundamental Theorem:

1. Derivative of an integral. 2.Derivative matches upper limit of integration. 3.Lower limit of integration is a constant. The long way: First Fundamental Theorem:

1. Derivative of an integral. 2.Derivative matches upper limit of integration. 3.Lower limit of integration is a constant.

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits.

Neither limit of integration is a constant. It does not matter what constant we use! (Limits are reversed.) We split the integral into two parts. (Chain Rule.)

or CHAIN RULE