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Section2.1

# Section2.1 - Section 2.1:The Tangent and Velocity Problems...

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Section 2.1:The Tangent and Velocity Problems The theory of di ff erential calculus historically stems from two di ff erent problems - trying to determine the slope of a tangent line from its equation and trying to find the velocity of a moving object given its position as a function of time. In this section, we shall explore these two problems and explain how a solution to one of these problems is in fact a solution to them both. 1. The Tangent Problem Defining the tangent line to a function at a point is di ffi cult because it requires calculus in order to make a formal definition. Therefore, we start with a naive definition and then develop the theory to make it formal. Definition 1.1. Suppose f ( x ) is a function. Then the tangent line to f ( x ) at x = a is the line which passes through the point ( a, f ( a )) which is the closest linear approximation of f ( x ) at that point. With at least a naive definition of tangent line, we can not formally state the Tangent Problem: Question 1.2. How do we find the equation for the tangent line to f ( x ) at x = a ? Answer. We need a point and the slope. Since we already know the tangent line passes through the point ( a, f ( a )), we are given a point for free. Therefore, the tangent problem translates into the following: Question 1.3. How do we find the slope of the tangent line to f ( x ) at x = a ? If we solve this problem, then we have solved the tangent problem, so we concentrate on this. We start with an example of how we can do this explicitly.

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Section2.1 - Section 2.1:The Tangent and Velocity Problems...

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