Ch. 2.1: The Derivative and the Tangent Line Problem

Four Major Math Problems that Made Calculus

• Tangent line problem

• Velocity and acceleration problem

• Minimum and Maximum problem

• Area problem

Each problem involves the notion of a limit → calculus is introduced with any of the four problems

Tangent Line Problem

• To solve the problem of finding the tangent line at point P, I need to fid the slope of the tangent line at point P

• Approximate the slope using a secant line through a point of tangency and second point on the curve

Slope Formula of Tangent Line Problem

• c,f(c) → point of tangency

• Second point on the graph of f (above)

• Slope of secant line (below)

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• Right-hand side of the equation → difference quotient

• Denominator is the change of x

• Numerator is the change in y

How to Find More Accurate Approximations of Slope of Tangent Line

By choosing points that are closer to the point of tangency

Definition of Tangent Line with Slope (m)

• Slope of the tangent line to the graph of f at the point of (c,f(c)) is also called the slope of the graph of f at x=c

Example of Slope of a Graph of a Linear Function

f(x) = 2x-3 when c=2

Vertical Tangent Line

If f is continuous at c and (below) is true

• Vertical line x=c passing through (c,f(c)) is a vertical tangent line to the graph of f

• Example

• When the domain of f is the closed interval [a,b], the definition of the vertical tangent line can include the endpoints by considering continuity and limits from the right (for x=a) and from the left (x=b)