Presentation on theme: "Derivative and the Tangent Line Problem"— Presentation transcript:
1Derivative and the Tangent Line Problem The beginnings of Calculus
2Tangent Line ProblemDefinition of Tangent to a Curve Now to develop the equation of a line we must first find slope
3Definition of Tangent Line with Slope m Slope of Secant LineIf f is defined on an open interval containing c, and if the limitexists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).
4Definition of Tangent Line The limit is telling us that the distance between the two points is getting smaller and smaller. If we let the limit approach zero, we are actually approaching being on one point and not two. That is how we can say the line is tangent and no longer a secant line.
5Definition of Tangent Line Find some slopes using this definition.
6Definition of the Derivative of a Function Demonstration
7Symbols for Derivative Sir Isaac Newton:f’(x) or y’Gottfried Leibniz:This is read as “the derivative of y with respect to x.”
8Finding the Derivative by the Limit Process Examples:Quadratic EquationSquare Root FunctionRational Function
9Finding the Derivative Summation A very good summation of this information
10Differentiation Definition Alternative form of derivative: (This is when given one point)
11When is a Function not Differentiable When the graph has a sharp point. (This is because the derivative (slope) from the right and the derivative from the left are different values.
12When is a Function not Differentiable When the graph has a vertical line tangent. Remember the slope of a vertical line is undefined.
13When is a Function not Differentiable Where the function is not continuous at that point.
14Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c.The converse is not true. Just because a function is continuous does not make it differentiable everywhere. (See the prior slides)