Table of Contents 9. Section 3.1 Definition of Derivative.

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Presentation transcript:

Table of Contents 9. Section 3.1 Definition of Derivative

Section 3.1 Definition of Derivative Essential Question – What is a derivative?

Calculus overview reminder 3 main concepts Limits Derivatives Integrals

Slope of secant line Remember back to Section 2.1…. As h shrinks to zero, slope of secant line approaches slope of tangent line http://faculty.wlc.edu/buelow/calc/nt2-1.html Slope of tangent line =

Difference quotient The formula for slope of a secant line (without the limit) is called the difference quotient We will use it A LOT when we differentiate

Derivative The formula for slope of a tangent line (with the limit) is called the derivative Can also be written

All these have the same meaning Slope of f(x) at x = a Slope of tangent line to f(x) at x = a Instantaneous rate of change of f(x) at x = a Derivative at x = a

Example Find the slope of y=x2 at the point x=2 and the equation of the tangent line at this point. This is the slope, use it to find the equation of the tangent line.

Example cont…..

Example Find the derivative of f(x) = x2

Example Find the derivative of f(x) = 4x2-7x at x=3

Example cont. Find the derivative of f(x) = 4x2-7x at x=3 (use second equation)

Example Find the derivative of f(x) = 1/x at x=2

Example cont. Find the derivative of f(x) = 1/x at x=2 (use second equation)

Example cont…. Now find the equation of the tangent line at x=2

Example cont…. Now find the equation of the normal line at x=2 Normal lines are perpendicular to a point

Derivatives of constant functions If f(x) = b is a constant function, f’(a)=0 for all a f(x)=b

Derivatives of linear functions If f(x) = mx + b is a linear function, f’(a)=m for all a f(x)=mx + b

Assignment Pg 124: #1-7 odd, 8, 9-29 odd, 41, 43, 52, 53-57 odd, 61, 63