Chapter 3,4,5,6,7
Differentiation
Chapter 3 • What Can be Done With a Derivative? – Find Slope Of A Tangent Line. – Maximum And Minimum On Graph Of A Function. – Analyze Rates Of Change. – Analyze Motion On Object. – Optimize Word Problems.
Derivative by Definition • FIRST FORM OF THE DEFINITION The derivative of a function f at a number c, denoted by f '(c), is given by
Example • Using the definition above, find the derivative of f(x) = x2 – 5x + 3 at x = 2; that is, find f '(2).
SECOND FORM OF THE DEFINITION • The derivative of a function f at a number c, denoted by f '(c) is given by:
Example • Using the definition above, find the derivative of f(x) = x2 – 5x + 3 at x = 2, that is, find f '(2).
Example • For f(x) = 3x2 – 12x + 9, find f '(x), the derivative at any point.
Find the Equation of a Line Tangent to a Curve • to find an equation of the line tangent to the graph of f(x) = x3 – 6x2 + 9x – 13 at the point with x-coordinate 2.
Horizontal Tangents • Find Points on a Curve at Which Tangent Line is Horizontal • Example : Find the coordinates of each point on the graph of f(x) = x3 – 6x2 + 9x – 13 at which the tangent line is horizontal.
Alternate Notations for a Derivative
Differentiability and Continuity
Chapter 4
Example
Rolle’s Theorem Then, there exists at least one number c in • (a,b) for which f '(c) = 0.
Then, there exists at least one number c in (a,b) for which f '(c) = 0.
Example For the function f(x) = x2 – 8x + 19, find the value of c in the open interval (2, 6) that is mentioned in Rolle’s Theorem.
The Mean Value Theorem
Example For f(x) = x3 – x2 – 2x, find the value of c in the interval (–1,1), which is mentioned in the Mean Value Theorem. Solution :
Limits: Indeterminate Forms and L’Hopital’s Rule
Example 1
Example 2
Chapter 5