Unit 3: Differentiation Sec 1: Limit Definition of Derivative What can the Derivative do for you? Find the slope of the tangent line. Find velocity and acceleration given position.
Find Maximums and Minimums. I. Limit Definition of Derivative The derivative of f at a values of x is given by ( + ) ( ) ( )=
or ( )= ( +) ( )
provided the limit exists. For all x for which this limit exists, f is a function of x. Ex 1 Find the slope of the graph of f(x) = 2x 3. Slope = 2
Ex 2 Find the slope of the tangent line to the graph of f(x) = x + 1 at the points (0, 1) and (-1, 2). Slope = 2x Slope at (0, 1) = 0 Slope at (-1, 2) = -2
Draw the parabola and the tangent line at each of the two points to demonstrate how the slope is different. Ex 3 Find the derivative of f(x) = x + 2x.
f(x) = 3x + 2 Ex 4 Find f(x) for f(x) = x. Then find the slope of the graph of f at the point (4, 2). f(4) = 1/4
AP Practice What At is f(x)? what value is f(x) being found? HOMEWORK
Pg 102 #5 24 odds II. Alternate Form of the Derivative To find the derivative of a function when x = c, then
provided that the limit exists and f(x) is continuous at c. This form is only useful if you know the value of c and it is differentiable at that point. Ex 1: Alternate Form Find
the derivative of f(x) = x(x 1) when x = 1 (will sometimes say when c = 1). Examples of Non-Differentiability If a function is Not Continuous at x = c, then the function is Not Differentiable at x = c.
If a function has a sharp turn (different slopes) at x = c, then the function is Not Differentiable at x = c. If a function has a vertical tangent line (m is undefined) at x = c, then the function is Not
Differentiable at x = c. AP Practice Draw the graph of . At what points is f(x) discontinuous? A. -1 B. 1 C. 3 D. All
None At what points is f(x) nondifferentiable? E. Ex 2: Determine if the function is differentiable. If so, find the derivative. If not, tell where it is not
differentiable. Ex 3: Find the derivative for c = -1 Ex 3: Using the Definition of Derivative Does f(x) have a derivative at x
= 1? Must test to see if it is continuous first and then make sure the derivatives (right & left limits) are equal. HOMEWORK Pg 86
103 #61-67 odds, 68-80, 85,