Section 2.1: The Derivative and the Tangent Line Problem Goals for this Section:

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1 Section 2.1: The Derivative and the Tangent Line Problem Goals for this Section: Find the slope of the tangent line to a curve at a point. Day 1 Use the limit definition to find the derivative of a function. Day 1 Understand the relationship between differentiability and continuity. Day 2 Chapter 2 1

2 Remember from Chapter 1... Tangent Line Problem In the tangent line problem, you are given a function fand a point P on its graph and are asked to find an equation of the tangent line to the graph at point P (figure page 45). The problem of finding the tangent line at a point P is equivalent to finding the slope of the tangent line at P. You can approximate this slope by using a line through the point of tangency and a second point on the curve. This line is called a secant line. As the slope of the secant line approaches the slope of the tangent line, a "limiting position" exists. Therefore, the slope of the tangent line is said to be the limit of the slope of the secant line. Chapter 2 2

3 Definition of Tangent Line with Slope m If f is defined on an open interval containing c and if the limit lim y = lim f (c + x) - f (c) = m x 0 x x 0 x exists, then the line passing through (c, f (c)) with slope m is the tangent line to the graph of f (c) at the point (c, f (c)) Example: Find the slope of the graph of f(x) = 2x - 3 at the point (2, 1). Chapter 2 3

4 Definition of the Derivative of a Function The derivative of f at x is given by f '(x) = lim f (x + x) - f (x) x 0 x provided the limit exists. For all x for which this limit exists f ' is a function of x. According to Wikipedia... In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position of a car at some point in time is the velocity, or speed, at which that car is traveling (conversely the integral of the velocity is the car's position or distance traveled). Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x. Other notations used to denote the derivative of y = f (x): f '(x) ; y' ; d [ f (x)] ; D x [y] ; dy dx dx dy = lim y = lim f (x + x) - f (x) = f '(x) dx x 0 x x 0 x Chapter 2 4

5 Example: Find the derivative of f(x) = x 3 + 2x Chapter 2 5

6 Chapter 2 6

7 Section 2.1: The Derivative and the Tangent Line Problem Goals for this Section: Find the slope of the tangent line to a curve at a point. Day 1 Use the limit definition to find the derivative of a function. Day 1 Understand the relationship between differentiability and continuity. Day 2 Chapter 2 7

8 Alternative Form of a Derivative: f '(c) = lim f (x) - f (c) x c x - c **Use this form when you are trying to find the derivative at a point... Example: Find the slope of the tangent line graph of f(x) = x 2-3 at the point (2, 1). Chapter 2 8

9 One-Sided Derivatives: A function y = f (x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits lim f (x) - f (c) and lim f (x) - f (c) x c - x - c x c + x - c exist and are equal. Chapter 2 9

10 Example: If f (x) = x 2-9 find f '(3) Chapter 2 10

11 Theorem 2.1: Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c. differentiable at point c continuous at c continuous at c differentiable at c Example: If f (x) = (2x)/(x-1) find f '(1) Chapter 2 11

12 USING YOUR GRAPHING CALC... If you were to graph the f(x) in your calculator... 2nd TRACE 6: dy/dx type in the "x" you are looking for From the home screen... MATH 8:nDerv screen should appear as follows nderv( type the equation,type the variable,type the "x") for example: nderv(x 2,x,4) = 8 Chapter 2 12