Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Secant and Tangent Lines http://www.mathscoop.com/applets/tangent-line-applet.php f (a + h) − f (a) h Slope of secant line: Slope of tangent line: f (a) f (a + h) f (a + h) − f (a) lim h!0 h Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y y = f (x) x a a + h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 f (a + h) − f (a) h Slope of secant line: Slope of tangent line: f (a) f (a + h) f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y y = f (x) x a a + h f (a + h) − f (a) h Slope of secant line: Slope of tangent line: f (a) f (a + h) f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y y = f (x) x a a + h f (a + h) − f (a) h Slope of secant line: Slope of tangent line: f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y y = f (x) f (a) f (a + h) x a a + h f (a + h) − f (a) h Slope of tangent line: f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y Slope of secant line: y = f (x) f (a) f (a + h) x a a + h h Slope of tangent line: f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y Slope of secant line: y = f (x) f (a + h) − f (a) f (a) f (a + h) x a a + h Slope of tangent line: f (a + h) − f (a) lim h!0 h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y Slope of secant line: y = f (x) f (a + h) − f (a) h f (a) f (a + h) x a a + h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y Slope of secant line: y = f (x) f (a + h) − f (a) h Slope of tangent line: f (a) f (a + h) f (a + h) − f (a) lim h!0 h x a a + h Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and Tangent Line Slope Recall, the slope of a line is given by any of the following: rise ∆y y2 − y1 run ∆x x2 − x1 Suppose we're interested in the slope of the secant line to f (x) through the points where x = a and x = a + h. y Slope of secant line: y = f (x) f (a + h) − f (a) h Slope of tangent line: f (a) f (a + h) f (a + h) − f (a) lim h!0 h x a a + h increasing It \points up." decreasing It \points down." If f 0(a) > 0, then f is at a. If f 0(a) < 0, then f is at a. If f 0(a) = 0, then f looks constant at a. It looks flat. Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. decreasing It \points down." increasing It \points up." If f 0(a) < 0, then f is at a. If f 0(a) = 0, then f looks constant at a. It looks flat. Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. If f 0(a) > 0, then f is at a. decreasing It \points down." If f 0(a) < 0, then f is at a. If f 0(a) = 0, then f looks constant at a. It looks flat. Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. If f 0(a) > 0, then f is increasing at a. It \points up." decreasing It \points down." If f 0(a) = 0, then f looks constant at a. It looks flat. Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. If f 0(a) > 0, then f is increasing at a. It \points up." If f 0(a) < 0, then f is at a. If f 0(a) = 0, then f looks constant at a. It looks flat. Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. If f 0(a) > 0, then f is increasing at a. It \points up." If f 0(a) < 0, then f is decreasing at a. It \points down." Chapter 2: Derivatives 2.2: Definition of the Derivative Derivative at a Point Definition Given a function f (x) and a point a, the slope of the tangent line to f (x) at a is the derivative of f at a, written f 0(a). f (a + h) − f (a) So, f 0(a) = lim . h!0 h f 0(a) is also the instantaneous rate of change of f at a. If f 0(a) > 0, then f is increasing at a. It \points up." If f 0(a) < 0, then f is decreasing at a. It \points down." If f 0(a) = 0, then f looks constant at a. It looks flat. a b c d Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) < 0? Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) < 0? f 0(b) < 0 Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) > 0? Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) > 0? f 0(a) > 0 and f 0(d) > 0 Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) = 0? Chapter 2: Derivatives 2.2: Definition of the Derivative Practice: Increasing and Decreasing y x a b c d Where is f 0(x) = 0? f 0(c) = 0 f (3 + h) − f (3) f 0(3) = lim h!0 h f (3 + h) (3 + h)2 − 5 − (32 − 5) = lim h!0 h 9 + 6h + h2 − 5 − 4 = lim h!0 h h2 + 6h = lim h!0 h = lim h + 6 h!0 f (3) = 6 3 + h Chapter 2: Derivatives 2.2: Definition of the Derivative Use the definition of the derivative to find the slope of the tangent line to f (x) = x 2 − 5 at the point x = 3.