Aim: What do slope, tangent and the derivative have to do with each other? Do Now: What is the equation of the line tangent to the circle at point (7, 8)?

A tangent to a circle is a line in the plane of Tangents & Secants A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. A secant of a circle is a line that intersects the circle in two points. B C

Tan  1 y radius = 1 center at (0,0) (x,y) cos , sin  -1 sin  1 -1

slope is falling: m is (-) Tangents to a Graph (x3, y3) slope is level: m = 0 (x2, y2) slope is falling: m is (-) (x4, y4) (x1, y1) slope is steep! Unlike a tangent to a circle, tangent lines of curves can intersect the graph at more than one point.

Finding the Slope (tangent) of a Graph at a Point 2 (1, 1) 1 This is an approximation. How can we be sure this line is really tangent to f(x) at (1, 1)?

Slope and the Limit Process A more precise method for finding the slope of the tangent through (x, f(x)) employs use of the secant line. (x + h, f(x + h)) f(x + h) – f(x) f(x + h) – f(x) is the change in y x, f(x) h h is the change in x If h is change in x, what are coordinates of second point? What is expression for change in y? Difference Quotient! How can the approximation get better? This is a very rough approximation of the slope of the tangent at the point (x, f(x)).

Slope and the Limit Process h is the change in x f(x + h) – f(x) is the change in y (x + h, f(x + h)) f(x + h) – f(x) x, f(x) As (x + h, f(x + h)) moves down the curve and gets closer to (x, f(x)), the slope of the secant more approximates the slope of the tangent at (x, f(x). h

Slope and the Limit Process h is the change in x f(x + h) – f(x) is the change in y (x + h, f(x + h)) What is happening to h, the change in x? x, f(x) f(x + h) – f(x) h It’s approaching 0, or its limit at x as h approaches 0.

Slope and the Limit Process As h  0, the slope of the secant, which approximates the slope of the tangent at (x, f(x)) more closely as (x + h, f(x + h)) moved down the curve. At reaching its limit, the slope of the secant equaled the slope of the tangent at (x, f(x)).

Definition of slope of a Graph The slope m of the graph of f at the point (x, f(x)) , is equal to the slope of its tangent line at (x, f(x)), and is given by provided this limit exists. difference quotient

Find the slope of the graph f(x) = x2 at the point (-2, 4). Model Problem Find the slope of the graph f(x) = x2 at the point (-2, 4). set up difference quotient Use f(x) = x2 Expand Simplify Factor and divide out Simplify Evaluate the limit

Slope at Specific Point vs. Formula What is the difference between the following two versions of the difference quotient? (1) Produces a formula for finding the slope of any point on the function. (2) Finds the slope of the graph for the specific coordinate (c, f(c)).

Definition of the Derivative The function found by evaluating the limit of the difference quotient is called the derivative of f at x. It is denoted by f ’(x), which is read “f prime of x”. The derivative of f at x is provided this limit exists. The derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x,f(x)).

Find the derivative of f(x) = 3x2 – 2x. Finding a Derivative Find the derivative of f(x) = 3x2 – 2x. factor out h

Aim: What is the connection between differentiability and continuity? Do Now: Find the equation of the line tangent to

Differentiability and Continuity What is the relationship, if any, between differentiability and continuity? f(x) is a continuous function (x, f(x)) f(x) – f(c) x – c x c (c, f(c)) Is there a limit as x approaches c? YES alternative form of derivative

Differentiability and Continuity Is this step function differentiable at x = 1? By definition the derivative is a limit. If there is no limit at x = c, then the function is not differentiable at x = c. Does this step function, the greatest integer function, have a limit at 1? NO: f(x) approaches a different number from the right side of 1 than it does from the left side.

Differentiability and Continuity If f is differentiable at x = c, then f is continuous at x = c. NO Is the Converse true? If f is continuous at x = c, then f is differentiable at x = c.

Graphs with Sharp Turns – Differentiable? alternative form of derivative f(x) = |x – 2| m = 1 m = -1 Is this function continuous at 2? YES Is this function differentiable at 2? One-sided limits are not equal, f is therefore not differentiable at 2. There is no tangent line at (2, 0)

Graph with a Vertical Tangent Line f(x) = x1/3 Is f continuous at 0? YES Does a limit exist at 0? NO f is not differentiable at 0; slope of vertical line is undefined.

Differentiability Implies Continuity corner vertical tangent a b c d f is not continuous at a therefore not differentiable f is continuous at b & c, but not differentiable f is continuous at d and differentiable

Summary 1. If a function is differentiable at x = c, then it is continuous at x = c. Thus, differentiability implies continuity. 2. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Thus continuity does not imply differentiability.