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Published byLeila Axley Modified over 3 years ago

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**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)?

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**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

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

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**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.

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**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)?

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**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)).

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**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

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**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.

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**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)).

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**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

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**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

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**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)).

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**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)).

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**Find the derivative of f(x) = 3x2 – 2x.**

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

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**Aim: What is the connection between differentiability and continuity?**

Do Now: Find the equation of the line tangent to

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**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

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**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.

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**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.

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**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)

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**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.

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**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

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**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.

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