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

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Aim: The Tangent Problem & the DerivativeCourse: Calculus 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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Aim: The Tangent Problem & the DerivativeCourse: Calculus Tan y x 1 1 radius = 1 center at (0,0) (x,y) cos , sin cos sin 1

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Aim: The Tangent Problem & the DerivativeCourse: Calculus slope is steep! slope is level: m = 0 Tangents to a Graph (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) (x 4, y 4 ) slope is falling: m is (-) Unlike a tangent to a circle, tangent lines of curves can intersect the graph at more than one point.

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Finding the Slope (tangent) of a Graph at a Point 1 2 This is an approximation. How can we be sure this line is really tangent to f(x) at (1, 1)? (1, 1)

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Slope and the Limit Process (x + h,f(x + h)) A more precise method for finding the slope of the tangent through (x, f(x)) employs use of the secant line. h h is the change in x f(x + h) – f(x) f(x + h) – f(x) is the change in y This is a very rough approximation of the slope of the tangent at the point (x, f(x)). x, f(x)

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Slope and the Limit Process x, f(x) h f(x + h) – f(x) (x + h,f(x + h)) h is the change in x f(x + h) – f(x) is the change in y 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).

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

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

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Aim: The Tangent Problem & the DerivativeCourse: Calculus 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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Aim: The Tangent Problem & the DerivativeCourse: Calculus Definition of the Derivative 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)). 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”.

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Finding a Derivative Find the derivative of f(x) = 3x 2 – 2x. factor out h

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Do Now: Find the equation of the line tangent to Aim: What is the connection between differentiability and continuity?

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

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Differentiability and Continuity 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. By definition the derivative is a limit. If there is no limit at x = c, then the function is not differentiable at x = c. Is this step function differentiable at x = 1?

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Differentiability and Continuity If f is differentiable at x = c, then f is continuous at x = c. Is the Converse true? If f is continuous at x = c, then f is differentiable at x = c. NO

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

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Graph with a Vertical Tangent Line f(x) = x 1/3 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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Aim: The Tangent Problem & the DerivativeCourse: Calculus Differentiability Implies Continuity abcd f is not continuous at a therefore not differentiable f is continuous at b & c, but not differentiable corner vertical tangent f is continuous at d and differentiable

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Aim: The Tangent Problem & the DerivativeCourse: Calculus Summary

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