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The Derivative and the Tangent Line Problem Section 2.1

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After this lesson, you should be able to: find the slope of the tangent line to a curve at a point use the limit definition of a derivative to find the derivative of a function understand the relationship between differentiability and continuity

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Tangent Line A line is tangent to a curve at a point P if the line is perpendicular to the radial line at point P. P Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve.

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The Tangent Line Problem Find a tangent line to the graph of f at P. f P Why would we want a tangent line??? Remember, the closer you zoom in on point P, the more the graph of the function and the tangent line at P resemble each other. Since finding the slope of a line is easier than a curve, we like to use the slope of the tangent line to describe the slope of a curve at a point since they are the same at a particular point. A tangent line at P shares the same point and slope as point P. To write an equation of any line, you just need a point and a slope. Since you already have the point P, you only need to find the slope.

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Definition of a Tangent Let Δx shrink from the left

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Definition of a Tangent Line with Slope m

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The Derivative of a Function Differentiation- the limit process is used to define the slope of a tangent line. Definition of Derivative: (provided the limit exists,) = slope of the line tangent to the graph of f at (x, f(x)). = instantaneous rate of change of f(x) with respect to x. This is a major part of calculus and we will differentiate until the cows come home! Also, Really a fancy slope formula… change in y divided by the change in x.

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Definition of the Derivative of a Function

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Notations For Derivative If the limit exists at x, then we say that f is differentiable at x. Let

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dx does not mean d times x ! dy does not mean d times y !

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does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

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(except when it is convenient to treat it that way.) does not mean times !

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The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

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A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

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Theorem 2.1 Differentiability Implies Continiuty

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function Example: Given, find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

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The Slope of the Graph of a Non-Linear Function b) x = -2 Example: Given, find f ’(x) and the equation of the tangent line at:

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The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

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The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

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The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

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The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

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Example-Continued If x = 2, the slope is, -¼. So, y = 1/4x + b. Going back to the original equation of y = 1/x, we see if x = 2, y = 1/2. So:

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Derivative Example: Find the derivative of f(x) = 2x 3 – 3x.

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Derivative Example: Find the derivative of f(x) = 2x 3 – 3x.

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Derivative Example: Find the derivative of f(x) = 2x 3 – 3x.

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Derivative Example: Find the derivative of f(x) = 2x 3 – 3x.

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Derivative Example: Find the derivative of f(x) = 2x 3 – 3x.

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Derivative Example: Find for

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Derivative Example: Find for

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Derivative Example: Find for

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Derivative Example: Find for THIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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Example-Continued Let’s work a little more with this example… Find the slope of the graph of f at the points (1, 1) and (4, 2). What happens at (0, 0)?

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Example-Continued Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1. Graph the function on your calculator. (I changed my window) Now, hit DRAW Select 5: Tangent( Type the x value, which in this case is 1, and then hit Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

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Differentiability Implies Continuity If f is differentiable at x, then f is continuous at x. Some things which destroy differentiability: 1.A discontinuity (a hole or break or asymptote) 2.A sharp corner (ex. f(x)= |x| when x = 0) 3.A vertical tangent line (ex: when x = 0)

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Where a Function is Not Differentiable : 1)A function f(x) is not differentiable at a point x = a, if there is a “corner” at a. 2.1 Differentiation Using Limits of Difference Quotients

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Where a Function is Not Differentiable: 2)A function f (x) is not differentiable at a point x = a, if there is a vertical tangent at a. 2.1 Differentiation Using Limits of Difference Quotients

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3. Find the slope of the tangent line to at x = 2. This function has a sharp turn at x = 2. Functions are not differentiable at a.Discontinuities b.Sharp turns c.Vertical tangents Therefore the slope of the tangent line at x = 2 does not exist.

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2.1 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable: 3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a. Example: g(x) is not continuous at –2, so g(x) is not differentiable at x = –2.

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4.Find any values where is not differentiable. This function has a V.A. at x = 3. Theorem: If f is differentiable at x = c, then it must also be continuous at x = c. Therefore the derivative at x = 3 does not exist.

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Example Find an equation of the line that is tangent to the graph of f and parallel to the given line. f(x) = x 3 + 2Line: 3x – y – 4 = 0

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Example Find an equation of the line that is tangent to the graph of f and parallel to the given line. f(x) = x 3 + 2Line: 3x – y – 4 = 0 Taking my word for it, the derivative of the function is This is 3 when x is

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Definition of Derivative The derivative is the formula which gives the slope of the tangent line at any point x for f(x) Note: the limit must exist –no hole –no jump –no pole –no sharp corner A derivative is a limit !

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