2 1 The Derivative and the Tangent Line

After this lesson, you will be able to: • find the slope of the

Tangent Line A line is tangent to a curve at a point P if

The Tangent Line Problem Find a tangent line to the graph of f at

Definition of a Tangent • Let Δx shrink from the left

Definition of a Tangent Line with Slope m p. 125

The Derivative of a Function Differentiation- the limit process is used to define the

Definition of the Derivative of a Function p. 127

Notations For Derivative Let If the limit exists at x, then we say that

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

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

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

The derivative is the slope of the original function. The derivative is defined at

A function is differentiable if it has a derivative everywhere in its domain. It

The Slope of the Graph of a Line Example: Find the slope of the

The Slope of the Graph of a Non-Linear Function Example: Given , find f

The Slope of the Graph of a Non -Linear Function Example: Given , find

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

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

Example-Continued Let’s work a little more with this example… Find the slope of the

Example-Continued Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x

Differentiability Implies Continuity If f is differentiable at x, then f is continuous at

2. 1 Differentiation Using Limits of Difference Quotients • Where a Function is Not

2. 1 Differentiation Using Limits of Difference Quotients • • • Where a Function

3. Find the slope of the tangent line to This function has a sharp

2. 1 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable:

4. Find any values where is not differentiable. This function has a V. A.

DEFINITION OF DERIVATIVE • The derivative is the formula which gives the slope of

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2. 1 The Derivative and the Tangent Line Problem pp. 124 -125

After this lesson, you will 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

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.

The Tangent Line Problem Find a tangent line to the graph of f at P. Why would we want a tangent f line? ? ? P 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.

Definition of a Tangent • Let Δx shrink from the left

Definition of a Tangent Line with Slope m p. 125

The Derivative of a Function Differentiation- the limit process is used to define the slope of a tangent line. P. 127 Definition of Derivative: This is a major part of calculus and we will differentiate a lot!!!!! Also, (provided the limit exists, ) Really a fancy slope formula… change in y divided by the change in x. = 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.

Definition of the Derivative of a Function p. 127

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

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

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

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

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

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. p

Differentiability Implies Continuity

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

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:

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

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

Derivative Example: Find for

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

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)?

Example-Continued Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1. 1 Graph the function on your calculator. 3 4 2 Select 5: Tangent( Type the x value, which in this case is 1, and then hit (I changed my window) Now, hit DRAW Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

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)

2. 1 Differentiation Using Limits of Difference Quotients • 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 • • • 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.

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

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.

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

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 !