The Derivative and the Tangent Line Problem Section 2.1

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

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. Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve. P

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

Definition of a Tangent Let Δx shrink from the left

Definition of a Tangent Line with Slope m

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

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

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

Note: does not mean ! 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.)

Note: 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

Theorem 2.1 Differentiability Implies Continiuty

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

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

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 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 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 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 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 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 lines at: a) x = 1 a) x = 1: b) x = -2

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

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

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

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

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

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:

Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

Derivative Example: Find for

Derivative Example: Find for

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. Graph the function on your calculator. 1 3 Select 5: Tangent( Type the x value, which in this case is 1, and then hit  4 (I changed my window) 2 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: A discontinuity (a hole or break or asymptote) A sharp corner (ex. f(x)= |x| when x = 0) 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 at x = 2. 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 Discontinuities Sharp turns Vertical tangents

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.

Example Find an equation of the line that is tangent to the graph of f and parallel to the given line. f(x) = x3 + 2 Line: 3x – y – 4 = 0

Example Find an equation of the line that is tangent to the graph of f and parallel to the given line. f(x) = x3 + 2 Line: 3x – y – 4 = 0 Taking my word for it, the derivative of the function is This is 3 when x is

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 !