Presentation on theme: "The Derivative and the Tangent Line Problem"— Presentation transcript:
1 The Derivative and the Tangent Line Problem Section 2.1
2 After this lesson, you should be able to: find the slope of the tangent line to a curve at a pointuse the limit definition of a derivative to find the derivative of a functionunderstand the relationship between differentiability and continuity
3 Tangent LineA 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
4 The Tangent Line Problem Find a tangent line to the graph of f at P.Why would we want a tangent line???fRemember, 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
5 Definition of a Tangent Let Δx shrink from the left
7 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.
15 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
43 DerivativeExample: Find forTHIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
44 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)?
45 Example-ContinuedLet’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1.Graph the function on your calculator.13Select 5: Tangent(Type the x value, which in this case is 1, and then hit 4(I changed my window)2Now, hit DRAWHere’s the equation of the tangent line…notice the slope…it’s approximately what we found
46 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)
47 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.
48 2.1 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable:2) A function f (x) is not differentiable at a pointx = a, if there is a vertical tangent at a.
49 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 atDiscontinuitiesSharp turnsVertical tangents
50 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 notcontinuous at –2,so g(x) is notdifferentiable at x = –2.
51 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.
52 ExampleFind an equation of the line that is tangent to the graph of f and parallel to the given line.f(x) = x Line: 3x – y – 4 = 0
53 ExampleFind an equation of the line that is tangent to the graph of f and parallel to the given line.f(x) = x Line: 3x – y – 4 = 0Taking my word for it, the derivative of the function isThis is 3 when x is
54 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 existno holeno jumpno poleno sharp cornerA derivative is a limit !