Find the slope of the tangent line to a curve at a point.
Use the limit definition to find the derivative of a function.
Understand the relationship between differentiability and continuity.
2.1 The Derivative and the Tangent Line Problem
The need for calculus:
To find the changing slope for a curve
To find velocity and acceleration for moving objects
To find maximum and minimum points
To find area of an irregular shape
This section will focus on finding slope of a tangent line.
The tangent line to a circle is the line perpendicular to the radius through the point. A tangent is ??? Touching but not crossing the curve? Touching curve at only one point?
In 1.1 we approximated the slope of the tangent line through a point on a curve by approximating the slope of the secant line through the point and a nearby point. (Review powerpoint 1.1 if needed). The denominator is Δ x, the small change in x, and Δ y = f(c+ Δ x) – f(c) is the small change in y.
As you let Δ x get infinitely small, through the limiting process, you get the slope. The slope of the tangent line to the graph of f at the point (c, f(c)) is called the slope of the graph of f at x=c
Ex 1 p. 98 The slope of the graph of a linear function Find the slope of the graph of f(x) = 3x – 5 at the point (2, 1) But we should already have known this, right?
Ex 2 p. 98 Tangent lines to the graph of nonlinear functions To avoid doing the same kind of thing twice, let (c, f(c)) be some arbitrary point on graph. Find the slopes for at (0, 1) and (-1, 2) So for (0, 1), c=0 and m=2 ٠ 0; for (-1, 2), c= -1 and m=2 ٠ -1
This definition doesn’t cover possibility of vertical tangent lines. For vertical tangent lines, if f is continuous at c, and or Then you know you have a vertical tangent line.
Now we are ready for a big breakthrough . . . (drum roll please) Be sure to notice that the derivative also results in a function, f ′ ( x) , which is read “f prime of x”
Other notations for f ′(x) are: The process of finding a derivative is called differentiation. dy/dx is read as the derivative of y with respect to x, or dy dx
Ex 3 p. 100 Finding derivative by the limit process Find the derivative of f(x) = x 3 - 5x This is a function which can output slopes for any x chosen on the graph of f(x); In other words, a function machine!
Ex 4 p 100 Using derivative to find the slope at a point. Then find the slope of f at points (1,1), (4,2). Describe the behavior of f at (0,0)
Ex 5 p. 101 Finding the derivative of a function Find the derivative with respect to t for the function y = 3/t. Find dy/dt Multiply by LCD of all denominators, t(t+ Δ t) as “one”
Differentiability and Continuity If a function is not continuous at x=c it is also not differentiable at x=c. Alternate form: derivative at x=c is and provided the one-sided limits exist and are equal.
It is true that differentiability implies continuity, but the converse is not necessarily true. If a function is continuous at a point, it is not always differentiable at that point. Examples 6, 7 will demonstrate that.
Ex 6 p. 102 Graph with sharp turn f(x) = | x-3 | this is continuous at x=3 but not differentiable at x=3 These one-sided limits are not equal, so f(x) is not differentiable.
Ex 7 p. 102 Graph with vertical tangent line This is continuous at x=0, but The tangent line is vertical and the function is not differentiable at x=0