9/7/07

Sometimes Limits Don't Exist
Provided the limit exists, the definition of a derivative is

However, the limit might fail to exist in four conditions:
1. Corner – one-sided derivatives are different
2. Cusp – the slopes of the secant lines approach ∞ from one side and –∞ from the other
3. Vertical tangent – the slopes of the secant lines approach either ∞ or –∞ from both sides
4. Discontinuity

For examples of what each of these conditions look like, see pages 105-106 in the textbook.

Don’t worry about this until it comes up in problems – you can still take the derivative of the equations!

Differentiability
Differentiability implies two things, which will become more important later:
1. Local Linearity
Graph a function, say y=(x-2)² + 4. Zoom in a lot on one area of this parabola and it looks linear. 2. Local continuity

Data
If you are given a table of values instead of an equation, you can still estimate the derivative of the midpoint between two data points by finding the average rate of change using (y2-y1)/(x2-x1) between the two points.

To see an example of this, refer to pages 99-100 in the textbook. First, they made a scatter plot of Table 3.1 (Figure 3.6) Next, they found the average rate of change and midpoint between the first two points, the 2nd and 3rd points… This is in Table 3.2. The midpoint became the x value and the average rate of change became the y value when they made a graph of the derivative (see page 100).

Numerical Derivatives
This is a calculator trick so that you don’t have to go through all the steps to find the slope of a tangent line.

On the calculator, press MATH then 8:nDeriv(. Fill in the equation, the variable that you want to take the derivative with respect to, and the value given in the problem.Ex. Given y=x³-3x²+4, find the tangent line at x=1. You can press nDeriv(x³-3x²+4,x,1) and get -2.999 which you would round to -3.To do this without a calculator, find the derivative of the equation which in the above example is y’=3x²-6x. Plug the x value into this equation to find the slope y’(1)=3(1)²-6(1)=–3.

If you want to find the y intercept of the equation of the tangent line, you would first plug 1 in the original equation to find f(x) which is f(1)=2. Then use this point (1,2) which you know is on the tangent line, along with the slope you found previously to solve for b.2=–3(1)+b → 5=b Therefore, the equation of the tangent line is y=–3x+5.

Homework Due Tuesday
Worksheet - 1-19 odd (front page) - just apply the rule, don't simplify.
Pg. 101 - exercises 7-12, 15, 17-19, 26
Pg. 111 - 1-10, 11-21 odd
Pg. 120 - 27, 29, 31