2. 11.1 An Introduction to Limits Lim f(x) = L as x  a x  a - is as x approaches a from the left x  a + is as x approaches a from the right In order for there to be a limit for x  a the limits from the left and right must be the same

3. 11.1 An Introduction to Limits Polynomials –Evaluate f(x) for a –Lim (x 2 -3x+4) as x  1 is (1) 2 -3(1)+4 –Create a chart with values at different distances from a Piecewise functions –Same as polynomials

4. 11.1 An Introduction to Limits When is there no limit? –When x  a - ≠ x  a + –F(x) becomes infinitely large –F(x) oscillates

5. 11.2 Techniques for Calculating Limits Rules for Limits –(Constant) for any constant –(Limit of x) –(Multiple) –(Sum) –(Difference) –(Product) –(Quotient)

6. 11.2 Techniques for Calculating Limits Limits (continued) –(Power) –(Polynomial) –(Rational) –(Logarithm) lim[log b f(x)] = log b [lim f(x)] when f(x)>0 & b>0 & b ≠ 1

7. 11.3 One-Sided Limits; Limits Involving Infinity if and only if AND Limits at Infinity of 1/x n

8. 11.3 One-Sided Limits; Limits Involving Infinity If n>m, the limit does not exist If n=m, the limit is equal to a/b If n<m, the limit is equal to 0 a^n+k b^m+k

9. 11.4 Tangent Lines and Derivatives Slope of the tangent line of y=f(x) at (a, f(a)) has the slope provided the limit as x  a exists If a is in the domain of f, then the derivative of f at a is defined by the previous formula, provided the limit x  a exists.

10. 11.4 Tangent Lines and Derivatives Velocity as a derivative –If an object is moving along a straight line and its position on the line at time t is s(t), then the velocity at time a is s’(a) = lim t  a [s(t) – s(a)]/[t-a]

11. 11.4 Tangent Lines and Derivatives Geometric definition of f’(a): f’(a) is the slope of the tangent line to f(x) at x=a Algebraic definition of f’(a): Interpretations of f’(a): f’(a) represents velocity when f(x) is a position function and marginal change when f(x) is an economics function.

12. Links http://www.math.uri.edu/~bkaskosz/flashm o/derplot/ - tangent demohttp://www.math.uri.edu/~bkaskosz/flashm o/derplot/ http://cow.math.temple.edu/~cow/cgi- bin/manager - COWhttp://cow.math.temple.edu/~cow/cgi- bin/manager http://archives.math.utk.edu/visual.calculu s/http://archives.math.utk.edu/visual.calculu s/