Math Camp 2005 Instructor: Udi Sommer

1. Math Camp – 2005Instructor: Udi Sommer Fourth Class (Tuesday, August 23, 10a-1p) Applied problems Maxima / minima Concavity, convexity, inflection Curve sketching Integration Indefinite integrals Definite integrals

2. Optimization – Maxima and minima – Extrema of functions Points of maximum and minimum value are a useful tool in analysis of data. Differentiation is a handy mathematical instrument to find these values.

3. Optimization – Maxima and minima – Extrema of functions positive slope (increasing – upward slope), negative slope (decreasing – downward slope), slope zero (stationary - maxima, minima or inflection points) the slope of the tangent is zero, and the same is true for the value of the first derivative. At these points the tangent is horizontal. These are called critical points of the curve. These points are the maxima, minima, or inflection points.

4. Optimization – Maxima and minima – Extrema of functions if we take the derivative of a function and find the points where it equals zero, we can find the maxima, minima, and inflection. These would be the local or relative minima or maxima. They might also be the absolute or global minima and maxima.

5. Definitions Let f(x) be defined on Df ? ? Maximum – c ? Df is a maximum point if f(x) ? f(c) for all x ? Df Minimum - c ? Df is a minimum point if f(x) ? f(c) for all x ? Df

6. Definitions How do we find those points mathematically? 1. First derivative test / First order condition Set first derivative ƒ ?(x) = 0 w. r. t. the variable of interest (simply x in uni-variate functions) Find the x-values for which this is true 2. Second derivative test Assume ƒ is twice differentiable on Df Then, for some c, d, g in interior of Df:a) ƒ?(c)=0 and ƒ??(c) < 0 ? c is a local maximum of ƒb) ƒ?(d)=0 and ƒ??(d) > 0 ? d is a local minimum of ƒc) ƒ?(g)=0 and ƒ??(g) = 0 ? g is an inflection point

7. Definitions Inflection point -An inflection point is where the sign of the second derivative changes. The function is concave where ƒ ?? (x) < 0 The function is convex where ƒ ?? (x) > 0

8. Summary To Locate Relative and Absolute (Local and global) Extrema of a Continuous Function on [a, b]: Find the critical points - Locate the points in (a, b), where ƒ ? (x) = 0 (First derivative test) Run the Second derivative test for those points. Those for which ƒ ?? (x) < 0 are local maxima Those for which ƒ ?? (x) > 0 are local minima Those for which ƒ ?? (x) = 0 are inflection points Evaluate f at the maxima, the minima and at the endpoints a, b. The largest of these values is the absolute maximum value. The smallest of these values is the absolute minimum value. Accordingly determine the local minima and maxima

9. The 6 steps for curve sketching Intercepts – y and x intercepts Extent – horizontal extent (possible values of x) and vertical extent (possible values of y) Asymptotes – vertical and horizontal asymptotes Maxima, minima, inflection Estimate values of y in significant points Sketch