- Higher Order Derivatives - Graph Sketching - Applications in Graphs Derivatives: Part 3 - Higher Order Derivatives - Graph Sketching - Applications in Graphs

Review What are? Tangent lines - a straight line that touches a function at only one point. Tangent Line = Instantaneous Rate of Change = Derivative Secant lines -cuts at 2 points -average rate of change

Higher Order Derivatives

Higher Order Derivatives: Why? First order – general tangent equation - rate of change Second order – rate of change of the tangent lines (curve up or down) - Max/Min/Point of Inflection point on a graph - rate of change of a rate of change Second and Third – Seen/Unseen point of inflection on a graph - motion along a straight line - rate of change of a rate of change of a rate of change

Higher Order Derivatives: Examples Find first, second and third order derivatives

Graph Sketching You need to know x x2 x3 ln x ex Why? So that you can visualise your functions better. Use your calculators and imagination.

Curve Sketching: Quadratic Curves STEPS STEP 1: Differentiate equation STEP 2: y’ = 0, obtain x value STEP 3: Substitute x value into y equation to obtain y value  Stationary point STEP 4: Determine max or min point using y’’ STEP 5: Determine x- and y- intercepts by substituting x = 0 and y = 0 STEP 6: Plot points and sketch graph accordingly ensuring x- and y-intercepts (if any), x- and y-axes, stationary points and curves are labelled.

Curve Sketching: Cubic Curves STEPS STEP 1: Differentiate equation STEP 2: y’ = 0, obtain x values by factorising. STEP 3: Substitute x value into y equation to obtain y values  Stationary points STEP 4: Determine max or min point using y’’ by substituting stationary point x value into the y’’ equation. Positive value = min point; Negative value = max point STEP 5: Determine x- and y- intercepts by substituting x = 0 and y = 0 STEP 6: Plot points and sketch graph accordingly ensuring x- and y-intercepts (if any), x- and y-axes, stationary points and curves are labelled.

Graph Sketching: In Class Exercise

Applications in Graphs: Extremes Absolute maximum & minimum values are called absolute extrema (plural of the Latin extremum). Absolute extrema are also called global extrema, to distinguish them from local extrema.

Application in Graphs: Extremes

Different Situations

Finding Extremes The first derivative theorem for local extreme values If f has a local max/min value at an interior point c of its domain, and if f ’ is defined at c, then f ’(c) = 0. Critical Point   An interior point of the domain of a function f where f ’ is zero or undefined is a critical point of f. *The only domain points where a function can assume extreme values are critical and endpoints.

Step-by-Step Method Evaluate f at all critical points and endpoints How to find the absolute extrema of a continuous function f on a closed interval. Evaluate f at all critical points and endpoints Draw/Sketch what you interpret Take the largest and smallest of these values

Extrema: Examples Review on Tangent lines.

hELp?!? In drawing graphs:- http://my.hrw.com/math06_07/nsmedia/tools/Graph_ Calculator/graphCalc.html Microsoft Math v16.0 (part of Microsoft Student package) In Calculus:- Text books and any other calculus books out there. Do and practice Questions!!! Hughes-Hallet et al. 2003. Calculus: 4th Ed. John-Wiley