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

2. 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

3. Higher Order Derivatives

4. 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

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

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

7. 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.

8. 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.

9. Graph Sketching: In Class Exercise

10. 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.

11. Application in Graphs: Extremes

12. Different Situations

13. 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.

14. 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

15. Extrema: Examples
Review on Tangent lines.

16. hELp?!?
In drawing graphs:- 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 Calculus: 4th Ed. John-Wiley