W9_Extrema and Curve Sketching

Fundamentals of Mathematics 1

• Session 9: Extrema and Curve Sketching

• Instructor: Erwan Lamy

• Institution: ESCP BUSINESS SCHOOL

• Locations: Berlin, London, Madrid, Paris, Turin, Warsaw

Objectives

• Finding critical values and determining relative maxima and minima of functions.

• Identifying extreme values on closed intervals.

• Testing functions for concavity and inflection points.

• Sketching function graphs that have asymptotes.

• Modeling situations to maximize or minimize quantities.

• Importance of graph sketching for problem clarification and extremum finding.

Outline

1. Relative Extrema (chp 13.1)

2. Absolute Extrema on a Closed Interval (13.2)

3. Concavity (13.3)

4. The Second-Derivative Test (13.4)

5. Asymptotes (13.5)

6. Applied Maxima and Minima (13.6)

Relative Extrema (13.1)

• A function is increasing if its derivative, f'(x), is greater than zero (f'(x) > 0).

• A function is decreasing if its derivative is less than zero (f'(x) < 0).

• Illustrative graphs show positive and negative slopes.

Definitions of Extrema

• A function f has a relative maximum at a if for some interval around a, f(a) is greater than or equal to f(x) for all x in that interval.

• A function f has a relative minimum at a if for some interval, f(a) is less than or equal to f(x) for all x in that interval.

• A function f has an absolute maximum at a if f(a) is greater than or equal to f(x) in the entire domain.

• A function f has an absolute minimum at a if f(a) is less than or equal to f(x) across the entire domain.

• Both relative and absolute extrema are critical characteristics in calculus.

Critical Values and Extrema

• A value a is a critical value if f'(a) = 0 or f'(a) does not exist.

• If a is a critical value, the point (a, f(a)) is a critical point.

• Relative extrema occur where the sign of f'(x) changes around a critical value a.

First-Derivative Test for Relative Extrema

1. Find the derivative f'(x).

2. Determine all critical values of f.

3. For continuous critical values, check the change of sign of f' around those points.

4. For non-continuous critical values, apply definitions of extrema directly.

Sign-Chart Method

• To find the sign of f'(x), solve inequality f'(x) > 0.

• A sign-chart method can help analyze intervals based on critical points.

• The sign chart provides a visual representation of how signs change across intervals formed by critical points.

Example of Critical Values

• Given the derivative, analyze it through a sign chart to confirm relative extrema.

• Use visual aids (tables/charts) for clarity.

Absolute Extrema on a Closed Interval (13.2)

• Continuous functions on a closed interval guarantee minimum and maximum values.

• Extreme-Value Theorem: uses critical values from the interval and endpoints to find absolute extrema.

Finding Absolute Extrema Steps

1. Find critical values of f.

2. Evaluate f at critical values and endpoints of the interval [a, b].

3. Identify the maximum and minimum values from evaluations.

Example of Absolute Extrema

• Given a function, determine critical values, evaluate at interval endpoints, and find absolute extrema.

Concavity (13.3)

• A curve is concave up if the slope (derivative) is increasing.

• A curve is concave down if the slope is decreasing.

• Higher derivatives (second derivative) provide insight on concavity: f''(x) > 0 indicates concavity upward, and f''(x) < 0 indicates concavity downward.

• Inflection points occur where the concavity changes; conditions for identifying inflection points include continuity and changes in concavity.

Second-Derivative Test (13.4)

• The second-derivative test assists in confirming relative extrema.

• If f'(a)=0 and f''(a)<0, a is a relative maximum.

• If f''(a)>0, a is a relative minimum.

Asymptotes (13.5)

• Asymptotes are lines that graph behavior near undefined points.

• Different types: vertical, horizontal, and oblique.

• The establishment of asymptotes involves limits; conditions defined for vertical and horizontal asymptotes to confirm presence.

Applied Maxima and Minima (13.6)

• Steps for solving real-life max-min problems often include diagramming, defining the function based on given constraints, finding critical points, and ultimately determining extrema in context.

• Example problem with maximizing revenue illustrates application of learned skills.