Lecture 2 Implicit Differentiation Inverse Trigonometric Functions Higher

Lecture 2: Implicit Differentiation, Inverse Trigonometric Functions, Higher Derivatives, Curve Sketching, and Min/Max Problems

Part I: Exponentials, Implicit Differentiation, and Inverse Trigonometric Functions

Objectives •

Differentiating Implicit Functions •

Example •

The Power Law Revisited •

Part II: Higher Derivatives

The Second Derivative •

• The Second Derivative 4 3 2 1 0 -1 -2 -3 -4 -3 -2 -1 0 1 2 3 4 x

Higher Derivatives •

Part III: Curve Sketching

Objectives • Be able to sketch functions, including their critical points, discontinuities, zeros, and asymptotes. Corresponding Sections in Simmons: 4. 1, 4. 2

Critical Points •

Examples: 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x The critical point (0, 0) is a minimum

Examples: 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x The critical point (0, 0) is not a minimum or maximum

Examples: 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x (0, 0) is a minimum but not a critical point.

Information for Sketching Functions •

• Examples

Examples: 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x

• Examples

6 Examples: 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 6

• Examples

6 Examples: 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 6

Asymptotes and Inflection Points •

Part IV: Min/Max Problems

Objectives • Know how to solve min/max problems by looking at the critical points and other points of interest Corresponding sections in Simmons: 4. 3, 4. 4

Finding Absolute Minima/Maxima •

• Example

Example Word Problem • What is the minimum perimeter of a rectangle with area 25? w l

Example Word Problem Continued •

Solving Word Problems • To solve word problems, we often: 1. Draw a picture for the problem 2. Find equations for the relevant variables 3. Use these equations to re-express what we’re trying to minimize/maximize as a function in one variable. 4. Minimize or maximize this function.

The second derivative test •

Reflection • What is the minimum length of a path from A to B that bounces off the mirror? A B a x b c-x

A Reflection B a x b c-x

Refraction • A a x c-x b B

Refraction A a x c-x b B