 # Lecture 2 Implicit Differentiation Inverse Trigonometric Functions Higher

• Slides: 43 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