Calculus I (MAT 145) Dr. Day Friday September 29, 2017 Derivative Shortcuts (Chapter 3) Sums & Differences, Products & Quotients, and More! (3.1- 3.2) Derivatives of Trig Functions (3.3) Derivatives of Composite Functions (3.4) Friday, September 29, 2017 MAT 145

Warm up! . Find the derivatives. Use correct notation! Friday, September 29, 2017 MAT 145

Using Derivative Patterns MAT 145 Suppose s(x), shown below, represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds, for x > 0. Calculate the object’s velocity and acceleration functions. Is the object moving left or right at time x = 1? Justify. Determine the object’s velocity and acceleration at time x = 2. Based on those results, describe everything you can about the object’s movement at that instant. Write an equation for the tangent line to the graph of s at time x = 1. Friday, September 29, 2017

Using Derivative Patterns Determine the equation for the line tangent to the graph of g at x = 4. Determine the equation for the line normal to the graph of g at x = 1. At what points on the graph of g, if any, will a tangent line to the curve be parallel to the line 3x – y = –5? Friday, September 29, 2017 MAT 145

Sums, differences, exponentials, & products of constants and functions Friday, September 29, 2017 MAT 145

Derivatives of Trig Functions Friday, September 29, 2017 MAT 145

Derivatives of Composite Functions Friday, February 24, 2017 MAT 145

Derivatives of Composite Functions (3.4) Friday, February 24, 2017 MAT 145

Here's a function in a function And your job here is to find THE CHAIN RULE WORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE" Here's a function in a function And your job here is to find The derivative of the whole thing With respect to x inside. Call the outside f of u And call the inside u of x. Differentiate to find df/du And multiply by du/dx. Use the chain rule. Use the chain rule when e'er you find The derivative of a function compositionally defined. Friday, February 24, 2017 MAT 145

Derivatives of Composite Functions (3.4) Friday, February 24, 2017 MAT 145

Derivatives of Composite Functions Friday, February 24, 2017 MAT 145

Composite Functions Friday, February 24, 2017 MAT 145

Practice Derivative Rules Friday, February 24, 2017 MAT 145

Using Derivative Patterns For s(t) = cos(2t): Calculate s’(t) and s’’(t). Determine an equation for the line tangent to the graph of s when t = π/8. Determine the two values of t closest to t = 0 that lead to horizontal tangent lines. Determine the smallest positive value of t for which s’(t) = 1. If s(t) represents an object’s position on the number line at time t (s in feet, t in minutes), calculate the object’s velocity and acceleration at time t = π/12. Based on those results, describe everything you can about the object’s movement at that instant. Friday, February 24, 2017 MAT 145