Calculus I (MAT 145) Dr. Day Wednesday Feb 27, 2013

1. Calculus I (MAT 145)Dr. Day Wednesday Feb 27, 2013 • Return Gateway Derivatives Quiz #1 (#2 tomorrow!) • Derivatives of Composite Functions: The Chain Rule (3.4) • Assignments MAT 145

2. Detecting Derivative Patterns • The Derivative of a Constant Function • The Derivative of a Power Function • The Derivative of a Function Multiplied by a Constant • The Derivative of a Sum or Difference of Functions • The Derivative of a Polynomial Function • The Derivative of an Exponential Function MAT 145

3. Detecting Derivative Patterns • Derivative of a Product of Functions • Derivative of a Quotient of Functions MAT 145

4. Detecting Derivative Patterns • Derivatives of Trigonometric Functions MAT 145

5. Using Derivative Patterns For f(x) = 2x2 – 3x + 1: • Calculate f’(x). • Determine an equation for the line tangent to the graph of f when x = -1. • Determine all values of x that lead to a horizontal tangent line. • Determine all ordered pairs of f for which f’(x) = 1. MAT 145

6. Using Derivative Patterns Suppose s(x) = 3x4 − 2x2 +5/x represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds. • 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. MAT 145

7. Using Derivative Patterns Forg(x) = x√x: • Where x = 4, determine the tangent-line equation. • Where x = 1, determine the normal-line equation. • 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? MAT 145

8. Using Derivative Patterns For h(x) = (x+3)/(x2-1): • Calculate h’(x). • Determine an equation for the line tangent to the graph of f when x = 0. • Determine all values of x that lead to a horizontal tangent line. • Determine all ordered pairs of h for which h’(x) = 2. • Identify, with justification, all values of x for which h is discontinuous and for which h is not differentiable. • Identify, with justification, all vertical and horizontal asymptotes. MAT 145

9. MAT 145

10. 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. MAT 145

11. Derivatives of Composite Functions (3.4) MAT 145

12. Derivatives of Composite Functions (3.4) MAT 145

13. 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. Use the chain rule whene'er you find The derivative of a function compositionally defined. MAT 145

14. Derivatives of Composite Functions (3.4) MAT 145

15. Assignments WebAssign • 3.2 (Part 2) through 3.4 assignments available this week. 3.5 and 3.6 on the way • Gateway Derivatives Quiz #1 returned today. GDQ #2 tomorrow! MAT 145