Calculus I (MAT 145) Dr. Day Thur sday March 7, 2013

1. Calculus I (MAT 145)Dr. Day Thursday March 7, 2013 • Logarithmic Differentiation (3.6) • Revisiting: Position, Velocity, Acceleration (3.7) • Assignments • Test #3: Tomorrow! MAT 145

2. Derivatives of Logarithmic Functions (3.6) And extend this to other functions: MAT 145

3. 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 • The Derivative of a Logarithmic Function MAT 145

4. Detecting Derivative Patterns • Derivative of a Product of Functions • Derivative of a Quotient of Functions • Derivatives of Trig Functions • Derivatives of Composite Functions (Chain Rule) • Implicit Differentiation • Logarithmic Differentiation MAT 145

5. 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

6. Position, Velocity, Acceleration (3.7) Here’s a function describing an object’s position on the number line: MAT 145

7. Position, Velocity, Acceleration (3.7) If s(t) represents the position of an object moving back and forth on a number line, v(t) = s’(t) is the object’s velocity, and a(t) = v’(t) = s’’(t) is the object’s acceleration, then: The object is moving in a positive direction when: The object is moving in a negative direction when: The object is speeding up when: The object is slowing down when: The object is changing directions when: The average velocity over a time interval is found by: The instantaneous velocity at a specific point in time is found by: The net change in position over a time interval (displacement) is found by: The total distance traveled over a time interval is found by: MAT 145

8. Position, Velocity, Acceleration (3.7) The object is moving in a positive direction when v(t) > 0. The object is moving in a negative direction when v(t) < 0. The object is speeding up when v(t) and a(t) share same sign. The object is slowing down when v(t) and a(t) have opposite signs. The object is changing directions when v(t) = 0 and v(t) changes sign. The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!). The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time. The net change in position over a time interval is found by calculating the difference in the positions at the stat and end of the interval. The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements. MAT 145

9. Assignments WebAssign • 3.6 (part 2) due tonight • Gateway Derivatives Quiz #3 today! • Test #3: Tomorrow! • Part 1: No calculator (30 pts) • Part 2: Calculator OK (70 pts) MAT 145