Calculus I (MAT 145) Dr. Day Wednesday Oct 11, 2017 Derivative Shortcuts (Chapter 3) Position, Velocity, Acceleration Derivatives of Implicitly Defined Functions (3.5) Friday: Test #2, STV 308 Help Session: Thursday, 6:15 p – 7:15 pm STV 347A Wednesday, October 11, 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. Wednesday, October 11, 2017 MAT 145

Derivatives of Implicitly Defined Functions (3.5) An implicitly defined function is: A function whose relation among the variable is given by an equation for which the function has not been explicitly stated. In the equation x2 + y2 = 25, y is an implicit function of x because the equation doesn’t explicitly express y in terms of x. Wednesday, October 11, 2017 MAT 145

Wednesday, October 11, 2017 MAT 145

Implicit Differentiation Find dy/dx when . Take derivative of each term. Use chain rule. X changes identically to itself; dx/dx=1. Solve for dy/dx. Simplify. Wednesday, October 11, 2017 MAT 145

Derivatives of Implicitly Defined Functions (3.5) To calculate the derivative of an implicitly defined function: Assume a functional connection among the variables. If x and y are present, assume y is a function of x. Calculate the derivative of each term in the equation. Because we don’t explicitly know how y is determined by x, when calculating the derivative of y, we use the chain rule and write dy/dx as the derivative of y. After determining term-by-term derivatives, carry out all necessary algebra steps to isolate dy/dx. That’s our goal! Wednesday, October 11, 2017 MAT 145

Implicit Differentiation Find dy/dx. Wednesday, October 11, 2017 MAT 145