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Calculus I (MAT 145) Dr. Day Thursday Feb 28, 2013

Calculus I (MAT 145) Dr. Day Thursday Feb 28, 2013. Gateway Derivatives Quiz #2 Derivatives of Composite Functions: The Chain Rule (3.4) Implicit Differentiation (3.5) Assignments. Using Derivative Patterns. For s(t) = cos (2t): Calculate s’(t) and s’’(t).

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Calculus I (MAT 145) Dr. Day Thursday Feb 28, 2013

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  1. Calculus I (MAT 145)Dr. Day Thursday Feb 28, 2013 • Gateway Derivatives Quiz #2 • Derivatives of Composite Functions: The Chain Rule (3.4) • Implicit Differentiation (3.5) • Assignments MAT 145

  2. MAT 145

  3. 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

  4. Derivatives of Composite Functions (3.4) MAT 145

  5. Derivatives of Composite Functions (3.4) MAT 145

  6. 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

  7. Derivatives of Composite Functions (3.4) MAT 145

  8. 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. MAT 145

  9. 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 isolatedy/dx. That’s our goal! MAT 145

  10. Derivatives of Implicitly Defined Functions (3.5) MAT 145

  11. Derivatives of Implicitly Defined Functions (3.5) Try These: MAT 145

  12. Derivatives of Logarithmic Functions (3.6) What is the derivative of the natural log function y = ln(x)? MAT 145

  13. Derivatives of Logarithmic Functions (3.6) Now apply this to other log functions: MAT 145

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

  15. Assignments WebAssign • 3.4 (Part 2) and 3.5 assignments up for completion; 3.6 on the way • Gateway Derivatives Quiz #2 today. GDQ #3 next week! • Test #3: Friday, March 8. MAT 145

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