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Calculus I (MAT 145) Dr. Day Mon day February 18, 2019

Explore derivative shortcuts, practice derivative rules, and apply the chain rule in calculus MAT.145. Calculate derivatives, find tangent lines, and analyze motion using derivative patterns. Includes exercises on sums, differences, exponential functions, trig functions, and composite functions.

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Calculus I (MAT 145) Dr. Day Mon day February 18, 2019

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  1. Calculus I (MAT 145)Dr. Day Monday February 18, 2019 • Derivative Shortcuts (Ch 3) • Sums, Differences, Product, Quotients • Constants, Powers, Exponential Functions • Trig Functions • The Chain Rule! • Return Quiz #5 MAT 145

  2. Warm up! Calculate each derivative. Use correct notation! • . MAT 145

  3. Sums, differences, exponentials, & products of constants and functions MAT 145

  4. Derivatives of Trig Functions MAT 145

  5. MAT 145

  6. MAT 145

  7. Practice Derivative Rules MAT 145

  8. 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

  9. Using Derivative Patterns Supposes(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. MAT 145

  10. 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? MAT 145

  11. Practice Derivative Rules - Answers MAT 145

  12. Practice Derivative Rules MAT 145

  13. Practice Derivative Rules - Answers MAT 145

  14. Derivatives of Composite Functions MAT 145

  15. Derivatives of Composite Functions (3.4) MAT 145

  16. 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 when e'eryou find The derivative of a function compositionally defined. MAT 145

  17. Derivatives of Composite Functions (3.4) MAT 145

  18. Derivatives of Composite Functions MAT 145

  19. Composite Functions MAT 145

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