Calculus I (MAT 145) Dr. Day Wednesday February 13, 2019

1. Calculus I (MAT 145)Dr. Day Wednesday February 13, 2019 • The limit definition of derivative (2.8) • Properties and Characteristics of Derivatives (2.8) • Derivative Shortcuts (Ch 3) • Quiz #5 on Friday! • Info from a graph: limits, function values, derivatives • Asymptotes defined by limits • Calculate a derivative using the Limit Definition • Contextual Meaning of the derivative MAT 145

2. The function f ’(x) is called: • the derivative of f at x, • the instantaneous rate of changefat x, • the slope of f at x, and • the slope of the tangent line to fat x. MAT 145

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4. Now we have a derivativefunction f that will be true for anyx value where a derivative exists! MAT 145

5. Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination. MAT 145

6. Here is the graph of the function y= |x|. Why does the derivative NOT exist at x = 0? MAT 145

7. Three situations for which a derivative DOES NOT EXIST! MAT 145

8. For each graphed function, state points at which the function is NOT differentiable. Explain your choices! MAT 145

9. Function Graphs and Their Derivatives http://www.maa.org/sites/default/files/images/upload_library/4/vol4/kaskosz/derapp.html MAT 145

10. Match each function, a-d, with its derivative, I-IV. MAT 145

11. Here are the graphs of four functions. One repre-sents the position of a car as it travels, another represents the velocity of that car, a third repre-sents the acceleration of the car, and a fourth graph represents the jerk for that car. Identify each curve. Explain your choices. MAT 145

12. Here is the graph of a function f. Use it to sketch the graph of f ’. MAT 145

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

14. Derivatives of Trig Functions MAT 145

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17. Practice Derivative Rules MAT 145

18. 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

19. 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

20. 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

21. Warm up! Find the derivatives. Use correct notation! • . • . • . MAT 145

22. Practice Derivative Rules - Answers MAT 145

23. Practice Derivative Rules MAT 145

24. Practice Derivative Rules - Answers MAT 145