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
Wednesday, February 13, 2019
MAT 145

2. The function f ’(x) is called:
the derivative of f at x,
the instantaneous rate of change f at x,
the slope of f at x, and
the slope of the tangent line to f at x.
Wednesday, February 13, 2019
MAT 145

3. Wednesday, February 13, 2019
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4. Now we have a derivative function f that will be true for any x value where a derivative exists!
Wednesday, February 13, 2019
MAT 145

5. Here is a graph of the function y = g(x)
Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination.
Wednesday, February 13, 2019
MAT 145

6. Here is the graph of the function y = |x|.
Why does the derivative NOT exist at x = 0?
Wednesday, February 13, 2019
MAT 145

7. Three situations for which a derivative DOES NOT EXIST!
Wednesday, February 13, 2019
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8. For each graphed function, state points at which the function is NOT differentiable. Explain your choices!
Wednesday, February 13, 2019
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9. Function Graphs and Their Derivatives
Friday, Sept 21, 2018
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10. Match each function, a-d, with its derivative, I-IV.
Wednesday, February 13, 2019
MAT 145

11. Identify each curve. Explain your choices.
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.
Wednesday, February 13, 2019
MAT 145

12. Here is the graph of a function f. Use it to sketch the graph of f ’.
Wednesday, February 13, 2019
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13. Sums, differences, exponentials, & products of constants and functions
Friday, Sept 21, 2018
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14. Derivatives of Trig Functions
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15. MAT 145

16. Friday, Sept 21, 2018
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17. Practice Derivative Rules
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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.
Friday, Sept 21, 2018
MAT 145

19. Using Derivative Patterns
MAT 145
Suppose s(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.
Friday, Sept 21, 2018

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?
Friday, Sept 21, 2018
MAT 145

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

22. Practice Derivative Rules - Answers
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MAT 145

23. Practice Derivative Rules
Friday, Sept 21, 2018
MAT 145

24. Practice Derivative Rules - Answers
Friday, Sept 21, 2018
MAT 145