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Derivatives and Graphs

Derivatives and Graphs. Tutorial. By Courtney Keirn. Introduction. Lesson. Quiz. Tutorial. This arrow will take you back a slide. This arrow will take you forward a slide. This icon will take you back to the first slide. Introduction.

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Derivatives and Graphs

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  1. Derivatives and Graphs Tutorial By Courtney Keirn Introduction Lesson Quiz

  2. Tutorial This arrow will take you back a slide. This arrow will take you forward a slide. This icon will take you back to the first slide.

  3. Introduction • Throughout this PowerPoint, you will view a lesson on the first derivative test. • This is for AP Calculus. • The learning objectives are as follows: • Students will learn the first derivative test. • Students will be able to compute this test. • Students will be able to analyze the graphs of the function using the first derivative.

  4. Ohio Content Standards • It is important for students to practice mathematical writing skills to help communicate their reasoning and explanations to the Readers who will be scoring their answers. • Students should have experience throughout the year with justifying conclusions using mathematical (calculus) arguments, such as use of the First Derivative Test or Second Derivative Test.

  5. Lesson • Here are some important facts to remember: • The original function is labeled as f(x) [f of x]. The first derivative is labeled as f’(x) [f prime of x]. • When the first derivative is positive (+), then the function is increasing. • When the first derivative is negative (-), then the function is decreasing. • When the first derivative has a zero, there is a local extrema. • Local extrema is either a minimum or a maximum of the original function.

  6. Lesson • First, let’s start out with a function and find it’s derivative. f(x) = 3x2 + 4 • To find the derivative, we will need to use the power rule [(xn)’=nxn-1]. f’(x) = 6x + 0 [Reminder: All constants have a derivative of zero.]

  7. Lesson • Next, we will find out where the first derivative has zero values. f(x) = 3x2 + 4 f‘(x) = 6x • If we set the derivative to zero, we can compute. 0 = 6x 0∕6 = 6x/6 0 = x x = 0 [This is our only zero for this function.]

  8. Lesson • Now that we know our zeros, we must find out where the first derivative is increasing and decreasing. • We will create a linear graph and plug in numbers on either side of our zeros to find if the first derivative is positive or negative. • We can plug in these x values of -1 and 1 to check to see if it is positive or negative. • f‘(-1) = 6(-1) = -6 Therefore, from -∞ to 0 the graph is decreasing. • f‘(1) = 6(1) = 6 Therefore, from 0 to ∞ the graph is increasing. 1 -1 0

  9. Lesson • f‘(-1) = 6(-1) = -6 Therefore, from -∞ to 0 the graph is decreasing. • f‘(1) = 6(1) = 6 Therefore, from 0 to ∞ the graph is increasing. • We are able to draw the graph now and find out if at x=0 there is a maximum or minimum. f(x) is decreasing f(x) is increasing 0

  10. Lesson • To double check, if we plug the function in to our calculators, we see that we have a minimum at x=0 for the function f(x) = 3x2 + 4.

  11. Are you ready for a quiz? • You may click any of the buttons to return to • Home: • Lesson: • Or to take the quiz:

  12. Quiz: Question #1 Is the following statement true or false? When the first derivative is zero, the original function is increasing. • True • False

  13. Yes! Next Question That is correct!

  14. Quiz: Question #2 • Find the derivative of the function and it’s zeros. f(x) = x5 + 3x2 + 500 A. 5x4 + 6x ; x = 0, x = 2 B. 5x4+ 6x + 1 ; x = -1.06, x = 0 C. 5x4 + 6x ; x = 0, x = -1.06 D. 5x4 + 6x ; x = 0

  15. Yes! Next Question That is correct!

  16. Quiz: Question #3 • In the equation f(x) = 4x3 - 2x + 8 where is the function increasing and decreasing? A. Increasing: -∞ to √(1/6), 0 to ∞ Decreasing: √(1/6) to 0 B. Increasing: √(1/6) to 0 Decreasing: -∞ to √(1/6), 0 to ∞ C. Increasing: -∞ to √(1/6) Decreasing: √(1/6) to ∞ D. Increasing: -∞ to -√(1/6), √(1/6) to ∞ Decreasing: -√(1/6) to √(1/6)

  17. Yes! That is correct!

  18. You have finished your quiz! You now know how to computethe first derivative test! Click to see credits 

  19. Credits • Congratulations image was from Microsoft clipart. • Sounds were from Microsoft clipart. • Rage comic images were from http://knowyourmeme.com. • Ohio Content Standards and lesson plan:http://apcentral.collegeboard.com/apc/members/repository/ap07_calculus_teachersguide_2.pdf

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