Sketching Curves Using Derivatives

1. Sketching Curves Using Derivatives By: Sarah Carley For Teachers For Students

2. Table of Contents Audience Who this lesson was prepared for Where this lesson occurs What should be accomplished Environment Objective

3. Audience 12th Grade Students Honors Calculus or AP Calculus Students Students who wish to learn how to hand draw sketches of curves Anyone who wishes to review curve sketching. Table of Contents

4. Environment Activity is to be done individually or in a small group of 2 or 3 students, In a computer lab, And with desk Space to allow taking notes and working out problems. Table of Contents

5. Objective After completing the Sketching a Curve Using Derivatives PowerPoint tutorial, students will be able to sketch a given curve with 100% accuracy as measured by a quiz. Table of Contents

6. t that point. Sketching a Curve Using Derivatives

7. Hi! My name is Apple! I’ll be helping you through sketching curves today. How can I learn this all by myself!?

8. I’m Ready to Take the Quiz! These are all of the things you will need to find in order to sketch a curve! When you’re finished click the quiz button to show what you learned. Orientation Find the Y-intercept Find the X-intercept Identify the Asymptotes Find the Extrema Intervals of Increasing or Decreasing Values Find Inflection Points Intervals of Concave up and/or concave down

9. Finding the Y-Intercept 1 In a given equation, find the value of ‘y’ when the value ‘0’ is substituted for ‘x’. Now let’s try it!

10. What was your answer? A. B. C. D. Find the Y-Intercept (1,0) (0,1) Undefined (0,0)

11. Not quite! Let me show you. ** Anything divided by 0 goes to infinity

12. Great Job!!! click Congratulations!

13. 2 Finding the X-Intercept Lets try it! The opposite of finding the y-intercept! In a given equation, find the value of ‘x’ when the value ‘0’ is substituted for ‘y’.

14. What was your answer? A. B. C. D. Find the x-intercept (2,0) (0,0) (2,0) and (-5,0) (-5,0)

15. Not quite! Let me show you. AND

16. 3 Identify All Asymptotes Horizontal Vertical An asymptote is a point or line that the curve approaches but never quite gets to (like a limit). This includes the horizontal asymptote/s, the vertical asymptote/s, and/or the oblique asymptote/s. Oblique

17. Horizontal Asymptote is at y=1 *The limit is the coefficients because the degree of the numerator and the degree of the denominator are equal. (This should be review) Finding the Horizontal Asymptote/s Take the limit of the equation as x approaches infinity. Example:

18. *x values must be then plugged into the numerator to show the numerator is NOT equal to 0. Finding the Vertical Asymptote/s Vertical Asymptotes at and The value of x when the denominator of the function is set equal to 0. Example:

19. Finding the Oblique Asymptote/s No Oblique Asymptote! Using long division, divide out the function if and only if the degree of the numerator is greater than the degree of the denominator. Example:

20. Now you try one! Find the Asymptotes

21. What was your answer? A. B. C. D. HA: none VA: OA: HA: none VA: none OA: HA: HA: none VA: VA: OA: OA:

22. Your horizontal asymptote is not quite right! Let me show you! Undefined! Therefore there is no horizontal asymptote

23. Your vertical asymptote is not quite right! Let me show you! Vertical Asymptote: x=-2

24. Your oblique asymptote is not quite right! Let me show you!

25. 4 Finding Extrema Using the first derivative test and a sign chart. Extrema: These are the local minimums and maximums on the graph.

26. First Derivative Test Step 1: Find the First Derivative Example: This should be a review of the quotient rule!

27. First Derivative Test Step 2: Set the numerator of the first derivative equal to 0, and then solve for x. Example:

28. First Derivative Test Step 3: Put the x values found back into the original function in order to find the critical points. Example: Critical Point: (0,2)

29. -1 0 1 Making a Sign Chart Step 1: Make a number line using all critical numbers Critical Numbers: These are all the critical values of x, including the x-intercept, any asymptote (with an x value), and values found using the first derivative test.

30. On f(x) Not on f(x) Ø O O -1 0 1 Making a Sign Chart Step 2: Determine which points are actually on the function f(x) Put all critical values of x into the original equation. If a y value is undetermined then the point is not on the curve.

31. - Ø + O + O - -1 0 1 Making a Sign Chart Step 3: Pick a point between the critical values on the number line. Place this x value into the equation f’(x), and determine if the y value is + or -. Place your finings on your chart like so:

32. - Ø + O + O - -1 0 1 What are the Extrema? All of the critical values of x that are on the curve f(x) (these are the ones with O), which have a “change in sign” on the sign chart. This means that the values change from – to +, or the values change from + to -. In this example there is one critical value that : x=1 The extrema is (1,y)! Note: This value of y is from the original equation f(x)

33. Now it’s your turn to try! Find the extrema!

34. What was your answer? A. B. C. D.

35. Not quite! Let me show you. The x-intercept is (0,0) If this does not make sense, please go back to the section on the x-intercept.

36. Help Continued…

37. Help Continued…

38. Help Continued… - O + O - 0

39. 5 Intervals of Increasing and Decreasing Value Here’s an easy one! Take your sign chart from the previous step. The intervals that have a – sign are decreasing on f(x), while the intervals that have a + sign are increasing on f(x).

40. - Ø + O + O - -1 0 1 Add the new information to your sign Chart This is what it should look like: We can now start to see what the curve is looking like.

41. 6 Finding the Inflection Points Once you’ve mastered the first derivative test, the one is the same, except you use the second derivative instead of the first. Simple right?

42. The Second Derivative Test These are the same steps as the first derivative test: Find the second derivative Set the derivative equal to 0 Find the critical values of x Make a new sign chart using the same steps as before: Make a number line with the critical x values Determine if the point is on f(x) Look at a point in the intervals between critical points, and determine if they are positive (+) or negative (-).

43. What Points are Inflection Points? Just like when we found the extrema! Like before, at the points where the sign changes, these are the x values for the inflection points. Place these x values into the original f(x) to get the points.

44. 7 Intervals of Concave Up and/or Concave Down Again, once you have done the second derivative test, this is an easy one! Using the sign chart, the intervals with a + sign are concave up, while the intervals with a – sign are concave down.

45. What is this concave thing? It’s easier to think of concave up as a “bowl”, and concave down as a “hill”. Concave Up Concave Down *When you think about this, the inflection point is where the curve changes from concave up, to concave down or vice versa.

46. I’m not ready! I need to go back an review sections first. Are You Ready? Now you have everything you need to draw the curve. Just put it all together! I’m ready for that quiz!

47. QUIZ TIME! Use everything you have learned to draw this curve. Quick hints! Remember what your basic functions look like Remember how to do your derivatives Most of this is review, take your time and put it all together! Take your time, you can do this!

49. I’m so proud of you! CONGRADULATIONS!!!! Check your work? You may now exit the PowerPoint (click esc)

50. X-intercept: (0,0) Y-intercept: (0,0) Vertical Asymptote: x=1,-1 Horizontal Asymptote: none Oblique Asymptote: y=x Summary