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

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

4. Audience 12 th 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

5. 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

6. 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

7. Sketching a Curve Using Derivatives t that point.

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

9. Orientation 1) Find the Y-intercept 2) Find the X-intercept 3) Identify the Asymptotes 4) Find the Extrema 5) Intervals of Increasing or Decreasing Values 6) Find Inflection Points 7) Intervals of Concave up and/or concave down 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. I’m Ready to Take the Quiz!

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

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

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

13. Congratulations! Great Job!!! click

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

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

16. Not quite! Let me show you. AND

17. Identify All Asymptotes 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. 3 Horizontal Vertical Oblique

18. Finding the Horizontal Asymptote/s Take the limit of the equation as x approaches infinity. Example: 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)

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

20. Finding the Oblique Asymptote/s 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: No Oblique Asymptote!

21. Find the Asymptotes Now you try one!

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

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

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

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

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

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

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

29. 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)

30. 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. -1 0 1

31. 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. -1 0 1 Ø O O Not on f(x) On f(x)

32. 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: -1 0 1 - Ø + O + O -

33. 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) -1 0 1 - Ø + O + O -

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

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

36. 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.

37. Help Continued…

39. 0 - O + O -

40. 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). 5

41. 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. -1 0 1 - Ø + O + O -

42. 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? 6

43. 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 (-).

44. 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.

45. 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. 7

46. 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.

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

48. QUIZ TIME! 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! Use everything you have learned to draw this curve. Take your time, you can do this!

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

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

52. Summary + O - Ø - O - Ø - O +

53. Summary -1 0 1 - Ø + O - Ø +

54. Sorry Go Back and Review the Lesson Again!