Guiding Discussion Questions: • Does change sign at each critical point? If so, does it change from negative to positive, or from positive to negative? Why is this important for classifying extrema? • On what intervals is the function increasing? On what intervals is it decreasing? • Classify the local extrema based on your sign analysis of • Are there global extrema on [0, 2π]? Why or why not? If so, what are they? How do you know? . 6. Add the extrema to your graph. You can use technology to evaluate the function values at the extrema, and to get decimal approximations of the critical points, to help you plot the points on the graph. 7. The next step is to find the intervals of concavity. Guiding Discussion Questions: • Find What equation do we need to solve to get information about concavity? dx² • What identity can we use to help us solve that equation? Use a similar process to how you found the x-intercepts, to solve this equation. • How can you use the information about the local extrema you've already classified using the First Derivative Test, dx² to infer the concavity of the original function on the intervals created by the zeros of -? What are the intervals of concavity? What are the inflection points? 8. Add the inflection points to your graph. You can use technology to evaluate the function values at the inflection points. 9. Sketch the graph, using all the information you have gathered about the function and its derivatives. 10. Bonus: Sketch additional periods of the graph, perhaps on [-67, 6π] to illustrate the pattern. Follow the curve sketching process for y = cos 2x + sinx on the interval [0, 2π]. Remember that the goal is to be able to use information about the derivatives, along with other algebraic information we can get from the function, to sketch a graph of the function without using technology. 1. Our first step is finding any intercepts and/or asymptotes. Guiding Discussion Questions: • Does this function have asymptotes? Why or why not? • How do we find the y-intercept? What is it? • What equation do we need to solve to find the x-intercepts? 2. To solve this equation, we'll need to use some trigonometric identities, so let's review how to do that! Recall that because of the Pythagorean identity cos²x+ sin²x = 1, there are three different ways to express the cosine double angle identity: cos 2x = cos²x - sin² can be rewritten in only sines by replacing cos² with 1 - sin² x and simplifying, or in only cosines by replacing sin² x with 1- cos²x. We then get three choices for the double angle identity for cosine: • cos²x - sin² x . 1-2 sin² x • 2 cos² - 1 Guiding Discussion Questions: • Which of the three choices will be the most useful for us, to do a substitution in the equation we need to solve, to get the x-intercepts? • What is the new equation after we do this substitution? • What strategy can we use to solve this equation, to get the x-intercepts? 3. After solving the equation, add all the intercepts to a blank set of coordinate axes, to begin your graph. You can use technology to get decimal approximations for the exact values you already calculated if needed to help you plot these points. 4. Next we need to find the critical points of the function. Guiding Discussion Question: • What are the two types of critical points that a function can have? Why do we need to find critical points? to find both types of Find the derivative of the function. Guiding Discussion Questions: • Which type of critical point, do we know that we do not have any of that type? Why? • For the type of critical point that we do actually have for this function, what equation do we need to solve to find the critical points of the function? • What identity can we use to help us solve this equation to get critical points? What other strategy can help here? (Note: Depending on whether you took the derivative of the original function, or the version of the function you got after using the cosine double angle identity to get intercepts, your derivative will look a little different. If you took the derivative of the original version, you'll need to use an identity to help you use another strategy. If you took the derivative of the version you got after using the identity, you should already have a form you can use that strategy on to solve the equation.) 5. Once we have the critical points, we need to determine whether or not the critical points are extrema, and if so, classify them. Guiding Discussion Question: • How can we determine whether or not the derivative changes sign at each critical point? If the thought of choosing numbers in various intervals and plugging them into the derivative to check if it's positive or negative sounds tedious, you might consider making a sign chart as a shortcut. Across the top of your chart, use the critical points to divide [0, 2π] up into several interesting intervals. On the left of your chart, list the factors of the derivative (that you used to find the zeros). Then make a quick sketch of the graph of each of the factors to determine on what intervals the factors are positive and negative, and mark this in the chart. For the bottom row of your chart, multiply the sign of the factors to find the sign of the derivative on each interval.