Topic 27 = Stewart 4.5 (Summary of Curve Sketching) = AP Classroom 5.8 & 5.9

Summary of Curve Sketching: we are going to combine all the techniques we have learned to use Calculus (and some pre-Calculus skills) to analyze the behavior of graphs to generate accurate sketches of curves (theoretically without using a calculator).

The problems are multiple parts. For each, before sketching, you are asked to:

State the domain of the function.
Find the x- and y- intercepts of the graph (if any).
Test symmetry. Recall:
- If $f (x) = f (- x)$ , the function is even and its graph has y-axis symmetry (think of a parabola or the cosine curve).
- If $f (- x) = - f (x)$ , the function is odd, and its graph has origin symmetry (think of a cubic or the sine curve).
- Note: even/odd can not be determined simply by the degree of a function. It has everything to do with what happens when you put in an x-value and its opposite.
Find horizontal and vertical asymptotes. Recall:
- vertical asymptotes are the result of finding the limit of the function at the domain's excluded values;
- horizontal asymptotes are the result of finding the limit of the function as x approaches negative and positive $LaTeX: \infty.$
Find intervals where the function is increasing and/or decreasing using the first derivative.
Find extreme values (minima and maxima) by using the FDT (first derivative test). Remember to use correctly stated claims when establishing extrema.
Find intervals of concavity (concave up or concave down) and point(s) of inflection using the second derivative. Again, make clear statements supporting your conclusions. Always practice writing correct mathematics!
Based upon the information you generate, sketch a curve of the graph.

Action Items:

1. Take notes in class. If absent, review the powerpoints below and/or the extra resources, take notes as necessary (use the Notes document if you prefer), and make note of any questions you may have.

2. Book Practice: 4.5.1: p. 317, #3, 9, 13; *AND 15 OR 21 OR 33. Practice is structured a little differently this time.

You are responsible for #3, 9, and 13 on your own.
With your group, you will develop a presentation for your assigned problem (15, 21, or 33). We will do a gallery walk to look at your group's presentation of your assigned problem during the second half of the day Tuesday.
If you are working from home attempt one of the problems (15, 21, or 33) on your own.

3. Highly recommended: watch and annotate the AP Classroom video for 5.8 and 5.9.

Resources:

Topic 27: Derivatives and Graph Shape
Topic 27: Derivatives and Graph Shape Notes (Optional)
AP Classroom Daily Videos: 5.3 through 5.7
Stewart, Chapter 4, Section 5
Homework Solutions