Review Topic 27 = Stewart 4.5 (Derivatives & the Shape of a Graph) = AP Classroom 5.7

There really is a TON of information in these sections, and all of it is tested in multiple ways on the AP test, because it is one of the "big ideas."

In Summary of Curve Sketching: you 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 have 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 $LaTeX: f\left(x\right)=f\left(-x\right)$ the function is even and its graph has y-axis symmetry (examples are the quadratic parent function or the cosine curve);
- If $LaTeX: f\left(-x\right)=-f\left(x\right)$ the function is odd, and its graph has origin symmetry (examples are the cubic parent function or a sine curve);
- Note: whether or not a function is even or odd cannot be generalized from its leading degree (if it is a polynomial.
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 WITHOUT using a calculator or Desmos or another graphing tool! 😳🥸 (You may check yourself once you have attempted to graph without a graphing tool).

Action Items:

1. Take notes in class (notes are likely to be me leading you through one of the homework problems as an example). 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.

3. *Optional: WaMap, 3.5: Sketching Graphs and Optimization

Resources:

Topic 27: Derivatives and Graph Shape
Topic 27: Derivatives and Graph Shape Notes (Optional)
AP Classroom Daily Videos: 5.7
Stewart, Chapter 4, Section 3
Homework Solutions