Learning Outcomes

1. Explain how the sign of the first derivative affects the shape of a function’s graph.
2. Use the first derivative test for local extrema.
3. Explain how the sign of the second derivative affects the shape of a function’s graph.
4. Find the inflection points of a function.
5. Use the second derivative test for local extrema.

Textbook

• Chapter 4.5  Derivatives and the Shape of a Graph

Textbook Assignment

• p. 403:   223-226 all, 229

WeBWork Assignment

• Application-Monotonicity
• Application-Shape of Polynomials

Exit problems of the session

1. For the functions :

(a). Determine the intervals of increase and decrease

(b). Determine the local maxima and minima

(c). Determine the intervals of concavity

(d). Determine the points of inflection

(e). Sketch the graph with the above information indicated on the graph

Key Concepts

• First derivative and the shape of a function:
• If over an interval I, then increases over I.
• If over an interval I, then decreases over I.
• First Derivative Test for local extrema:
• If is a critical point of and for and for , then has a local maxima at • If is a critical point of and for and for , then has a local minima at • Second derivative and the shape of a function:
• If over an interval I, then is concave up over I.
• If over an interval I, then is concave down over I.
• Inflection point: if is continuous at and changes concavity at , the point is an inflection point of .
• Second Derivative Test for local extrema:
• If and , then has a local minima at .
• If and , then has a local maxima at .

#### Videos and Practice Problems of Selected Topics

• The first derivative test
1. The first derivative test (5:24) A discussion on finding relative extrema using derivatives.
2. Finding relative extrema (7:48) Find the relative maximum point of .
3. Analyzing mistakes when finding extrema I (4:32) Pamela was asked to find where has a relative extremum.  Is her work correct?
4. Analyzing mistakes when finding extrema II (4:41) Erin was asked to find if has a relative maximum.  Is her work correct?
5. * Finding relative extrema A text with detailed examples and questions.
6. * Practice: Finding relative extrema.
7. * Relative minima and maxima review A text with detailed examples and questions.
• Determining concavity of intervals and inflection points: graphical
1. Concavity introduction (9:53) Finding where the function is concave up/down using derivatives.
2. Analyzing concavity graphically (2:22) A function is plotted. Highlight were and .
3. * Practice: Concavity intro. (4 problems)
4. Inflection points (2:33)  A discussion based on a graph of , and .
5. Inflection points (graphical) (3:20) The graph of a differentiable function over is given. How many inflection points does the graph of have?
6. * Practice: Inflection points. (4 problems)
• Determining concavity of intervals and inflection points: algebraic
1. Analyzing concavity (9:15) Find the intervals where is concave up/down.
2. Inflection points (5:34) Find the points of inflection of .
3. Mistakes when finding inflection points (6:10) Robert was asked to find where has inflection points. Is his work correct?
4. Mistakes when finding inflection points (4:01) Olga was asked to find where has inflection points. Is her work correct?
5. * Analyzing the second derivative to find inflection points. A review with detailed examples and questions.
6. * Practice: Analyze concavity. (4 problems)
7. * Practice: Find inflection points. (4 problems)
• Using the second derivative test to find extrema
1. The second derivative test (6:12) A discussion on the second derivative to find relative extrema.
2. * Practice: The second derivative test. (4 problems)