MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - - PowerPoint PPT Presentation

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Julia Kim December 3rd, 2020 Warm-up question: What is special about the points P and Q in the picture?

y=g(x) P Q

Find the coordinates of P and Q

g(x) = x4 − 6x2 + 9

y=g(x) P Q

True or False – Concavity and inflection points

Let f be a differentiable function with domain R. Let c ∈ R. Let I be an interval. Which implications are true?

1 IF f is concave up on I,

THEN ∀x ∈ I, f ′′(x) > 0.

2 IF ∀x ∈ I, f ′′(x) > 0,

THEN f is concave up on I.

3 IF f is concave up on I

THEN f ′ is increasing on I.

4 IF f ′ is increasing on I,

THEN f is concave up on I.

5 IF f has an I.P. at c,

THEN f ′′(c) = 0.

6 IF f ′′(c) = 0,

THEN f has an I.P. at c.

7 IF f has an I.P. at c,

THEN f ′ has a local extremum at c

8 IF f ′ has a local extremum at c,

THEN f has an I.P. at c. I.P. = “inflection point"

A polynomial from 3 points

Construct a polynomial that satisfies the following three properties at once:

1 It has an inflection point at x = 2 2 It has a a local extremum at x = 1 3 It has y-intercept at y = 1.

Before next class...

• Watch videos 6.6, 6.7, and 6.8, and then 6.9 and 6.5.
• Download the next class’s slides (no need to look at them!)