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Warm Up

Warm Up. With a partner, do problem #40 on page 207. 5.3 B – Curve Sketching (concavity). Goal. Use concavity test to find info about the graph of f(x). Definition: concavity. concave up. concave down. f(x) is concave up if y’ is increasing. (y’’>0)

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Warm Up

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  1. Warm Up With a partner, do problem #40 on page 207

  2. 5.3 B – Curve Sketching (concavity)

  3. Goal • Use concavity test to find info about the graph of f(x).

  4. Definition: concavity concave up concave down • f(x) is concave up if y’ is increasing. (y’’>0) • f(x) is concave down if y’ is decreasing. (y’’<0) • “Smiley face” = concave up • “Frowny face” = concave down

  5. Definition points of inflection • The place on a graph where concavity changes. • Find where second derivative equals 0 or DNE.

  6. Summary is positive is negative is zero Second derivative: Curve is concave up. Curve is concave down. Possible inflection point (where concavity changes).

  7. Example • Find the concavity of • Solution: the function is concave up everywhere

  8. Example • A particle is moving along the x-axis with position function • Find the velocity, acceleration, and describe the motion of the particle.

  9. Solution • To describe the motion, we want to set up sign charts for both v(t) and a(t), so we need the critical points for each (set equal to 0 and solve for t)

  10. Solution Continued Remember, when velocity is positive, object moves right, when it is negative it moves left. The particle moves right in the time intervals and left in . The accelerating force is directed to the left in the time interval , is momentarily 0 at and is directed toward right at

  11. Estimate where f’ and f’’ are 0, positive and negative. Solution: f’ : 0 at x=-1 and x=1 positive at negative at (-1, 1) f’’ : 0 at x=0 (point of inflection) positive at (concave up) negative at (concave down) Example

  12. Homework • 5.3 Day 2: 8, 12, 15, 16, 20-30

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