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Section 4.3

Section 4.3. Relating the Graphs of f, f’ and f’’. First Derivative Test for Local Extrema. TBT: What is a critical point of a function? Point at which f’ is zero or undefined. At a critical point c…..

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Section 4.3

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  1. Section 4.3 Relating the Graphs of f, f’ and f’’

  2. First Derivative Test for Local Extrema • TBT: What is a critical point of a function? • Point at which f’ is zero or undefined. • At a critical point c….. • 1. If f’ changes sign from positive to negative at c, then f has a local maximum value at c.

  3. First Derivative Test • 2. If f’ changes sign from negative to positive at c, then f has a local minimum at c. • 3. If f’ does not change sign at c (same sign on both sides), then f has no local extreme value at c.

  4. How can you tell if an endpoint is a max or min numerically? • Analyze the slope near the endpoints.

  5. Examples • (a) Find local max and mins. • (b) Identify the intervals in which the function is increasing or decreasing. • 1. f(x) = x3 – 6x2 + 9x + 1 • 2.

  6. The Second Derivative • What does y’’ tell you about the graph of y’? • When y’’ is positive, y’ is increasing. • When y’’ is negative, y’ is decreasing. • What does y’’ tell you about the graph of y? • When y’’ is positive, the slope of y is increasing. • When y’’ is negative, the slope of y is decreasing. • What does increasing/decreasing slope look like? • When y’’ is positive, y is concave up. • When y’’ is negative, y is concave down.

  7. Concavity on a Graph • Where does the concavity change on the graph below? These points are called points of inflection and they are critical points for y’’.

  8. Examples • 3. Use the function from example 1: • f(x) = x3 – 6x2 + 9x + 1 • Determine where this function is concave up and concave down. • Combine the information learned from f’’ with the info learned from f’ into one chart. • Sketch the graph of f(x) using the information in the chart.

  9. Additional Example • f(x) = x2ex • 1. Find the x-coordinates for any critical points. • 2. Determine where f is increasing/decreasing. • 3. Determine the max and mins. • 4. Find where the function is concave up/down. • 5. Locate any points of inflection. • 6. Make a rough sketch based on your answers from 1-5.

  10. Application to Motion • Position = s(t) • Velocity = v(t) = s’(t) • Acceleration = a(t) = v’(t) = s’’(t) • Remember that these are all vectors. Positive values generally indicate to the right or up. Negative values usually refer to left or down. • Speed is not a vector. If acceleration and velocity have the same sign, the object is speeding up (either in a + or – direction. • If a(t) and v(t) have opposite signs, the object is slowing down.

  11. Example • s(t) = 3t4 – 16t3 + 24t2 • 1. When is the object moving left/right? • 2. When does the object reverse direction? • 3. When is the velocity increasing/decreasing? • 4. Describe the motion of the object (including its speed) in words • 5. Sketch a graph of the position curve. • 6. Sketch a graph of the velocity curve. • 7. Verify answers using the graphing calculator.

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