Second derivative test ap calc

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  • Second derivative test ap calc

    1. 1. Warm up 1. Look at the intervals where the function is increasing 2. Try to describe what looks different about the regions where the function is increasing y = x 4 − 12 x 3 + 48 x 2 − 64 x
    2. 2. Chapter Three: Section Four    The word we will use to describe the different behaviors of the increasing regions of the function is the word concavity. When the second derivative of a function is positive, we say that the function is concave up. What this means physically is that the movement of the graph has positive acceleration. When the second derivative is negative then the function is said to be concave down and this means that the acceleration of the graph is negative.
    3. 3. Concavity
    4. 4. First derivative: y′ is positive Curve is rising. y′ is negative Curve is falling. y′ is zero Possible local maximum or minimum. Second derivative: y′′ is positive Curve is concave up. y′′ is negative Curve is concave down. y′′ is zero Possible inflection point (where concavity changes). →
    5. 5. Example Determining Concavity Use the Concavity Test to determine the concavity of f ( x) = x on the interval (2,8). 2 Since y " = 2 is always positive, the graph of y = x is concave up on any interval. In particular, it is concave up on (2,8). 2
    6. 6. Inflection Point 1. Point where concavity changes 2. Point where second derivative changes sign 3. POSSIBLY happen when f “ = 0 or f ’’ is undefined
    7. 7. Find the points of inflection and discuss the concavity 1 4 3 f ( x) = x + 2 x 2
    8. 8. Find the points of inflection and discuss the concavity f ( x) = x − 4 x 4 3
    9. 9. Ex01: Determining Concavity 6 f ( x) = 2 x +3
    10. 10. Ex02: Determining Concavity x +1 f ( x) = 2 x −4 2
    11. 11. Learning about Functions from Derivatives

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