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Ex: Find the second derivative for f(x)=x 3 +6x 2 -4x+12 f’(x)= 3x 2 + 12x - 4

Ex: Find the second derivative for f(x)=x 3 +6x 2 -4x+12 f’(x)= 3x 2 + 12x - 4 f’’(x) = 6x + 12. F’(x). F’’(x). F(x). Ex3, pg101) f(x) = x 2 Determine whether each of the quantities is pos., neg., or zero f’(1) Positive, increases from left to right b) f’(-1)

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Ex: Find the second derivative for f(x)=x 3 +6x 2 -4x+12 f’(x)= 3x 2 + 12x - 4

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  1. Ex: Find the second derivative for f(x)=x3+6x2-4x+12 f’(x)= 3x2 + 12x - 4 f’’(x) = 6x + 12 F’(x) F’’(x) F(x)

  2. Ex3, pg101) f(x) = x2 Determine whether each of the quantities is pos., neg., or zero • f’(1) Positive, increases from left to right b) f’(-1) Negative, decreases from L to R c) f’(2) Pos, increases from L to R Higher than f’(1) b/c the line is steeper d) f’(0) Is zero since the graph has a horizontal tangent at x = 0

  3. Which graph is the derivative? Answer: Graph B GRAPH A GRAPH B

  4. Page 120 “If we think the derivative as a rate of change, then the second derivative is the rate of change of the rate of change”

  5. The second derivative is written as f’’ If y =f(x), then the second derivative can be written as d2y which means the derivative of dy/dx dx2

  6. REVIEW • If f’>0, f is increasing • If f’<0, f is decreasing • If f’’>0, then f is concave up • If f’’<0, then f is concave down

  7. F’(X) F(X) F’’(X) • F is concave up, f’(x) increases from negative to positive (left to right) and f’’(x) is positive

  8. F is concave down, f’(x) decreases from negative to positive (left to right) and f’’(x) is negative F’’(X) F(X) F’(X)

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