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Calculus Chapter 4

Calculus Chapter 4. 4.1 If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b), . Rolle’s Therem. and if f(a) = f(b), then there is at least one number c between a and b at which f’(c)=0. Calculus Chapter 4. 4.1

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Calculus Chapter 4

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  1. Calculus Chapter 4 4.1 If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b),

  2. Rolle’s Therem and if f(a) = f(b), then there is at least one number c between a and b at which f’(c)=0.

  3. Calculus Chapter 4 4.1 If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b),

  4. Mean Value Theorem then there is at least one number c between a and b at which f’ ( c) = f(b) – f(a) . b-a

  5. Def. Extrema • Absolute minimum • Absolute maximum • Relative minimum (local min) • Relative maximum (local max) • Note: our book classifies abs min/max as local too

  6. Critical numbers are found • When f’(x)=0 • When f’(x) DNE

  7. Critical points arepossible extrema Other extema may occur at endpoints of the interval.

  8. Theorem If a function has a min or max at c, c in (a,b), then f’ ( c) exists and f’ ( c ) =0.

  9. Calculus Chapter 4 4.2 If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b),

  10. First Derivative Test If f’ (x) >0 at each point of (a,b), Then f increases on (a,b). If f’ (x)<0 at each point of (a,b), Then f decreases on (a,b).

  11. “Definition” • Concave up • Concave down

  12. Def. Point of inflection occurs when the graph changes concavity.

  13. Possible point of inflection • When f’’(x)=0 • When f’’(x) DNE

  14. warning • A function may have a local min or max without the graph having a horizontal tangent. • A function may have a horizontal tangent without the function having a local min or max.

  15. More info on f’’ • If f’(x)=0 and f’’(x)>0,then f(x) is a rel min • If f”(x)=0 and f’’(x)<0, then f(x) is a rel max. • If f’(x)=0 and f’’(x)=0 then the f’’ test fails and you must use the first derivative test.

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