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IB Maths. Turning points. First derivative test

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By the end of the lesson, students will be able to use derivatives to find maximum and minimum points of a function, and use second derivatives to determine the nature of stationary points and points of inflection. Specifically, they will learn that: (1) if the first derivative is zero at a point, it is a stationary point; (2) the second derivative test can determine if it is a local max, min or point of inflection; and (3) points of inflection occur when the curve changes concavity. Students will apply these concepts to find the stationary points of sample functions and classify their nature.

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By?the?end?of?the?lesson?you?will?be?able?to: ? Use?the?derivative?of?a?function?to?find? ????maximum?and?minimum?points. ? Use?the?second?derivative?to?test?the?nature?of?a? stationary?point?and/or?point?of?inflexion.
increasing increasing decreasing stationary stationary http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html
? ? ? ? ? ? ? ? A B P Q At??A???? ?f?'(x)?>?0?? f?is?increasing ????????? At??P???? ?f?'(x)?<?0?? f?is?decreasing ????????? At??B?and?Q???? ?f?'(x)?=?0?? B?and?Q?are? stationary?points.????????? f?'??=?0 f?'??<?0 f?'??>?0 f?'??>?0 f?'?=?0
If the derivative is positive then the function is increasing. If the derivative is negative then the function is decreasing. ?
f?'(?a?)?=?0???? (a,?f(a))???is?a?stationary?point ? ? ? ? ? ? ? ? A B P Q A point on a curve at which the gradient is zero is called a stationary point. At a stationary point, the tangent to the curve is horizontal.
? ? ? ? ? ? Local Maximum point ?f?'???>?0 ?f?'???=?0 ?f?'???<?0 P To?the?left?of?P At?point?P To?the?right?of?P ?f?'???>?0 ?f?'???<?0?f?'???=?0 P??is?a?local?maximum?point
Local Minimum point ?f?'???>?0 ?f?'???=?0 ?f?'???<?0 To?the?left?of?P At?point?P To?the?right?of?P ?f?'???>?0?f?'???<?0 ?f?'???=?0 P??is?a?local?minimum?point P ? ? ? ?? ? Maximum?and?minimum?points?are?also?called? turning?points.
Point of inflexion ? ? ? ? ? ? f?'?=0 f?'?>?0 f?'?>?0P ? ? ? ? ? ? f?'?=?0 f?'?<?0 f?'?<?0 P f?'?(?a)?=?0?but??f?'?has?the?same?sign?to?the?right? and?left?of?a,??a?is?called?a?horizontal?point?of? inflexion.(because?the?tangent?is?horizontal?at?P) Point?of?inflection.ggb
? ? non?horizontal?point?of?inflexion?(?tangent?is?not? horizontal) f?'?<?0 f?'?<?0 tangent ? ?
Find?the?coordinates?of?the?stationary?points?on?the? curve?y=?x3 +3x2 +1?and?determine?their?nature.
y=?x3 +3x2 +1?
Find?the?coordinates?of?the?stationary?points?on?the? curve?y=?x4? ??4?x3 ??and?determine?their?nature.
y=?x4? ??4?x3 ?
Attachments Point?of?inflection.ggb

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