Ch 2 Differentiation Math 12 Class

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If ?(?) =?/?, then a critical point of ? is

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Let ? be a differentiable function such that ?′(?) = ? then if ?′(?) does not change sign i.e.,before and after ? = ? , then it occurs ______ at ? = ?

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Let ? be a differentiable function such that ?′(?) = ? then if ?′(?) changes sign from -iv to+iv i.e., before and after ? = ? , then it occurs relative ______ at ? = ?

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Let ? be a differentiable function such that ?′(?) = ? then if ?′(?) changes sign from +iv to –iv i.e., before and after ? = ? , then it occurs relative ______ at ? = ?

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If ?′(?) does not change before and after ? = ? , then this point is called_______

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If ?′(?) = ? or ?′(?) is undefined , then the number ? is called critical value and thecorresponding point is called_______

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A stationary point is called ______ if it is either a maximum point or a minimum point

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A function ?(?) is such that , at a point ? = ? , ?′(?) = ? at ? = ? , then ? is said to be

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A function ?(?) is such that , at a point ? = ? , ?′(?) < 0 at ? = ? , then ? is said to be

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A function ?(?) is such that , at a point ? = ? , ?′(?) > 0 at ? = ? , then ? is said to be

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If ? is differentiable function at the point ? and ? is differentiable at point ?(?), then (???)′(?) ???/??(???)(?) =

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If ?(?) = ?^? + ??^? + ?^? then ?′(?) =

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If ? be a real valued function , continuous in interval ]?, ??[∈ ?? and if ??? ??→? (?(??)−?(?))/(??−?)exists, then the quotient is called

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If ?′(?) = ?, ?′′(?) ≤ ? at a point P, then P is called

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? − ? + ?? − ?? + ?? + ⋯ . +(−?)^?.?^? + ⋯ is the expansion of

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The function ?(?) = ??^? + ?? + ? has minimum value if:

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?/?? (?(−?) )

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A function ? is neither increasing nor decreasing at a point , provided that ?′(?) = ? at thatpoint , then it is called:

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?/?? (??????) is equal to

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If ?′(?) = ?, then ? has relative maximum value at ? = ?, if

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. If ? = ?^??, then ?? =

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?^? ?/??(????) =

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?/?? ?^(????) is equal to

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If ?(?) = ?^(??), then ?′′′(?) =

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?/?? (?/????) =

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? = ?^? has the value

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Let ? be differentiable function in neighborhood of ? and ?′(?) = ? then ?(?) has relative maxima at ? if

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At ? = ? , the function ?(?) = ? − ?^?has

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?/?? (??^(????)) =

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If ?(?) = ???? then ?′(???^−?(??)) =

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If ?(?) = ?? + ?? + ? then ?′′(?) =

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?/??(????) −?^?/??^? (????) =

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The maximum value of the function ?(?) = ?^? − ? − ? is

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?(?) = ???? is

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A point where 1st derivative of function is zero , is called

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If a function ? is decreasing within ]?, ?[ ,then slope of tangent to its graph within]?, ?[remains

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If a function ? is increasing within ]?, ?[ ,then slope of tangent to its graph within]?, ?[remains

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? is said to be decreasing on ]?, ?[ if for ??, ?? ∈]?, ?[

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? is said to be increasing on ]?, ?[ if for ??, ?? ∈]?, ?[

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? − ? + ?? − ?? + ?? − ⋯ =

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?(?) = ?(?) + ??′(?) +(?^?/?!)?′′(?) +(?^?/?!)?′′′(?) + ⋯ … . + (?^?/?!) ??(?) … is called_____ series.

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If ???(?? + ?), ???? ?? =

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If ? = ?^−??, ????? ??/?? =

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If ? = ?^−?? , ???? ?? =

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If ????−?(????), ???? ??/?? =

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If ? = ????−?(?? + ?), then ??/?? =

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? = ??????−?? if and only if ? = ??????? is valid when

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? = ????−?? if and only if ? = ????? is valid when

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? = ????^−?(?) if and only if ? = ????? is valid when

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? = ????^−? (? ) if and only if ? = ????? is valid when

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?/?? ??[?(?)] =

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???? (?) =

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If ? > 0 ,? ≠ ?, and ? = ?^? then the function defined by ? = ????^?(? > 0) is called alogarithmic function with base

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The function ?(?) = ?^?, ? > 0 , ? ≠ 0, and ? is any real number is called

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?/?? (?(?))^?=

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(???)′(?) =

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. If ?(?) =?/?, ???? ?′′(?) =

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?/?? (?/?(?))

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[?(?)?(?)]′=

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???[?(?) + ?(?)] =

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?/??(????) =

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?/??(?? + ?)^? = ??(?? + ?)^(?−?) is valid only when ? must be:

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?/??(?^?) = ??^(?−?) is called

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????→? ?(?)−?(?)/(?−?)=

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The notation ??(?) or ?? is used by

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The notation ?′(?) or ?′ is used by

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The notation ?̇(?) is used by

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The notation ??/?? or ??/?? is used by

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The change in variable ? is called increment of ?.It is denoted by ?? which is

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?/??(?? + ?)^? =(

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If ? = ?^??, then ?? =

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?/?? (??) =

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?/?? (?????) =

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Ali Durrani