حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  إجابات كتاب التمارين

إجابات كتاب التمارين

الشرط الأولي

في كل مما يأتي المشتقة الأولى للاقتران begin mathsize 20px style bold f bold left parenthesis bold x bold right parenthesis end style، ونقطة يمر بها منحنى begin mathsize 20px style bold italic y bold equals bold f bold left parenthesis bold x bold right parenthesis end style أستعمل المعلومات المعطاة لإيجاد معادلة الاقتران begin mathsize 20px style bold f bold left parenthesis bold x bold right parenthesis end style:

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 3 x minus 2 semicolon left parenthesis negative 1 comma 2 right parenthesis end style (1)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral left parenthesis 3 x minus 2 right parenthesis d x equals 3 over 2 x squared minus 2 x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 3 over 2 x squared minus 2 x plus C end cell row blank cell f left parenthesis negative 1 right parenthesis equals 2 not stretchy rightwards double arrow 3 over 2 plus 2 plus C equals 2 not stretchy rightwards double arrow C equals negative 3 over 2 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 3 over 2 x squared minus 2 x minus 3 over 2 end cell end table end style

 begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator square root of x end fraction semicolon left parenthesis 4 comma 5 right parenthesis end style (2)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral fraction numerator x plus 1 over denominator square root of x end fraction d x equals integral left parenthesis fraction numerator x over denominator square root of x end fraction plus fraction numerator 1 over denominator square root of x end fraction right parenthesis d x equals integral left parenthesis x to the power of 1 half end exponent plus x to the power of negative 1 half end exponent right parenthesis d x end cell row blank cell equals 2 over 3 x to the power of 3 over 2 end exponent plus 2 x to the power of 1 half end exponent plus C equals 2 over 3 square root of x cubed end root plus 2 square root of x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 2 over 3 square root of x cubed end root plus 2 square root of x plus C end cell row blank cell f left parenthesis 4 right parenthesis equals 5 not stretchy rightwards double arrow 16 over 3 plus 4 plus C equals 5 not stretchy rightwards double arrow C equals negative 13 over 3 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 2 over 3 square root of x cubed end root plus 2 square root of x minus 13 over 3 end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals negative x left parenthesis x plus 1 right parenthesis semicolon left parenthesis negative 1 comma 5 right parenthesis end style (3)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral negative x left parenthesis x plus 1 right parenthesis d x equals integral left parenthesis negative x squared minus x right parenthesis d x equals negative 1 third x cubed minus 1 half x squared plus end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 1 third x cubed minus 1 half x squared plus C end cell row blank cell f left parenthesis negative 1 right parenthesis equals 5 not stretchy rightwards double arrow 1 third minus 1 half plus C equals 5 not stretchy rightwards double arrow C equals 31 over 6 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 1 third x cubed minus 1 half x squared plus 31 over 6 end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals x cubed minus 2 over x squared plus 2 semicolon left parenthesis 1 comma 3 right parenthesis end style (4)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral left parenthesis x cubed minus 2 over x squared plus 2 right parenthesis d x equals integral subscript 0 left parenthesis x cubed minus 2 x to the power of negative 2 end exponent plus 2 right parenthesis d x end cell row blank cell space of 1em equals 1 fourth x to the power of 4 plus 2 over x plus 2 x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 1 fourth x to the power of 4 plus 2 over x plus 2 x plus C end cell row blank cell f left parenthesis 1 right parenthesis equals 3 not stretchy rightwards double arrow 1 fourth plus 2 plus 2 plus C equals 3 not stretchy rightwards double arrow C equals negative 5 over 4 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 1 fourth x to the power of 4 plus 1 over x plus 2 x minus 5 over 4 end cell end table end style

 begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals x plus square root of x semicolon left parenthesis 1 comma 2 right parenthesis end style (5)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral left parenthesis x plus square root of x right parenthesis d x equals integral left parenthesis x plus x to the power of 1 half end exponent right parenthesis d x equals 1 half x squared plus 2 over 3 x to the power of 3 over 2 end exponent plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 1 half x squared plus 2 over 3 x to the power of 3 over 2 end exponent plus C equals 1 half x squared plus 2 over 3 square root of x cubed end root plus C end cell row blank cell f left parenthesis 1 right parenthesis equals 2 not stretchy rightwards double arrow 1 half plus 2 over 3 plus C equals 2 not stretchy rightwards double arrow C equals 5 over 6 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 1 half x squared plus 2 over 3 square root of x cubed end root plus 5 over 6 end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals negative 10 over x squared semicolon left parenthesis 1 comma 15 right parenthesis end style (6)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral negative 10 over x squared d x equals integral negative 10 x to the power of negative 2 end exponent d x equals 10 x to the power of negative 1 end exponent plus C equals 10 over x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 10 over x plus C end cell row blank cell f left parenthesis 1 right parenthesis equals 15 not stretchy rightwards double arrow 10 plus C equals 15 not stretchy rightwards double arrow C equals 5 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 10 over x plus 5 end cell end table end style

(7) إذا كان ميل المماس لمنحنى الاقتران begin mathsize 20px style f left parenthesis x right parenthesis end styleهو begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals square root of x end style، فأجد قاعدة الاقتران begin mathsize 20px style f left parenthesis x right parenthesis end style، علماً بأن منحناه يمر بالنقطة begin mathsize 20px style left parenthesis 9 comma 25 right parenthesis end style.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral square root of x d x equals integral x to the power of 1 half end exponent d x equals 2 over 3 x to the power of 3 over 2 end exponent plus C equals 2 over 3 square root of x cubed end root plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 2 over 3 square root of x cubed end root plus C end cell row blank cell f left parenthesis 9 right parenthesis equals 25 not stretchy rightwards double arrow 54 over 3 plus C equals 25 not stretchy rightwards double arrow C equals 7 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 2 over 3 square root of x cubed end root plus 7 end cell end table end style

(8) إذا كان ميل المماس لمنحنى العلاقة begin mathsize 20px style y end style هو: begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 2 over x squared end style، فأجد قاعدة العلاقة begin mathsize 20px style y end style، علماً بأن منحناها يمر بالنقطة begin mathsize 20px style left parenthesis 2 comma 4 right parenthesis end style.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral 2 over x squared d x equals integral 2 x to the power of negative 2 end exponent d x equals negative 2 x to the power of negative 1 end exponent plus C equals negative 2 over x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 2 over x plus C end cell row blank cell f left parenthesis 2 right parenthesis equals 4 not stretchy rightwards double arrow negative 1 plus C equals 4 not stretchy rightwards double arrow C equals 5 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 2 over x plus 5 end cell end table end style

(9) إذا كان ميل المماس لمنحنى العلاقة begin mathsize 20px style y end style هو: begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 3 x squared minus 12 x plus 8 end style، ومر منحناها بنقطة الأصل، فأجد الإحداثي begin mathsize 20px style x end style لجميع نقاط تقاطع منحنى العلاقة مع المحور begin mathsize 20px style x end style، مبرراً إجابتي.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral left parenthesis 3 x squared minus 12 x plus 8 right parenthesis d x equals x cubed minus 6 x squared plus 8 x plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals x cubed minus 6 x squared plus 8 x plus C end cell row blank cell f left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow x cubed minus 6 x squared plus 8 x plus C equals 0 not stretchy rightwards double arrow C equals 0 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals x cubed minus 6 x squared plus 8 x end cell end table end style

لإيجاد الإحداثيات لنقاط تقاطع المنحني مع محورbegin mathsize 20px style x end style نفرض begin mathsize 20px style y equals 0 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 0 equals x cubed minus 6 x squared plus 8 x not stretchy rightwards double arrow x left parenthesis x squared minus 6 x plus 8 right parenthesis equals 0 not stretchy rightwards double arrow x left parenthesis x minus 2 right parenthesis left parenthesis x minus 4 right parenthesis equals 0 end cell row blank cell not stretchy rightwards double arrow x equals 0 comma x equals 2 comma x equals 4 end cell end table end style

(10) الإيراد الحدي: يمثل الاقتران: begin mathsize 20px style R to the power of straight prime left parenthesis x right parenthesis equals x squared minus 3 end style الإيراد الحدي (بالدينار) لكل قطعة تباع من منتجات إحدى الشركات، حيث begin mathsize 20px style x end style عدد القطع المبيعة، وbegin mathsize 20px style R left parenthesis x right parenthesis end style إيراد بيع begin mathsize 20px style x end style قطعة بالدينار. أجد اقتران الإيراد begin mathsize 20px style R left parenthesis x right parenthesis end style، علماً بأن begin mathsize 20px style R left parenthesis 0 right parenthesis equals 0 end style.

إرشاد: يمثل الإيراد الحدي مشتقة اقتران الإيراد.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell R left parenthesis x right parenthesis equals integral left parenthesis x squared minus 3 right parenthesis d x equals 1 third x cubed minus 3 x plus C end cell row blank cell not stretchy rightwards double arrow R left parenthesis x right parenthesis equals 1 third x cubed minus 3 x plus C end cell row blank cell R left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 0 minus 0 plus C equals 0 not stretchy rightwards double arrow C equals 0 end cell row blank cell not stretchy rightwards double arrow R left parenthesis x right parenthesis equals 1 third x cubed minus 3 x end cell end table end style

(11) يتحرك جسيم في مسار مستقيم، وتعطى سرعته المتجهة بالاقتران: begin mathsize 20px style v left parenthesis t right parenthesis equals 3 t squared minus 12 t plus 11 end style، حيث begin mathsize 20px style t end style الزمن بالثواني، وbegin mathsize 20px style v end style سرعته المتجهة بالمتر لكل ثانية. إذا بدأ الجسيم حركته من نقطة الأصل، فأجد موقعه بعد ثانيتين من بدء الحركة.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell s left parenthesis t right parenthesis equals integral left parenthesis 3 t squared minus 12 t plus 11 right parenthesis d t equals t cubed minus 6 t squared plus 11 t plus C end cell row blank cell not stretchy rightwards double arrow s left parenthesis t right parenthesis equals t cubed minus 6 t squared plus 11 t plus C end cell row blank cell s left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 0 plus 0 plus 0 plus C equals 0 not stretchy rightwards double arrow C equals 0 end cell row blank cell not stretchy rightwards double arrow s left parenthesis t right parenthesis equals t cubed minus 6 t squared plus 11 t end cell row blank cell not stretchy rightwards double arrow s left parenthesis 2 right parenthesis equals left parenthesis 2 right parenthesis cubed minus 6 left parenthesis 2 right parenthesis squared plus 11 left parenthesis 2 right parenthesis equals 8 minus 24 plus 22 equals 6 straight m end cell end table end style

(12) يتحرك جسيم في مسار مستقيم، ويعطى تسارعه بالاقتران: begin mathsize 20px style a left parenthesis t right parenthesis equals 6 t minus 30 end style، حيث begin mathsize 20px style t end style الزمن بالثواني ، وbegin mathsize 20px style a end style التسارع بالمتر لكل ثانية تربيع. إذا بدأ الجسيم حركته من نقطة الأصل بسرعة متجهة مقدارها 72m/s، فأجد موقعه بعد 3 ثوان من الحركة.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell v left parenthesis t right parenthesis equals integral left parenthesis 6 t minus 30 right parenthesis d t equals 3 t squared minus 30 t plus C end cell row blank cell not stretchy rightwards double arrow v left parenthesis t right parenthesis equals 3 t squared minus 30 t plus C end cell row blank cell v left parenthesis 0 right parenthesis equals 72 not stretchy rightwards double arrow 0 plus 0 plus 0 plus C equals 72 not stretchy rightwards double arrow C equals 72 end cell row blank cell not stretchy rightwards double arrow v left parenthesis t right parenthesis equals 3 t squared minus 30 t plus 72 end cell row blank cell s left parenthesis t right parenthesis equals integral left parenthesis 3 t squared minus 30 t plus 72 right parenthesis d t equals t cubed minus 15 t squared plus 72 t plus C end cell row blank cell not stretchy rightwards double arrow s left parenthesis t right parenthesis equals t cubed minus 15 t squared plus 72 t plus C end cell row blank cell s left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 0 plus 0 plus 0 plus C equals 0 not stretchy rightwards double arrow C equals 0 end cell row blank cell not stretchy rightwards double arrow s left parenthesis t right parenthesis equals t cubed minus 15 t squared plus 72 t end cell row blank cell not stretchy rightwards double arrow s left parenthesis 3 right parenthesis equals left parenthesis 3 right parenthesis cubed minus 15 left parenthesis 3 right parenthesis squared plus 72 left parenthesis 3 right parenthesis equals 27 minus 135 plus 216 equals 108 m end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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