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Advanced Calculus and Mathematical Analysis: Derivatives, Integrals, and Graphical Analysis

This solved assignment explores derivatives, integrals, and graphical analysis in advanced calculus.

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Advanced Calculus and Mathematical Analysis: Derivatives, Integrals,and Graphical Analysis1)Find theslope of curve as( )()()( )( )3322939339039dhtttdtdddttdtdtdttt=−−=−−=−−=−At3t=slope is( )( )233 3927918h=−=−=Write the equation of line with slope18m=and point()()11,3,3ty=−as()()()113183318541857yym ttytytyt−=−− −=−+=−=−Thus, the equation of tangent line is1857yt=−. Hence, the correct option isB.2)The volume of sphere at3.0r=is()()31343.0336 3.14113.04 cmV===The volume of sphere at3.1r=is()()()323343.134 3.143.13124.72 cmV==The change in volume is33213124.72 cm113.04 cm11.68 cmVV−=−=Hence, the correct option isC.3)Asxtens to2 from left, the function value is( )2 226+=and asxtens to 2 from right, thefunction value is( )246+=, so to remove the discontinuity()2fmust be equal to 6.Hence, the correct option isB.4)To find the velocity function, differentiate position vector with respect totas( )()222222122dv ttdttt=+=+=+At1t=,( )1122 1141 m/sec2v=+==Hence, the correct option isB.5)Since the slope of line is positive in interval()5,3−−and()0,3, so0f in this interval.Since the slope of line is negative in interval()3,0−, so0f in this interval.

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Since the slope of line is constant in interval()3,6, so0f =in this interval.Hence the correct graph isC.6)The derivative of20.05dvv=+with respect tovis()()( )()220.050.050.05 210.11dddvvvvdvdvdvvv+=+=+=+At46v=,()0.1 4614.615.6+=+=Hence the correct option isA.7)Let the length of rectangle isxfeet and width of rectangle isyfeet.Since the area is 680 square feet, so680680xyyx==The cost is given by()()7 26 214126801412816014Cxyxyxxxx=+=+=+=+Derivative of cost function is2816014Cx =−SetCequal to zero and solve forx222816014081601481601481601424.1xxxx−====So, value ofyis68024.1428.2y=Hence the correct option isC.8)Find the composite function as()()72727277xgfxgxxx−=−=+=−+=Hence, correct option isA.9)Integrate the function as
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