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Chapter 14: Q. 46 (page 897)

The volume V of a cube of side x meters isV=V(x)=x3. Find the instantaneous rate of change of the volume with respect to the side x

atx=3.

Short Answer

Expert verified

If the function for the volume of a cube isV=V(x)=x3then the

instantaneous rate of change of the volume with respect to the side x

at x=3will be27m3m.

Step by step solution

01

Step 1. Given information   

Function for the volume of a cube is

V=V(x)=x3

Side of cubex=3

02

Step 2. The Volume of the cube

Substitutex=3in the function for the volume of the cube

role="math" localid="1647033519318" V(x)=x3V(3)=33V(3)=27

So the volume of the cube at the siderole="math" localid="1647033175289" x=3isrole="math" localid="1647033531007" 27m3

03

Step 3. Instantaneous rate of change of volume 

Use instantaneous rate of change formula with respect to side x at x=3

role="math" localid="1647033651982" V'(c)=limx鈫�cV(x)-V(c)x-cV'(3)=limx鈫�3V(x)-V(3)x-3V'(3)=limx鈫�3x3-27x-3V'(3)=limx鈫�3(x-3)(x2+3x+9)x-3V'(3)=limx鈫�3x2+3x+9V'(3)=32+3(3)+9V'(3)=27

So Instantaneous rate of change of volume of the cube is27m3m

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Most popular questions from this chapter

In Problems 7鈥� 42, find each limit algebraically.

limx鈫�01-2x{"x":[[93,93,93,93,93,93,93,93,93,93,93,93,93,93,93],[115,115,115,115,115,115,115,115,115,115,115],[112],[112,112],[132,132,133,134,135,135,135,135,135,135,135,135,135,135,135,135,135,135,135,136,136,136,136,138,138,139,139,139,139,139,139,141,142,143,145,146,148,150,151,152,153,155,155,155,155,155,155,155,155,155,155,155,155,154,154,154,154,154,154,154,154,154,155,156,157,158,159,159,160,162,163,165,167,168,170,171,171,172,173,174,175,175,175,175,175,175,175,176,176],[90,91,91,95,98,100,103,104,106,107,107,107,107,107,106,105,104,104,104,104,104,104,104,104,104,104,104,104,104,104,104,104,104,105,106,106,107,108,110,111,113,115,115,117,118,119,118,116,113,111,109,106,105,103,103,102,101,101,101,101,102,104,105,107,109,111,111,113,114,115,115,116,118,119,119,120,121,122,123,123],[133,134,138,142,144,146,148,150,151,151,152,153,154,155,155,155,156,158,159,159,159,158,157,156,155,154,153,151,148,147,145,143,142,141,140,141,142,143,143,144,145,146,149,151,153,155,157,159,160,160,161,163,163,163,163,163,162,161,160,159,158,157,156,155,155,154,153],[182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182],[183,183,181,180,180,180,180,180,180,180,180,180,180,180,180,181,183,184,186,189,192,196,199,201,203,205,206,207,209,210,210,210,210,210,209,208,206,204,203,201,199,198,196,195,193,191,189,188,187,186,185,183,183],[212,213,215,219,223,225,228,230,231,233,234,234,235,235,235,235,234,231,231,229,228,227,227,227,227,227,227,227,227,227,227,227,227,227,229,231,237,250,261,273,287,295,302,313,320,327,337,340,343,347,348,351,353,355,356,357],[258,258,258,259,259,259,259,259,259,259,259,259,259,259,260],[298,304,310,316,319,321,325,327,328,329,330,331,331,331,332],[336,339,343,345,347,351,355,362,367,368,371,372,372,372,372,371,368,366,364,360,358,355,354,352,351,351,350,349,348,348,348,349,350,351,354,356,359,362,365,367,370,373,375,377,379,379,380,382,383],[383,384,386,388,390,391,393,395,395,396,396,396,396,396,396,396,396,396,396,396,396,396,396,396,395,394,393,392,391,391,391,391,391,391,392,393,394,395,397,399,400,402,403,404,405,406,405,404,403,403,402,401,399,397,396,395,395,394,393,392,392,392,392,392,392,393,395,396,397,398,399,402,403,403,404,405,406,407,407,408,409,410,411,411,412,413,414,415,415,416,417],[340,342,344,347,352,358,360,364,367,370,371,373,375,375,376,377,378,379,379,380,381,382,383,384,385,386,387,388,389,389,391,392,394,395,398,399,401,403,404,407,407,409,411,411,412,413,414]],"y":[[57.399993896484375,58.399993896484375,63.399993896484375,66.39999389648438,69.39999389648438,73.39999389648438,77.39999389648438,81.39999389648438,84.39999389648438,87.39999389648438,90.39999389648438,92.39999389648438,94.39999389648438,96.39999389648438,97.39999389648438],[81.39999389648438,83.39999389648438,85.39999389648438,90.39999389648438,93.39999389648438,97.39999389648438,99.39999389648438,101.39999389648438,104.39999389648438,105.39999389648438,106.39999389648438],[59.399993896484375],[59.399993896484375,59.399993896484375],[80.39999389648438,82.39999389648438,87.39999389648438,92.39999389648438,100.39999389648438,105.39999389648438,107.39999389648438,110.39999389648438,113.39999389648438,115.39999389648438,117.39999389648438,117.39999389648438,118.39999389648438,117.39999389648438,117.39999389648438,116.39999389648438,115.39999389648438,111.39999389648438,108.39999389648438,105.39999389648438,102.39999389648438,101.39999389648438,99.39999389648438,95.39999389648438,93.39999389648438,91.39999389648438,89.39999389648438,88.39999389648438,86.39999389648438,85.39999389648438,85.39999389648438,84.39999389648438,84.39999389648438,84.39999389648438,84.39999389648438,85.39999389648438,87.39999389648438,88.39999389648438,89.39999389648438,92.39999389648438,93.39999389648438,96.39999389648438,101.39999389648438,103.39999389648438,105.39999389648438,108.39999389648438,109.39999389648438,110.39999389648438,111.39999389648438,112.39999389648438,111.39999389648438,110.39999389648438,109.39999389648438,105.39999389648438,100.39999389648438,96.39999389648438,93.39999389648438,90.39999389648438,89.39999389648438,87.39999389648438,86.39999389648438,85.39999389648438,85.39999389648438,84.39999389648438,83.39999389648438,83.39999389648438,83.39999389648438,83.39999389648438,83.39999389648438,83.39999389648438,83.39999389648438,84.39999389648438,84.39999389648438,85.39999389648438,86.39999389648438,87.39999389648438,89.39999389648438,91.39999389648438,92.39999389648438,95.39999389648438,97.39999389648438,100.39999389648438,103.39999389648438,105.39999389648438,105.39999389648438,108.39999389648438,109.39999389648438,109.39999389648438,110.39999389648438],[136.39999389648438,136.39999389648438,136.39999389648438,138.39999389648438,140.39999389648438,143.39999389648438,146.39999389648438,149.39999389648438,150.39999389648438,153.39999389648438,156.39999389648438,159.39999389648438,161.39999389648438,162.39999389648438,164.39999389648438,166.39999389648438,167.39999389648438,168.39999389648438,1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In Problems 7鈥� 42, find each limit algebraically

limx鈫�3(2-5x)

True or False Some rational functions have holes in their graph.

In Problems 7鈥�16, use a table to find the indicated limit.

13.limx鈫�0(ex+1).

In Problems 33-44, find the one-sided limit.

limx鈫�-4-x2+x-12x2+4x
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