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习题练习:IB MAI HL Calculus Topic 5.2 Integration



 作者: admin   总分: 52分  得分: _____________

答题人: 匿名未登录  开始时间: 24年03月14日 00:00  切换到: 整卷模式

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1#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The area A is defined as the region bounded c;5t6mw zbz ;.sc; k;hnf/ocb by the curve y=x26(x6) and the x -axis for 0x6 .
1. Sketch the curve on the diagram below, shading the area A .

2. Write down a definite integral that represents area A .
3. Find the area of A .
[Area A]=   units2

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2#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The area A is defined as the region boundesed:(dt-31mj rj/pp fd by the curve y=x25(2x5) and the x -axis for 0x52 .

1. Sketch the curve on the diagram below, shading the area A .

2. Write down a definite integral that represents area A .
Area A=05/2x25(2x5)dx ;a=  ,b=  .
3. Find the area of A .
Area A=ab units2 ; a=  ,b=  .

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3#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The equation of a curve is y:+j i,j :4re1osdaqf k=-x3+4x2+x4 . A section of the curve is shown on the diagram below, with the three x -intercepts labelled.

1. Find dy dx = - ax2+ bx+c ; a=  ,b=  ,c=  .
2. Write down the coordinates of the local maximum.
3. Write down an integral representing the area of the shaded region.
A=14(x3+ax2+xb)dx;a=  ,b=  .
4. Find the area of the shaded region.
A=ab;a=  ,b=  .

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4#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The equation of a curve isk(2la .k cn*x.rp o bvkdb-3f. y=-x2+8x12 . A section of the curve is shown on the diagram below, with the two x -intercepts labelled.

1. Find dy dx = -ax + b; a=  ,b=  .
2. Write down the coordinates of the local maximum, the local maximum is located at (  ,  )
3. Write down an integral representing the area of the shaded region.
4. Find the area of the shaded region.
A=ab units2;a=  ,b=  .

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5#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let f(x)=2xex and g(x)=-2 f(x)-1 .
The graphs of f and g intersect at x=a and x=b , where a < b

1. Find the value of a and of b .
2. Hence, find the area of the region enclosed by the graphs of f and g .
A=  .

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6#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A reservoir of water has a cross-sectional area shown in the shaded region2ue/6 9 x n,dqrndd;sj of the diagram below, where x and y values are in metres. The bottom of the reservoir (the curve shown)jux9q2dd/n,erdn ; 6s has an equation in the form

y=k(x8)2

1. Find the area of the rectangle OMNP, The area of the rectangle OMNP is determined by A1  m2.

2. Determine the value of k is ab;a=  ,b=  .
3. Find the cross-sectional area of the reservoir.
The number is    m2.

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7#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A section of the curve with equationq:q d r;cov+l5 y=2(x-3)(x+8) is shown.

1. Write an integral for the shaded area A.
[Area A] = ab2(x3)(x+8)dx; a=  ,b=  .
2. Find the area of A is    units2 .
3. Find the area of B is    units2 .

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8#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram shows th5u0 kw wtnd;(ce parabolic shape of a gateway arch that has a span of 12 me;0(uc5w ntwkdtres and a maximum height of 8 metres.

The curve has an equation in the form y=k(x6)2+8 .
1. Determine the value of k is ab;a=  ,b=   .
2. Write down an integral that represents the cross sectional area under the arch shown as OMN.
A=012[ab(x6)2+c]dx;a=  ,b=  ,c=  
3. Find the cross sectional area under the arch.Evaluating the integral in part (b), we get a=   m2.

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9#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let f(x)=92ln(x2+4), for xR. The graph of f passes through the point (p, 3) , where p>0 .
1. Find the value of p is   .

The following diagram shows part of the graph of f .

The region enclosed by the graph of f , the x -axis and the lines x=-p and x=p is rotated 360 about the x -axis.
2. Find the volume of the solid formed is   .

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10#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the region enclosed by the graph o;yk5fgssx6s+m3pln7qno1(d f the function f(x)=ex/2 , the x -axis and the vertical lines x=0 and x=3 .
1. Sketch the function f and shade in the region.

2. 1. Write down a definite integral that represents the shaded region, it is  1. a3ex/b dx,a=  ,b=  .
2. Calculate the area of the shaded region is   .
3. Use the trapezoidal rule with n=6 trapeziums to estimate the area of the region.

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11#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let f(x)=13x6 , for x>2
1. Find (f(x))2 dx = 1aln(bx6)+C;a=  ,b=  .

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f , the x -axis, and the lines x=3 and x=11 .
2. Find the exact volume of the solid formed when R is rotated 360 about the 3 x -axis.
V= πalnb,a=  ,b=  .

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12#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  On the Persian Gulf, there is a small peninsula that can be modelled by t0dk elc w(cpcoz(3;ra:l /f 9che equatif(c zcpwdl3:9 e; krc/a0(colon y=0.2x3+1.2x2x where x and y are measured in kilometres, as shown in the following diagram. Points P and Q are on the coastline.

1. Find the coordinates of P is (  ,  ) and Q is (  ,  ) .
2. Find the furthest distance that the land protrudes from the coastline.
3 . Find the area of the peninsula,The area of the peninsula is given by A=   km2.

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13#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The graphs of y=-0.5 x+2.5 and ufqsx ,b8kh/;0twk si; n.m krk2 pxtt--05gay=0.5x2+0.5x+1 intersect at (-3,4) and (1,2) , as shown in the following diagrams.

1. Calculate the shaded area in diagram 1 , the area enclosed by the lines y=-0.5 x+2.5 x=-3, x=1 and the x -axis, A1  units2
2. 1. Write down an integral for the shaded area in diagram 2 , the area enclosed by the curve y=0.5x2+0.5x+1 , the lines x=-3 and x=1 , and the x -axis.
A2=31(0.5x2+0.5x+a)dx;a=  .
2. Calculate the area of this shaded region,A2≈   units2
3. Hence, determine the area enclosed by the line y=-0.5 x+2.5 and the curve y= 0.5x2+0.5x+1

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14#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A curve y=f(x) has a gradient funh8pwp u/vk 4u*fq* s2zction of f^{\prime}(x)=a x+1 . The diagram below shows part of the curve, with the x and y intercepts labelled.

1. Find the equation of the curve is y=-ax2+bx+c; a=  ,b=  ,c=  .
2. Find the area under the curve from 0A=abunits2,a=  ,b=  .

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15#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A curve y=f(x) has a gradi ;7nv(fo1bunqkfz vrpn- )x42ent function of f(x)=ax+5. The diagram below shows part of the curve, with the x and y intercepts labelled.

1. Find the equation of the curve,y=ax2+bx+c;a=  ,b=  ,c=  .
2. Find the area under the curve from 0 ,b=  .

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16#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The shape of a bowl is given by revolving thstkgow3l;vj; csk :*oix/rv 1w5) qv)e curve y=13x74, 0x6through360 about the y -axis. The units are in centimetres.

1. Find the height of the bowl.
h≈   cm.(Two decimal places)
The bowl is completely filled with water.
2. Find the volume of water in the bowl.
V≈   cm3(round number)

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17#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram shows the graph of the functio g /as:ey-qqwgg3wf (tb +9p(nn f(x)=baa2x2

Find an expression, in terms of a and b , for the volume of the solid formed by rotating the curve 360 about the x -axis.
we get V=xπab2y;x=  ,y=  .

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18#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following table shows the rate, R , at which i w2u,3vlq 8mpu *i/zorubbish is produced by a town, in tonnes per day. t represents the time in days since the rate of rubbisuu2*qopi8, /mv iw zl3h production was first recorded.

The total amount of rubbish produced by the town over the 3-day period is equal to the area underneath the graph of the curve R(t) .

1. Use the trapezoidal rule with 6 intervals to estimate the total amount of rubbish produced by the town over the 3-day period;The total amount of rubbish produced is approximately    tonnes (round number please)

The actual amount of rubbish produced over the 3 days was 191 tonnes.
2. Find the percentage error between the estimate in part (a) and the actual amount of rubbish produced;The percentage error between the estimate in part (a) and the exact value can be found using the percentage error formula is close to   % (tow decimal please)

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19#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A curve y=f(x) passes through poi oj 4jj0 w8a*pmodw;4unt P(2,7) and has a gradient of f(x)=4x3.
1. Find the gradient of the curve at point P ;f(x)=  .
2. Find the equation of the tangent to the curve at point P; y=kx-b,k=  ,b=  .
3. Determine the equation of the curve; the equation of the curve is y=ax2-bx+c;a=  ,b=  ,c=  .

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20#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram shows part bwp :0n7jh eu(jze5ke:n82osof the graph of f(x)=\left(16-x^{2}\right)(x+ 3 ), x \in \mathbb{R} . The shaded region R is bounded by the x -axis, y -axis and the graph of f .

1. Write down an integral for the area of region R ;[Area R] = 04(ax2)(x+b)dx;a=  ,b=  .
2. Find the area of region R ; [Area R]=    units2

A trapezoid A B C D has four vertices with the coordinates shown as below.

3. Find the value of a , the x -coordinate of point D , such that the area of the trapezoid is equal to the area of region R .
Since the area of the trapezoid ABCD is equal to the area of region R, we get a is   .

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21#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram shows part 9mdm/vf btm6-l 3m2kn of the graph of f(x)=x33x24x+12, xR . The shaded region R is bounded by the x -axis, y -axis and the graph of f .

1. Write down an integral for the area of region R ;[Area A]=20(x3ax24x+b)dx;a=  ,b=  
2. Find the area of region R is    units2

A parallelogram A B C D has four vertices with the coordinates shown as below.

3. Find the value of a , the y -coordinate of points B and C , such that the area of the parallelogram is equal to the area of region R . a=  .

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22#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A company is going to make an excavation to a g/ liq:bo+rp;ccess a mine. The cost of excavating r/qp +bl io:;gis $ 12 per m3 . The following table shows an estimate of the areas of horizontal crosssections, at 5 -metre intervals, of the underground to be removed where the height is measured in metres below the ground level.


1. Use the trapezoidal rule to find an estimate of the volume of ground to be removed,we obtain the V ≈   m3 (round number)
2. Hence, or otherwise, estimate the total cost of excavation;The total cost of excavation is $   .


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23#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A piece of jewellery is being desig isyp3 +2x-bfnned for an auction. Its shape is based on the gras+2 fyb3xn-ip ph of

y=x50625x2 for 0x25

which is revolved 360 about the x -axis, where x and y are measured in mm .

1. Find the area bounded by the graph and the x -axis.
Area ≈   .
2. Hence or otherwise, find the volume of the jewel is(≈)   mm3.
3. If 1 cm3 of silver weighs 10.5 grams, and 1 gram of silver costs $1.50 , find the cost of the silver used for the jewel to the nearest dollar is(≈)   .

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24#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Rain is filling an empty water reservoihl3preuk ,hw*f4z;o8+t nf z5r at a rate of r litres per minute, where r=4 e pwofhznl8u +h3,zt*; krf5t5 and t is the time measured in minutes. The graph of this is shown below.

1. Find how much water has filled the reservoir after 1 hour ; we get V=    L
2. Determine an expression for the amount of water that has filled the reservoir for any given number of minutes, m ;The volume of water that has filled the reservoir after m minutes can be expressed as V=abm2 (L);a=  ,b=  .

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25#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram s1e *u0pwcvcat0+1 wta hows the graph of f(x)=4xx2+1 , for 0x6 , and the line x=6 .

Let R be the region enclosed by the graph of f , the x -axis and the line x=6 .
1. Show that f(x)dx=2ln(x2+1)+C .
2. The exact value for the area of the region R is lnk , where k is a positive integer. Find the value of k is   .

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26#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The diagram below shows thh /1vh+ i)wyhfgkj9 7l*wpyl 7e graphs of functions y=42x2 and y=2x3.

1. Find the volume resulting from a rotation of the shaded region shown in the diagram through 2π about the x -axis.
V≈   .( one decimal pleaces)
2. Find the volume resulting from a rotation of the region shown in the diagram through 2π about the y -axis.V≈   .( two decimal pleaces)

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27#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The diagram below shows the graphs of functions p5cfw507sky5djm8ohc 9 c9zw y=53x3 and y=2x4 .

1. Find the volume resulting from a rotation of the region shown in the diagram through 2π about the x -axis.V≈   (one decimal pleaces)
2. Find the volume resulting from a rotation of the region shown in the diagram through 2π about the y -axis.V≈   (two decimal pleaces)

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28#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Bellatrix Aviation manufactures bnfvd*sje ;jflu*-ant9 n.+1r usiness jets. The profit the company makes, P , in millions of dollars, changes based on n *uevj*- tf +1rs 9lnanf.;djthe number of jets, x , they manufacture per quarter.
The rate of change of the profit made from manufacturing x jets is modelled by

dP dx=0.4x+16,x0.


In the previous quarter, the company made a profit of 165 million dollars from manufacturing 20 jets.
1. Find an expression for P in terms of x .Therefore we obtain P= ax2+bxc;a=  ,b=  ,c=  .

The company has the capacity to increase the number of jets they manufacture.
2. Describe how the profit changes if the company increases manufacturing to over 40 jets per quarter and up to 75 jets per quarter.

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29#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The water in a pool is bdec 4dufz43 hw *y-n cg/ui5w/eing pumped out into a nearby lake at a rate of r litres per secfwcu4ed*3hzcd/ ny 5g4u w/-iond where r=2t275 and t is time measured in seconds. The graph of this is shown below.

1. Find out how much water has been pumped out after 1 minute,We get V≈   L
2. Determine an expression for the volume of water that has been pumped out of the pool for any given number of seconds, s ;The volume of water that has been pumped out of the pool after s seconds can be expressed as
v=abs2 (L) ;a=  ,b=  .

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30#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the region R enclotlc bp:9h 1d1osed by the curve y=3(x2)(x8) and the x axis.

1. Sketch the curve and shade the region R on the following axes.

2. 1. Write down an integral that represents the area of region R ;The area of region
R is determined by A=28a(x2)(xb)dx ;a=  ,b=  .
2. Find the area of region R is    units2.
3. Estimate the area of region R using five trapezoids.
4. Find the percentage error between the exact value found in part (b)(ii) and the estimation in part (c);The percentage error between the estimation in part (c) and the exact value in part (b)(ii) can be found using the percentage error formula [% error]=   %

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31#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let f(x)=x(x3)2 , for 0x4 . The following graph shows f .

Let R be the region enclosed by f , the x -axis and the lines x=0, x=3 .
1. Show that f(x)dx=x442x3+9x22+C.
2. Find the volume formed when R is rotated 360 about the x -axis ;The volume is given by the integral V=   .
3. The diagram below shows part of the graph of the quadric function g(x)=x(a-x) . The graph of g crosses the x -axis when x=a .

The area enclosed by the graph of g is equal to the area of R . Find the value of a is   .

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32#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  BioNature Inc. is considering the monthly i ; kh+t*job (6ql(/n mkvkn+rcost of producing straws made from paper. It is estimated that the rate of change in monthly cost with respect to the number of p*oh(ivqntn k kjl( r+6mb k+;/aper straws produced is modelled by

C(q)=0.006q20.48q12, for q>0

Where the monthly cost, $C , is in thousands and the number of straws, q , is also in thousands.
1. Find the number of paper straws that the company should produce in a month to minimize the cost.

The company produces 110000 paper straws with a cost of $88000 in a certain month.
2. Find an expression for C in terms of q ;we obtain C(q)=0.002q3aq212q+b;a=  ,b=  

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33#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A healthcare company is designing a new deodorant stick that can be modelled p6b iu; v- v,.8ld lgct:+gzlcby the shape of a parabola as the top, vertical lines as the sides and a horizontal line as the base, on the x -axisbi;v:gtg.,6v8l z- u+ dlpclc . The parabola has end points at (0,7) and (5,7) , and vertex at (2.5,8) .
This design is shown in the diagram below. All distances are measured in centimetres.

The quadratic curve can be expressed in the form y=ax2+bx+c for 0x5 .
1. 1. Find the value of c is  .
2. Using the points (2.5,8) and (5,7) , write two equations in a and b .
3. Hence, find the equation of the quadratic curve; we get y=-ax2+bx+c;a=  ,b=  ,c=  .
2. Find the area of the shaded region in the deodorant design.

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34#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Water is pumped from a river basin into an irrigated field at a rate of r=\fracp*r yhq.s*:o0j-3a zn-f qmghbq:4 fq*g o3yz04*sq:ma.jhfrqpf :qb q--nh{t^{2}}{25} where r is measured in cubic metres of water per minute and t is the time measured in minutes. The graph of this is shown below.


1. Write down an integral that represents the area of the shaded region. It is A =060t2a dt ;a=  .
2. 1. Find the area of the shaded region is    m2.
2. Interpret the meaning of the shaded area in the context of this question.
3. Find the number of minutes it takes to pump 500  m3 of water into the field.
m≈   min (one deciaml pleaces).

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35#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following diagram shows a section of a curve y=f(x) and the coord9xl8h6n mxx g7inates of six poinx hx7g896xmnlts that lie on the curve.


1. Estimate the area of the shaded region, giving your answer to two decimal places.
15y dx    units^2
The equation of the curve was found to be y=2 x24x+20 .
2. Write an integral that represents the area of the shaded region.
A= 15(ax2bx+c)dx; a=  ,b=  ,c=  .
3 . Find the area of the shaded region, giving your answer in exact values.
A=ab units2 ; a=  ,b=  .
4. Find the percentage error between the estimation in part (a) and the exact value found in part (c).

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36#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The diagram below shows part of the graphs3p 5r wuwx8 b c0haulmvjpf+/1ev3::y of y=(x+2)2 and y=(6x)2 .

1. Find the coordinates of the points P, Q and R .
P is (  ,  ) ;Q is (  ,   ) ;R is (  ,  )
The shaded region is rotated through 360 about the x -axis.
2. Calculate the volume of the solid obtained.

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37#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the region R mt rd7 4tsb -yf:z*; rj)dax-sdtt6l5 enclosed by the curve y=12(x+2)(x3) and the x -axis.
1. Sketch the curve and shade the region R on the following axes.


2. 1. Write down an integral that represents the area of region R .
2. Find the area of region R is A ≈   units2(one decimal please)
3. Estimate the area of region R using five trapezoids.
it is ≈    units2 (round number)
4. Find the percentage error between the exact value found in part (b)(ii) and the estimation in part (c).

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38#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  In a memory competition, the rate at which English speaking participants mem tb .f9bpr-sy*orise French vocabulary canys.fpbr -b9t* be modelled by the equation

dMdt=10.01t, t0,

where M(t) is the number of words memorised in t minutes.
The number of words participants have memorised at the beginning of the competition, t=0 , is zero.
1. Find the equation for the number of memorised words at time t . M(t)= t - a3 t2 ; a=  .

The competition ends when students get so tired that the number of additional memorised words per minute becomes zero.
2. Find the value of t when the competition will end.
t=   minutes.
3. Determine
1. the domain of M(t) is [  ,  ],
2. the range of M(t) is [  ,  ].

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39#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Kai launches stones from a slingshot into the air. Each time he launches a;v:g -x+;hsnqlia3e k nzo zxl:z49d* stone, he records the angle, \theta , in degrees, at which the stone is launched, and the horizontal distance, L , in metres, which the stone travels from his shooting position to where it first lands. The diagram below illustrates how th : l:3a;+zodxne hx vk;-4lqszz9i n*ge stones travel in the air when launched at different angles.


Kai analyses his results and concludes:

dL dθ=0.08θ+3.3,0θ90 .

1. Determine whether the graph of L versus \theta is    at θ=50 .

Kai observes that when the angle is 30, the sling stone will travel a horizontal distance of 90 m .
2. Find an expression for L in terms of θ .
L= -0.04θ2+ aθ+b ;a=  ,b=  .

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40#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The cross-sectional view of a tw4vn (uiu9 +c i8pmls+yo-lane road tunnel system is shown on the axes below. The lefn8mc4s+v y+pu9 (iluit and right lane tunnels are separated by a 2 metre thick concrete wall. The right-hand tunnel passes through the points A, B, C and D and its height, in metres, above the base of the tunnel, is modelled by f(x)=-0.04x3+ 0.41 x2, 4x10 , relative to an origin O.


Point A has coordinates (4,4) and point D has coordinates (10,1) .
1. Find the height of the right-hand tunnel when:
1. x=6 ; f(6)=   (m)
2. x=8 ; f(8)=   (m)
The left-hand tunnel can be modelled by a function g(x) , found by reflecting f(x) in the line x=3 .

2. Find the equation of g(x) .g(x)=0.04x3ax2bx+6.12;a=  ,b=  .

3. 1. Find g(x) .g(x)=ax2bx0.6;a=  ,b=  .
2. Hence find the maximum height of the left-hand tunnel.
4. 1. Write down an integral which can be used to find the cross-sectional area of the left-hand tunnel.
2. Hence find the combined cross-sectional area of both tunnels.

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41#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A fire truck is pumping water at a rate o eddc; mx+d;+mf r=2t235 where r is in litres per second and is the time measured in seconds.
The graph of r is shown below.

1. Write down an integral that represents the area of the shaded region.
A=xyat2b dt;a=  ,b=  ; x=  ,y=  .
2. 1. Find the area of the shaded region, rounding your answer to the nearest whole number.
2. Interpret the meaning of the shaded area in the context of this question.
3. Calculate the number of seconds it takes for the fire truck to pump 1000 L of water.
s≈   s (one decimal plaeaces)

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42#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The diagram below shows the vtow. .01mqflcross-sectional area of a mound of beach sand created after a high t. 1ql.vm0o fwtide.


The curve of the cross section can be modelled by the following equation

y=x2(90x)1800

where y represents the vertical height of the mound in cm and x denotes the horizontal width in cm , from the start of the mound.
1. At a horizontal width of x=30 , determine
1. The vertical height of the mound at this point; y=   cm.
2. The gradient of the mound curve at this point. y'=ab;a=  ,b=  .
2. 1. Find the value of x which corresponds to the maximum the vertical height of the mound;it is   .

2. Hence, find the maximum vertical height of the mound. y=   cm
3. Calculate the cross-sectional area of the mound, rounding your answer to one decimal place.

A child uses a toy shovel to remove the top of sand mound, as illustrated by the line segment MN below. Point M has coordinates at (30,30) .

4. Determine the coordinates of point N .

The cross-sectional area removed by the child can be expressed by the following integral

pqx2(90x)1800 dxR,

5. Determine the values of p, q and R . p=  ; q=  , R=  .

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43#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following table shows the x and y coordinates of five points thaw+j7uvz,y a14xxdx 9 ft lie on 14dwfxv9a+ ,zux xy j7a curve y=f(x) .


1. Estimate the area under the curve over the interval 2
The equation of the curve was found to be y=x37x2+14x2 ;it is Approximately equal to    units2( one decimal please)
2. 1. Write down an integral that represents the area under the curve over the interval.
2. Find the exact value of the area under the curve over the interval.A=ab;a=  ,b=  .
3. Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.

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44#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The following table shows the x and y coordinates of wl.+z5/69-entim2 uaz gi7 s h5icedhfive points that lie on a curve y=f/zisn95l+z 7a2td 5i.-6 gheimuew c h(x) .

1. Estimate the area under the curve over the interval 1
The equation of the curve was found to be y=-110(4x+5)(x8) .
it is approximately equal to    units2

2. Find the exact value of the area under the curve over the interval 1A=ab;a=  ,b=  .
3. Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.it is approximately equal to    %(two decimal places)

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45#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  During a trip to a forest to forage for mushrooms, Viviane finds a giant mushrb 9 ug0yjv)a0/akm m7room. She decides to model the shape of the musm00)7rj /mbyu vag9ak hroom to find its volume.

After taking a photo of the mushroom and zooming in to get its real size, she rotates the photograph and estimates that the cross-section passes through the points (0,3) (15,3),(15,15),(22,10),(23.5,6) and (24,0) , where all measurements are in centimetres. The cross-section is symmetrical about the x -axis, as shown below.

Viviane models the section from (0,3) and (15,3) with a straight line.
1. Write down the equation of the line that passes through these points.
y=  .
Next, Viviane models the section that passes through the points (15,15),(22,10) , (23.5,6) and (24,0) with a quadratic curve. it is (  ,  )
2. 1. Use your G.D.C. to find the equation of this quadratic curve.
y≈-1.82(x-a)2+a;a=  .
2. By considering the gradient of the curve at the point (15,15) , explain why this may not be a good model.

Viviane thinks she can obtain a better model if a quadratic passing through the point (24,0) with a maximum point at (15,15) is used.
3. Find the equation of this model, in the form y=a(x-h)^{2}+k .

Using this new model, Viviane proceeds to estimate the volume of the mushroom by finding the volume of revolution about the x -axis.
4. 1. Write down an expression for her estimate as a sum of two integrals.
2. Find the volume of the mushroom estimated by Viviane.
V≈   cm2 (round number)

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46#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Lohan receives an antique vase as a gift from her grandparents. ,l5 1j ,1pgr.lf7 m- fkjkdp2typ1wam She decides to modelljjk,-5wl mgmf t 112y7 ,p .k1apfrpd the shape of the vase to calculate its volume.

She places the vase horizontally on a piece of paper and uses a pencil to mark five representative points (0,5),(7.5,10),(17,7.5),(25,3) and (35,7.5) , as shown below. These points are connected to form a symmetrical cross-section about the x axis. All units are in centimetres.


Lohan initially uses a straight line to model the section from (25,3) to (35,7.5) .
1. Determine the equation of the line that passes through these two points.
y=ax-b;a=  ,b=  .
Lohan thinks that a quadratic curve might be a good model for the section from (0,5) to (25,3) . She carries out a least squares regression using this model for the points she has recorded.
2. 1. Determine the equation of the least squares regression quadratic curve found by Lohan.
2. By considering the gradient of the curve at the point where x=10 , determine whether the quadratic regression curve is a good model or not.

Lohan decides that a cubic curve for the entire section from (0,5) to (35,7.5) would be a better fit.

3. Find the equation of the cubic model.

Using this model, Lohan estimates the volume of the vase by calculating the volume of revolution about the x -axis.
4. Find the volume of the vase estimated by Lohan.

Lohan subsequently fills the vase with water and discovers that the true volume is 5500  cm3 .
5. Calculate the percentage error in Lohan's estimate of the volume.
Using the formula for percentage error, we have
ϵ=  %


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47#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A function is given 2j ta uz+,b0dkby f(x)=4x3xfor12x12 .
1. Sketch the graph of f for the given domain, labelling all intercepts and stationary points.

2. Find an approximation for I=\int_{0}^{1 / 2} f(x) \mathrm{d} x$$ using the trapezoidal rule with a strip width of 0.1 .
3. Find the exact value of I in part (b).The exact value of I is -  
4. Explain why the approximation found in part (b) is larger than the exact value found in part (c).
5. Write down the value of J=1/21/2f(x)dx without calculation. Give a reason for your answer.
I=  .

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48#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The table below shows the daily revenue, R , in thousands u+kuysrx+t7 ac/.-h3pg jly( of USD, of a toy store during the month of December. The first row of the table shows the day of the month (t ) and the second row the revenue of that day in tp ( h-3s+ r.a7cuu+x/yjgtyl khousands of USD.

1. 1. Generate a scatter diagram for this data on the axes below.

2. Identify a possible outlier and give a possible reason for this outlier;
3. Give a reason why plotting R vs log(t) is likely to be the best linearisation method.
2. Linearise the data and determine an appropriate linear model for the daily revenue R in terms of day of the month in December, t .
3. Using your model, predict the revenue after 3 weeks in December.
4. The total revenue for the month of December can be estimated by abRdt .
1. Write down the values of a and b .a=  ,b=  .
2. Estimate the total revenue in December using your model.
TR≈    thousads of USD

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49#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A company that manufactures and sells cardboard boxes hpq:)9lbwf5lxt)q2ucp 4) ,eb 5jmze tas a box with an open-top des bp5ltq9ec 5w )p)z2xfm:)jq e tul,b4ign. This box is constructed from a rectangular cardboard sheet with a length of 2 meters and a width of 1.2 meters, as illustrated in the diagram below. The box is formed by cutting squares of equal side length ( x meters) from each corner and folding up the sides.

1. Show that the volume of the box can be described by the function V(x)=4x36.4x2+2.4x.

2. 1. Find V(x)= ax2-bx+c;a=  ,b=  .c=  .
2. Hence or otherwise, find the value for x that maximises the volume of the box;
x=   m
3. Hence, find the maximum volume of the box.

3. Sketch the graph of V(x) on the axes below for the domain 0
4. 1. Write down an integral representing the area between the graph of V(x) and the x -axis;
2. Hence, find this area between the graph of V(x) and the x -axis.

Let A(x) be a function representing the outside surface area of the box.
5. Determine the function A(x) .
A(x)=-ax2+b;a=  ,b=  .
6. Given that the volume of the box is maximised, find the outside surface area of the box.

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50#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  All lengths in this question are in m 6zm(sud(:ggxetres.
Consider the function f(x)=164x27 , for 2x2. In the following diagram, the shaded region is enclosed by the graph of f and the x -axis.

A rainwater collection tank can be modelled by revolving this region by 360 about the x -axis.
1. Find the volume of the tank.
V=   m3
Rainwater in the tank is used for drinking, cooking, bathing and other needs during the week.
The volume of rainwater in the tank is given by the function g(t) , for 0t7 , where t is measured in days and g(t) is measured in m3 . The rate of change of the volume of rainwater in the tank is given by g(t)=1.54cos(0.12t2) .
2. The volume of rainwater in the tank is increasing only when a 1. Find the value of a and the value of b .
2. During the interval a d=   m3
When t=0 , the volume of rainwater in the tank is 8.2  m3 . It is known that the tank is never completely full of rainwater during the 7 day period.
3. Find the minimum volume of empty space in the tank during the 7 day period.

Vmin m3

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51#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the function defi0os7e g 9l2i,qh cchqbnx1 6:tned by f(x)=(1x)2xx2 where 0x2.
1. Show that f(1x)=f(1+x) , for 1x1 .
2. Find f(x)=2x2ax+b2xx2;a=   ,b=  .
3. Find the x -coordinates of any local minimum or maximum points.

4. Find the range of f is [-   ,   ]
5. Sketch the graph of y=f(x) , indicating clearly the coordinates of the x -intercepts and any local maximum or minimum points.
6 . Find the area of the region enclosed by the graph of y=f(x) on the x -axis, for 0x1 .

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52#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Water is flowing out of ak2xnu0+,vbt x+c/rwv6 axaz0 tank at a rate modelled by the function

R(t)=4sin(t100)


Water is flowing into the same tank at a rate modelled by the function

S(t)=12t21+t3


Both R^{\prime} and S^{\prime} are measured in m3 , and t in hours for 0t10 .
1. Find the interval on which the amount of water in the tank is increasing.

2. Find an expression, T , for the amount of water in the tank at time t if initially there was 25  m3 of water in the tank.
T(t)=4ln(t3+1)+400 cos(t100)-a; a=  .
3. Hence, or otherwise, find the value of the maximum amount of water in the tank and the time it occurs.

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