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MATH1052 Assignment
创建日期：2024-09-05 17:51:53
所属分类：数学mathematics
标签： ​MATH1052
Assignment

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MATH1052 Assignment 


● Marking:

✕ The maximum mark for the assignment is 50. ✕ Marking Scheme for questions worth 1 mark:

* Mark of 0:  You  have  not submitted a relevant answer, or you have no strategy present in your submission.

* Mark of 1/2: You have the right approach, but need to   ne tune some aspects of your justi  cation/calculations.

* Mark of 1: You have demonstrated a good understanding of the topic and techniques involved, with clear justi  cation and well-executed calculations.

✕ Marking Scheme for questions worth 2 marks:

* Mark of 0:  You  have  not submitted a relevant answer, or you have no strategy present in your submission.

* Mark of 1: You have the right approach, but need to   ne tune some aspects of your justi  cation/calculations.

* Mark of 2: You have demonstrated a good understanding of the topic and techniques involved, with clear justi  cation and well-executed calculations.

✕ Marking Scheme for questions worth 3 marks:

* Mark of 0:  You  have  not submitted a relevant answer, or you have no strategy present in your submission.

* Mark of 1:  Your  submission has some relevance, but does not demonstrate deep understanding or sound mathematical technique. This topic needs more attention!

* Mark of 2: You have the right approach, but need to   ne tune some aspects of your justi  cation/calculations.

* Mark of 3: You have demonstrated a good understanding of the topic and techniques involved, with clear justi  cation and well-executed calculations.

1. The equation of a quadratic curve is

2x2 + √3xy + y2 - 10 = 0.

(a)  (2 marks) Use MATLAB to plot the graph of the quadratic curve. What does the curve look like?

Just because the curve looks like ....  doesn't mean it is  ....  Seeing is not believing in maths!  We will now PROVE that it is indeed a ...  !  Let us begin with the general equation of a quadratic curve:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

We will eliminate the xy-term in the equation by a rotation of the coordinate axes.  Such a cool idea but it will take some work so brace yourself!

(b)  (3 marks) Suppose P = (x, y) is a point on the xy-plane.  Now rotate the coordinate axes anticlockwise about the origin through an angle Q. Let the coordinates of P with respect to the new coordinate axes be (x, , y, ) (see   gure below).  Show that

x = x, cos Q - y, sin Q, y = x, sin Q + y, cos Q.

Useful tip:  Recall the sum formula for cosine  and sine.  If you can't remember what these are, go to the FYLC. I've posted the formula on one of the walls in the room :). While you're there, say hello to the FYLC tutors - they can de  nitely help you with this question.

(c)  (2 marks) The equation Ax2 +Bxy+Cy2 +Dx+Ey+F = 0, B 0, maybe transformed into a new quadratic equation A,x,2 + B,x,y, + C,y,2 + D,x, + E,y, + F,  = 0. Determine an expression for B,  in terms of A,B, C and Q.

(d)  (2 marks) To eliminate the cross term in the new quadratic equation in (c), show that Q must satisfy

The angle Q gives the angle of rotation of the coordinate axes to produce a quadratic equation with no cross term.

(e)  (1 mark) Determine Q for the quadratic equation 2x2  + √3xy + y2  - 10 = 0.

(f)  (3 marks) Find an equation for 2x2  + √3xy + y2  - 10 = 0 in the x,y,-plane.  Describe the curve.

2. The   gure below shows a point P (x0 , y0 ) on the parabola y2  = 4px.  The line L is tangent to the parabola at P. The parabola's focus lies at F (p, 0). The ray L,  extending from P to the right is parallel to the x-axis.  We show that light from F to P will be re  ected out along L, by showing that β equals Q. This result establishes the re  ective property of parabolas.

(a)  (3 marks) Show that tan β = 2p/y0 .

(b)  (1 mark) Show that tan φ = y0 / (x0  - p).

(c)  (3 marks) Use the identity

to show that tan Q = 2p/y0 .  Hence show that Q = β .

(d)  (3 marks) You have just proved the re  ective property of parabolas!  This property is used in car headlights and satellite dishes.  Write an amusing essay  (maximum  150 words) on an interesting application of the re  ective property of parabolas.  You may include illustrations in your essay.

(e)  (2  marks)  The   gure below shows a method for constructing a parabola.   You  will require a ruler and a string the length of the ruler.  Draw a line on a piece of paper.

This is the directrix of the parabola. Choose a point F on the paper. This is the focus of the parabola.  Pin one end of the string to F and the other end of the string to the upper end of the ruler. Hold the string taut against the ruler with a pencil, and slide the ruler along the directrix. As the ruler moves, the pencil will trace a parabola. Explain

why. Hint: recall that a parabola is the set of points in a plane equidistant from a given xed point and a given   xed line in the plane.

(f)  (Optional)  Try  the  construction  in  (e)  for  fun!   Special  MATH1052  Prize  for  best parabola model based on this idea.   Submit  your  model  to  Poh  (Room  67-556)  by Friday 10 May, 5 pm.  If Poh is not in her room, leave the model on the table located outside her room. Please write your name and Student ID on the model.

3.  Let

(a)  (3 marks) Sketch (by hand) the graph of z = f(x, y).

(b)  (3 marks) Is f continuous at (0, 0)?  Justify your answer using the de  nition of continuity.

(c)  (3 marks) Use the sketch in (a) to  guess  and   at (0, 0). Check that your guess is correct for using the de  nition of the partial derivative.

(d)  (2 marks) What conclusion, if any, can you make about the existence of partial deriva- tives and continuity for a function f(x, y)?

4. Let's learn some vector calculus ahead of time!   Let r(t)  =  x(t)i + y(t)j + z(t)k be the position vector of particle at time t. The velocity vector is given by v(t) = i + j + k.

An important fact about the velocity vector is that it is tangent to the path of motion. That's all you need to know for this problem! In preparation for a spectacular Taylor Swift concert at UQ, MATH1052 students at UQ constructed a surface given by the equation xz2 - yz + cos(xy) = 1.

(a)  (1 mark) Show that  (0, 0, 1) is on the surface.  This is the point on the surface where Taylor Swift will land.

(b)  (3 marks) Construct the equation of the platform tangent to the surface at  (0, 0, 1). Hint: in page 171 of the workbook, you have the following geometrical property about gradient vectors in the plane:    If f  is di  erentiable  at  (a, b),  the gradient  ▽f (a, b) is perpendicular to the contour line through (a, b).” Generalise this result to gradient vectors in space to help you construct the equation of Swift's platform.

(c)  (3 marks) Taylor Swift will glide through UQ's exhibition hall along the path r(t) = (ln t)i + (tlnt)j + tk, t > 0. Show that the curve is tangent to the surface.  Phew! That was a smooth landing!

5. A MATH1052 trophy is constructed by rotating the following region about the x-axis.

(a)  (2 marks) Compute the volume of the trophy.

(b)  (2 marks) Find the volume V (r, s) of a similar solid created by rotating a region with dimensions r and s instead of 3 and 2 respectively.

(c)  (3 marks) Ooops!!  The original numbers  "2"  and "3" were rounded o   numbers.  The actual quantities are o   by up to 0.5 in either direction.  Use linear approximation to estimate the maximum possible error in your answer to (a).

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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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