CHAPTER 3. Cylinder. ) be any point on the cylinder. The equation of the generator through the point P and parallel to equation (2) are
|
|
- Shonda McDaniel
- 6 years ago
- Views:
Transcription
1 CAPTER 3 Cylinder 3.. DENTON A cylinder is a surface generated by a straight line which is parallel to a fixed line and intersects a given curve or touches a given surface. The fixed line is called the axis and the given curve is called the guiding curve of the cylinder. Any line on the surface of a cylinder is called its generator. 3.. EQUATON O A CYLNDER (a) To find the equation of a cylinder whose generators intersect the conic ax + hxy + by + gx + fy + c = 0, z = 0...() and are parallel to the line x/l = y/m = z/n....() Suppose P (x, y, z ) be any point on the cylinder. The equation of the generator through the point P and parallel to equation () are x x y y z z = =...(3) l m n The generator (3) meets the plane z = 0 in the point lz x n y mz,, 0 n K Since the generator (3) meets the given conic (), lz a x n K + h lz mz x y n K n K + b mz y n K + g lz x n K mz + f y n K + c = 0 or a (nx lz ) + h (nx lz ) (ny mz ) + b (ny mz ) + gn (nx lz ) + fn (ny mz ) + cn = 0 80
2 CYLNDER 8 Thus the locus of (x, y, z ) is a (nx lz) + h (nx lz) (ny mz) + b (ny mz) + gn (nx lz) + fn (ny mz) + cn = 0 This is the required equation of the cylinder. Corollary : The equation of the cylinder whose generators are parallel to the z-axis, then (l = 0, m = 0, n = 0) put in equation (4), we get ax + by + hxy + gx + fy + c = 0 Corollary : The equation of the form f(x, y) = 0 represents a cylinder whose generator are parallel to z-axis. Corollary 3: The equation of the cylinder whose axis is z-axis and whose generators intersection the circle x + y = a, z = 0, is given by x + y = a RGT CRCULAR CYLNDER A right circular cylinder is a surface generated by a straight line passing through the point on a fixed circle and is perpendicular to its plane. The normal to the plane of the circle through its centre is called the axis of the cylinder and the section by a plane which is perpendicular to the axis is called the normal section i.e., a circle. The radius of the normal section is also called the radius of the cylinder. The length of the perpendicular from any point on a right circular cylinder to its axis is equal to its radius EQUATON O A RGT CRCULAR CYLNDER To find the equation of a right circular cylinder whose axis is the line x x y y z z = = and whose radius is r. l m n Suppose P (x, y, z) be any point on the cylinder and PN be the length of the point perpendicular from the point P on a given line and the given line passes through the point A (x, y, z ). Let PN = r M r N P (x, y, z) A(x,y,z) AN = ( x x ) l + ( y y ) m + ( z z ) n l + m + n AP = distance between the point A and P = ( x x ) + ( y y ) + ( z z )
3 8 ENGNEERNG MATEMATCS Now, we have PN = AP AN = {(x x ) + (y y ) + (z z ) {( ) ( ) ( )} } l x x + m y y + n z z l + m + n ence PN = p = {(x x ) + (y y ) + (z z ) {( ) ( ) ( )} } l x x + m y y + n z z l + m + n Since, the radius of the given cylinder is r, so by definition of right circular cylinder we have p = r. i.e., r (l + m + n ) = {(x x ) + (y y ) + (z z ) } (λ + m + n ) {l(x x ) + m (y y ) + n (z z )} or r (l + m + n ) = [(y y ) n (z z ) m] + [(z z ) l (x x ) n] + [(x x ) m (y y ) l] or r (l + m + n ) = Σ[(y y )n (z z )m] This is required equation of right circular cylinder ENVELOPNG CYLNDER A cylinder whose generator touches a given surface and is directed in a given direction is called an enveloping cylinder EQUATON O AN ENVELOPNG CYLNDER To find the equation of the enveloping cylinder whose generator touch the sphere ax + by + cz =, and are parallel to line x y l = z m = n. Let P (x, y, z ) be any point on the given enveloping cylinder. The equation of the generator of the cylinder through the point P and parallel to the line x l = y z m = n is x x y y z z = = = r (say)...() l m n The coordinates if any point on the generator () are (lr + x, mr + y, nr + z ) Suppose the given sphere meets the point (lr + x, mr + y, nr + z ), then we have a (lr + x ) + b (mr + y ) + c (nr + z ) = or r (al + bm + cn ) + r (alx + bmy + cnz ) + ax + by + cz =...() The line () will touch given sphere if the equation () has equal roots. Therefore we get (alx + bmy + cnz ) = (al + bm + cn ) (ax + by + cz )
4 CYLNDER 83 ence the locus of (x, y, z ) is (alx + bmy + cnz ) = (al + bm + cn ) (ax + by + cz ) This is required equation of enveloping cylinder EQUATON O A TANGENT PLANE TO TE CYLNDER To find the equation of a tangent plane to the cylinder whose equation is ax + hxy + by + gx + fy + c = 0 at the point P (x, y, z ). Suppose the given equation of the cylinder is ax + hxy + by + gx + fy + c = 0...() Since, the point P (x, y, z ) line on (), then ax + hx y + by + gx + fy + c = 0...() Let the equation of a line which passes through the point P (x, y, z ) and whose direction x x y y z z cosine are l, m, n be = = = r...(3) l m n any point on the line (3) is (lr + x, mr + y, nr + z ) and the given cylinder () are given by a (lr + x ) + h (lr + x ) (mr + y ) + b (mr + y ) + g (lr + x ) + f (mr + y ) + c = 0 r (al + hlm + bm ) + r[l(ax + hy + g) + m (hx + by + f)] + (ax + hx y + by + gx + fy + c) = 0 Using equation (), we get r (al + hlm + bm ) + r [l(ax + hy + g) + m(hx + by + f)] = 0...(4) One root of this equation is zero. This given line (3) will be a tangent line at (x, y, z ). f the other root is also zero. Equation (4) other root is zero if l [hx + by + g] + m [hx + by + f}] = 0...(5) Eliminating l, m, n between equation (3) and (5), we get (x x )[hx + by + g] + (y y )[hx + by + f] = 0 or x (ax + hy + g) + y (hx + by + f) + gx + fy + c = ax + hx y + by + gx + fy + c = 0 Using equation (), we get x (ax + hy + g) + y (hx + by + f) + gx + fy + c = 0 This is the required equation of tangent plane to the cylinder.
5 84 ENGNEERNG MATEMATCS Corollary : The tangent plane at the point (x, y, z ) to the cylinder ax + hx y + by + gx + fy + c = 0...() x (ax + hy + g) + y (hx + by + f) + (gx + fy + c) = 0 or axx + h (xy + x y) + byy + g (x + x ) + f (y + y ) + c = 0 this equation is obtained by replacing x by xx, y by yy, x by x + x, y by y + y and xy by xy + yx in (). SOLVED EXAMPLES Example. ind the equation of a cylinder whose generators are parallel to the line x = y/ = z and passing through the curve is 3x + y =, z = 0. Sol. The equation of the giving curve is 3x + y =, z = 0...() The equation of the giving line is x y z = =...() Let us consider a point P (x, y, z ) on the cylinder. The equation of generator through the point P (x, y, z ) which is a line parallel to the given line () are x x y y z z = =...(3) The generator (3) meets the plane z = 0 in the point given by x x y y z z = = i.e., (x + z, y + z, 0) Since the generator (3) meets the conic (). ence the point (x + z, y + z, 0) will satisfy the equation of the conic given by (), we have 3 (x + z ) + (y + z ) = or 3 (x + x z + z ) + ( y + 4y z + 4z ) = x + 6x z + z + y + 8y z = 0 The locus of P (x, y, z ) is 3x + 6xz + z + y + 8yz = 0 This is required equation of the cylinder. Example. ind the equation of the circular cylinder whose generating lines have the direction cosines l, m, n and which pass through the fixed circle x + y = a in ZOX plane. Sol. The equation of the guiding curve (circle) are x + y = a, ZOX lane i.e., y = 0...()
6 CYLNDER 85 let us consider a point P (x, y, z ) on the cylinder. The equation of generator through the point P (x, y, z ) and with direction cosine l, m, n are x x y y z z = =...() l m n the generator () meet the plane y = 0 in point given by x x l y y z z = = m n i.e. x ly, 0, z m Since, the generator () meets the curve (). ence, the point x the equation of the curve given by (), we have ly x m K + ny z m K = a or (mx ly ) + (mz ny ) = a m The locus of P (x, y, z ) is (mx ly) + (mz ny ) = a m ny m K K ly ny, 0, z will satisfy m m This is the required equation of cylinder. Example 3. ind the equation of the cylinder which intersects the curve ax + by + cz =, lx + my + nz = p and whose generators are parallel to x-axis. Sol. The given equation of the guiding curve are ax + by + cz =...() and lx + my + nz = p...() Since, the generators of the cylinder are parallel to x-axis, so the equation of the cylinder will not contain terms of x. Thus the equation of the cylinder will be obtained by eliminates x between equation() and (), we get a l (p my nz) + by + cz = a(p my nz) + bl y + cl z = l or a (am + bl )y + (an + cl )z + amnyz amby anbz + (ap l ) = 0 This is the required equation of the cylinder. Example 4. ind the equation of a right circular cylinder described on the circle through the three points (, 0, 0), (0,, 0), (0, 0, ) are guiding circle. Sol. Let the given three points A (, 0, 0), B (0,, 0), C (0, 0, ). The equation of the sphere OABC is x + y + z x y z = 0 and the equation of the plan ABC is x + y + z =
7 86 ENGNEERNG MATEMATCS Therefore, the equation of the circle ABC is x + y + z x y z = 0 and x + y + z =...() Since, the cylinder is a right circular cylinder, then the axis of the given cylinder is perpendicular to the plane x + y + z. So direction ratio of the axis are (,, ). The generator through (x, y, z ) and parallel to the axis has equation x x y y z z = = = r Any point on this line (r + x, r + y, r + z )lies on the circle (), if r + x, r + y, r + z = or 3r = (x + y + z )...() and (r + x ) + (r + y ) + (r + z ) (r + x + r + y + r + z ) = 0...(3) Multiply by 3 in equation (3) and using (), we get [ (x + y + z )] + (x + y + z )[ (x + y + z )] + 3[ x + y + z ] = 0 or 3 (x + y + z ] (x + y + z ) + = 0 or x + y + z x y + y z x z = The locus of P (x, y, z ) is or x + y + z xy yz zx =. This is required equation of the right circular cylinder. Example 5. ind the tangent plane to the cylinder 3x + 8xy + 5y + x + 7y + 6 = 0 at the point (,, ). Sol. The tangent plane at the point (x, y, z ) to the cylinder ax + hxy + by + gx + fy + c = 0...() is axx + h (xy + x y) + byy + g (x + x ) + f (y + y ) + c = 0 The tangent plane the given cylinder at (,, ) is 3x () + 4[x ( ) + (y)] + 5y ( ) + (x + ) + 7 (y ) + 6 = 0 or x 5y + 6 = 0 This is the required equation of the tangent plane. Example 6. ind the equation of the sphere enveloping cylinder of the sphere x + y + z x + 4y = 0 having its generation parallel to the line x = y = z. Sol. The equation of the given sphere is x + y + z x + 4y = 0...() the generators of the enveloping cylinder are parallel to the line x = y = z...()
8 CYLNDER 87 let us consider a point P (x, y, z ) on the given enveloping cylinder. The equation of the generator through the point P (x, y, z ) and parallel to the line x = y = z is x x y = y = z z = r...(3) the point of intersection of the line (3) and the given () are given by (r + x ) + (r + y ) + (r + z ) (r + x ) + 4(r + y ) = 0 or 3r + (x + y + z + )r + (x + y + z x + 4y ) = 0...(4) or enveloping cylinder, the equation (4) must have equal roots. This requires (x + y + z + ) = 3(x + y + z x + 4y ). or x + y + z y z x y 4x + 5y z = 0 The locus of P (x, y, z ) is x + y + z yz xy 4x + 5y z = 0 This is the required equation of the enveloping cylinder. Example 7. ind the equation of the right circular cylinder whose axis is x = z, y = 0 and which passes through the point (3, 0, 0). Sol. The given equation of the axis of the cylinder is x y 0 z = = 0...() 0 We know r = the length of the perpendicular for a point (3, 0, 0) on the cylinder to the axis () = [( ) + {. 03 ( )} + { 0.( 3). 0} ] = () Let us consider a point P (x, y, z ) on the cylinder. The length of the perpendicular from the point P to the given axis () is equal to the radius of the cylinder. i.e., {. y 0. z} + {. z. (x )} + {0. (x ). y} = y + (z x + ) + y = x + y + z zx 4x + 4z + 3 = 0 This is the required equation of the right circular cylinder. G K J ( ) EXERCSE 3.. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and whose guiding curve is the ellipse x + y =, z = 0.
9 88 ENGNEERNG MATEMATCS. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and whose guiding curve is the ellipse x + y =, z = ind the equation of the cylinder whose generatoring lines have the direction cosines l, m, n and which passes through the fixed circle is the ellipse x + z = in the ZOX plane. 4. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and passes through the curve x + y = 6 and z = ind the equation of the cylinder whose generators are parallel to the line x/4 = y/ = z/3 and which intersects the ellipse 4x + y =, z = ind the equation of the surface generated by a straight line which is parallel to the line y = mx, z = nx and intersects the ellipse x /a + y /b =, z = ind the equation of the cylinder with generators parallel to z-axis and passes through the curve ax + by = cz, lx + my + nz = p. 8. ind the equation of the cylinder with generators parallel to x-axis and passing through the curve ax + by = cz, lx + my + nz = p. 9. ind the equation of the right circular cylinder of radius whose axis is the line (x )/ = y/3 = (z 3)/. 0. ind the equation of the right circular cylinder whose axis is (x )/ = (y ) = z/3 and which passes through (0, 0, ).. ind the equation of the right circular cylinder of radius 4 whose axis is the line x =y = z.. ind the equation of the right circular cylinder of radius 3 whose axis is the line (x )/ = (y 3)/ = (z 5) / ind the equation of the right circular cylinder whose guiding circle is x + y + z = 9, x y + z = Show that the coordinate of the foot of perpendicular from a point P (x, y, z ) on the line x = y = z are 3 (x + y z ), 3 (x + y z ), 3 (x + y z ). 5. ind the equation of the right circular cylinder whose guiding circle passes through the points (a, 0, 0), (0, b, 0), (0, 0, c). 6. ind the equation of the right circular cylinder with generators parallel to z-axis and intersect the surfaces ax + by + cz =, lx + my + nz = p. 7. ind the equation of the right circular cylinder of radius whose axis is the line (x )/ = (y ) = (z 3)/. 8. ind the equation of the right circular cylinder whose one section is the circle x + y + z x y z = 0, x + y + z = is x + y + z yz zx xy =. 9. ind the equation of the right circular cylinder of radius whose axis passes through (,, 3) and has direction cosines proportional to, 3, ind the equation of the cylinder whose generating line are parallel to the line x/l = y/m = z/n and which touches the sphere x + y + z = a.. ind the enveloping cylinder of the sphere x + y + z = having its generators parallel to the line x/ = y/ = z/3.. Show that the enveloping cylinder of the conicoid ax + by + cz = with generators perpendicular to the z-axis meets the plane z = 0 in parabolas.
10 CYLNDER ind the equation of a right circular cylinder which envelopes a sphere of centre (a, b, c) and radius r, and has its generator parallel to a line with direction ratio l, m, n. 4. ind the equation of the right circular cylinder of radius 5 and having for its axis the line x/ = y/3 = z/6. 5. ind the equation of a cylinder whose generator touches the sphere x + y + z + ux + vy + wz + d = 0 whose generators are parallel to the line x/l = y/m = z/n. 6. Prove that the enveloping cylinder of ellipsoid x y z = = =,whose generator are parallel to line a b c x y z = = meet the plane z = 0 in circles. 0 ± a b c ANSWERS. 9 (x + y + z ) 6xz + 4yz = 9.. 3x + 6y + 3z + 8yz zx + 6x 4y 8z + 4 = (mx ly) + (mz ny) = m. 4. 9x + 9y + 5z 6zx yz 44 = (3x 4z) + (3y + z) = b (nx z) + a (ny mz) = a b n. 7. anx + bny + c(lx + my) pc = (am + bl ) y + an z + amnyz apmy (apn + cl )z = x + 5y + 3z xy 6yz 4zx 8x + 30y 74z + 59 = x + 3y + 5z 4xy 6yz zx 36x 8y + 30z 35 = 0.. 5x + 8y + 5z 4xy + 4yz + 8zx 44 = 0.. 5x + 5y + 8z 8xy + 4yz + 4zx 6x 4y 96z + 5 = x + 5y + 5z + 4xy + 8yz 4zx 7 = 0. x y z x y z a b c K (x + y + z ax by cz) = + + a b c + + K a b c 6. (an + cl ) x + (bn + cm ) y + clmny cplx cpmy + cp n = x + 8y + 5z 4xy 4yz 8zx + x 6y 4z 0 = x + 40y + 3z 36yz 4zx + xy 4x 80y 6z + 94 = (lx + my + nz) = (l + m + n ) (x + y + z a ).. (x + y + 3z) = 4 (x + y + z ).. ab (mx ly) = (al + bm ), z = {l (x a) + m (y b) + n (z c)} = (l + m + n ){(x a) + (y b) + (z c) r } x + 40y + 3z xy 36yz 4zx 5 = {l (x + u) + m (y + v) + n (z + w)} = (l + m + n ) ( x + y + z + ux + vy + wz + d). 6. x + y = a, z = 0.
If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2)
Sphere Definition: A sphere is the locus of a point which remains at a constant distance from a fixed point. The fixed point is called the centre and the constant distance is the radius of the sphere.
More informationPARABOLA SYNOPSIS 1.S is the focus and the line l is the directrix. If a variable point P is such that SP
PARABOLA SYNOPSIS.S is the focus and the line l is the directrix. If a variable point P is such that SP PM = where PM is perpendicular to the directrix, then the locus of P is a parabola... S ax + hxy
More informationTHREE DIMENSIONAL GEOMETRY
For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z 1 1 1 (i) The coordinates of point
More informationQuadric Surfaces. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Quadric Surfaces Spring /
.... Quadric Surfaces Philippe B. Laval KSU Spring 2012 Philippe B. Laval (KSU) Quadric Surfaces Spring 2012 1 / 15 Introduction A quadric surface is the graph of a second degree equation in three variables.
More informationQuadric Surfaces. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24
Quadric Surfaces Philippe B. Laval KSU Today Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24 Introduction A quadric surface is the graph of a second degree equation in three variables. The general
More informationx 6 + λ 2 x 6 = for the curve y = 1 2 x3 gives f(1, 1 2 ) = λ actually has another solution besides λ = 1 2 = However, the equation λ
Math 0 Prelim I Solutions Spring 010 1. Let f(x, y) = x3 y for (x, y) (0, 0). x 6 + y (4 pts) (a) Show that the cubic curves y = x 3 are level curves of the function f. Solution. Substituting y = x 3 in
More informationChapter 15: Functions of Several Variables
Chapter 15: Functions of Several Variables Section 15.1 Elementary Examples a. Notation: Two Variables b. Example c. Notation: Three Variables d. Functions of Several Variables e. Examples from the Sciences
More informationDrill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3
Drill Exercise - 1 1. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ). 2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral
More informationFunctions of Several Variables
. Functions of Two Variables Functions of Several Variables Rectangular Coordinate System in -Space The rectangular coordinate system in R is formed by mutually perpendicular axes. It is a right handed
More informationMAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 covers essentially the same material as MAT201, but is more in depth and theoretical. Exam problems are often more sophisticated in scope and difficulty
More informationPART A (5x5M =25M) dx +2xy 4x2 = 0 and passing through the origin. using the method of multipliers. PART B (5x10M = 50M)
FIRST YEAR B.SC. MATHEMATICS PAPER I SEMESTER I DIFFERENTIAL EQUATIONS MODEL QUESTION PAPER (THEORY) Time: 3 Hours Max. Marks: 75 *This Paper Csists of Two parts. Follow the Instructis Carefully PART A
More informationREVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections
REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections 14.1-8. 1. Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable.
More information(c) 0 (d) (a) 27 (b) (e) x 2 3x2
1. Sarah the architect is designing a modern building. The base of the building is the region in the xy-plane bounded by x =, y =, and y = 3 x. The building itself has a height bounded between z = and
More informationMath 113 Calculus III Final Exam Practice Problems Spring 2003
Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross
More informationLagrange multipliers October 2013
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 1 1 3/2 Example: Optimization
More informationLagrange multipliers 14.8
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 Maximum? 1 1 Minimum? 3/2 Idea:
More informationDrill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3
Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.
More informationADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE
ADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE It is given that the straight line L passes through A(5, 5) and is perpendicular to the straight line L : x+ y 5= 0 (a) Find the equation of L (b) Find
More informationAnalytical Solid Geometry
Analytical Solid Geometry Distance formula(without proof) Division Formula Direction cosines Direction ratios Planes Straight lines Books Higher Engineering Mathematics By B S Grewal Higher Engineering
More informationwe wish to minimize this function; to make life easier, we may minimize
Optimization and Lagrange Multipliers We studied single variable optimization problems in Calculus 1; given a function f(x), we found the extremes of f relative to some constraint. Our ability to find
More informationSection 2.5. Functions and Surfaces
Section 2.5. Functions and Surfaces ² Brief review for one variable functions and curves: A (one variable) function is rule that assigns to each member x in a subset D in R 1 a unique real number denoted
More informationLagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers
In this section we present Lagrange s method for maximizing or minimizing a general function f(x, y, z) subject to a constraint (or side condition) of the form g(x, y, z) = k. Figure 1 shows this curve
More information12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d )
- THREE DIMENSIONAL GEOMETRY Page ( ) If the angle θ between the line x - y + x + y - z - and the plane λ x + 4 0 is such that sin θ, then the value of λ is - 4-4 [ AIEEE 00 ] ( ) If the plane ax - ay
More informationIf (x, y, z) are the coordinates of a point P in space, then the position vector of the point P w.r.t. the same origin is.
7. 3D GEOMETRY. COORDINATE OF A POINT IN SPACE Let P be a point in the space. If a perpendicular from that point is dropped to the xyplane, then the algebraic length of this perpendicular is considered
More information14.5 Directional Derivatives and the Gradient Vector
14.5 Directional Derivatives and the Gradient Vector 1. Directional Derivatives. Recall z = f (x, y) and the partial derivatives f x and f y are defined as f (x 0 + h, y 0 ) f (x 0, y 0 ) f x (x 0, y 0
More informationHOMEWORK ASSIGNMENT #4, MATH 253
HOMEWORK ASSIGNMENT #4, MATH 253. Prove that the following differential equations are satisfied by the given functions: (a) 2 u 2 + 2 u y 2 + 2 u z 2 =0,whereu =(x2 + y 2 + z 2 ) /2. (b) x w + y w y +
More information4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2
12.6 Quadrics and Cylinder Surfaces: Example: What is y = x? More correctly what is {(x,y,z) R 3 : y = x}? It s a plane. What about y =? Its a cylinder surface. What about y z = Again a cylinder surface
More informationAnalytical Solid Geometry
Analytical Solid Geometry Distance formula(without proof) Division Formula Direction cosines Direction ratios Planes Straight lines Books Higher Engineering Mathematics by B S Grewal Higher Engineering
More informationPartial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives
In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. By the definition of a derivative, we have Then we are really
More information12.6 Cylinders and Quadric Surfaces
12 Vectors and the Geometry of Space 12.6 and Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and We have already looked at two special types of surfaces:
More informationSection 12.2: Quadric Surfaces
Section 12.2: Quadric Surfaces Goals: 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an
More informationMultivariate Calculus: Review Problems for Examination Two
Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double
More informationRectangular Coordinates in Space
Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then
More informationQ-1 The first three terms of an AP respectively are 3y 1, 3y +5 and 5y +1. Then y equals
CBSE CLASS X Math Paper-2014 Q-1 The first three terms of an AP respectively are 3y 1, 3y +5 and 5y +1. Then y equals (A) -3 (B) 4 (C) 5 (D) 2 Q-2 In Fig. 1, QR is a common tangent to the given circles,
More information1.6 Quadric Surfaces Brief review of Conic Sections 74 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2
7 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = x 1.6 Quadric Surfaces Figure 1.19: Parabola x = y 1.6.1 Brief review of Conic Sections You may need to review conic sections for
More informationPractice problems from old exams for math 233 William H. Meeks III December 21, 2009
Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These
More informationMultivariate Calculus Review Problems for Examination Two
Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,
More information6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
More informationGrad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures
Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables
More informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
More informationThree-Dimensional Coordinate Systems
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will
More informationMath 209 (Fall 2007) Calculus III. Solution #5. 1. Find the minimum and maximum values of the following functions f under the given constraints:
Math 9 (Fall 7) Calculus III Solution #5. Find the minimum and maximum values of the following functions f under the given constraints: (a) f(x, y) 4x + 6y, x + y ; (b) f(x, y) x y, x + y 6. Solution:
More informationAssignment Assignment for Lesson 11.1
Assignment Assignment for Lesson.1 Name Date Conics? Conics as Cross Sections Determine the conic section that results from the intersection of the double-napped cone shown and each plane described. 1.
More informationDesign and Communication Graphics
An approach to teaching and learning Design and Communication Graphics Solids in Contact Syllabus Learning Outcomes: Construct views of up to three solids having curved surfaces and/or plane surfaces in
More informationTrue/False. MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY
MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY True/False 10 points: points each) For the problems below, circle T if the answer is true and circle F is the answer is false. After you ve chosen
More informationEquation of tangent plane: for implicitly defined surfaces section 12.9
Equation of tangent plane: for implicitly defined surfaces section 12.9 Some surfaces are defined implicitly, such as the sphere x 2 + y 2 + z 2 = 1. In general an implicitly defined surface has the equation
More informationMath 233. Lagrange Multipliers Basics
Math 33. Lagrange Multipliers Basics Optimization problems of the form to optimize a function f(x, y, z) over a constraint g(x, y, z) = k can often be conveniently solved using the method of Lagrange multipliers:
More informationYou may know these...
You may know these... Chapter 1: Multivariables Functions 1.1 Functions of Two Variables 1.1.1 Function representations 1.1. 3-D Coordinate System 1.1.3 Graph of two variable functions 1.1.4 Sketching
More informationMATH 116 REVIEW PROBLEMS for the FINAL EXAM
MATH 116 REVIEW PROBLEMS for the FINAL EXAM The following questions are taken from old final exams of various calculus courses taught in Bilkent University 1. onsider the line integral (2xy 2 z + y)dx
More information3D Coordinate Transformation Calculations. Space Truss Member
3D oordinate Transformation alculations Transformation of the element stiffness equations for a space frame member from the local to the global coordinate system can be accomplished as the product of three
More informationWhat is log a a equal to?
How would you differentiate a function like y = sin ax? What is log a a equal to? How do you prove three 3-D points are collinear? What is the general equation of a straight line passing through (a,b)
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13
MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus
More informationf xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
More informationChapter 15 Vector Calculus
Chapter 15 Vector Calculus 151 Vector Fields 152 Line Integrals 153 Fundamental Theorem and Independence of Path 153 Conservative Fields and Potential Functions 154 Green s Theorem 155 urface Integrals
More information13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution
13 Functions of Several Variables 13.1 Introduction to Functions of Several Variables Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives Understand
More information1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral
1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places. 2 2.5 3 3.5 4 When When When When When
More informationDifferentiability and Tangent Planes October 2013
Differentiability and Tangent Planes 14.4 04 October 2013 Differentiability in one variable. Recall for a function of one variable, f is differentiable at a f (a + h) f (a) lim exists and = f (a) h 0 h
More informationvolume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section)
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter December 25, 2016 volume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section) Harish
More informationUNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 5
UNIVERSITI TEKNOLOGI MALAYSIA SSE 189 ENGINEERING MATHEMATIS TUTORIAL 5 1. Evaluate the following surface integrals (i) (x + y) ds, : part of the surface 2x+y+z = 6 in the first octant. (ii) (iii) (iv)
More informationMath 253, Section 102, Fall 2006 Practice Final Solutions
Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they
More informationMAT175 Overview and Sample Problems
MAT175 Overview and Sample Problems The course begins with a quick review/overview of one-variable integration including the Fundamental Theorem of Calculus, u-substitutions, integration by parts, and
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationCBSE X Mathematics 2012 Solution (SET 1) Section C
CBSE X Mathematics 01 Solution (SET 1) Q19. Solve for x : 4x 4ax + (a b ) = 0 Section C The given quadratic equation is x ax a b 4x 4ax a b 0 4x 4ax a b a b 0 4 4 0. 4 x [ a a b b] x ( a b)( a b) 0 4x
More informationI IS II. = 2y"\ V= n{ay 2 l 3 -\y 2 )dy. Jo n [fy 5 ' 3 1
r Exercises 5.2 Figure 530 (a) EXAMPLE'S The region in the first quadrant bounded by the graphs of y = i* and y = 2x is revolved about the y-axis. Find the volume of the resulting solid. SOLUTON The region
More information7.3 3-D Notes Honors Precalculus Date: Adapted from 11.1 & 11.4
73 3-D Notes Honors Precalculus Date: Adapted from 111 & 114 The Three-Variable Coordinate System I Cartesian Plane The familiar xy-coordinate system is used to represent pairs of numbers (ordered pairs
More information27. Tangent Planes & Approximations
27. Tangent Planes & Approximations If z = f(x, y) is a differentiable surface in R 3 and (x 0, y 0, z 0 ) is a point on this surface, then it is possible to construct a plane passing through this point,
More informationCurves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent
Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( 11.3-11.4) Feb. 26, 2012 (Sun) Signs of Partial Derivatives on Level Curves Level curves are shown for a function
More information(1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives.
Section 11.3 Partial Derivatives (1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives. MATH 127 (Section 11.3) Partial Derivatives The University of Kansas
More informationB.Stat / B.Math. Entrance Examination 2017
B.Stat / B.Math. Entrance Examination 017 BOOKLET NO. TEST CODE : UGA Forenoon Questions : 0 Time : hours Write your Name, Registration Number, Test Centre, Test Code and the Number of this Booklet in
More information. Tutorial Class V 3-10/10/2012 First Order Partial Derivatives;...
Tutorial Class V 3-10/10/2012 1 First Order Partial Derivatives; Tutorial Class V 3-10/10/2012 1 First Order Partial Derivatives; 2 Application of Gradient; Tutorial Class V 3-10/10/2012 1 First Order
More informationThe Divergence Theorem
The Divergence Theorem MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Summer 2011 Green s Theorem Revisited Green s Theorem: M(x, y) dx + N(x, y) dy = C R ( N x M ) da y y x Green
More informationExample is the school of mankind, and they will learn at no other. Edmund Burke ( )
Examples Example is the school of mankind, they will learn at no other. Edmund Burke (79 797) This section, like the previous section, is organised into 9 groups:. Trigonometry. Circles. Triangles.4 Quadrilaterals.5
More informationElements of three dimensional geometry
Lecture No-3 Elements of three dimensional geometr Distance formula in three dimension Let P( x1, 1, z1) and Q( x2, 2, z 2) be two points such that PQ is not parallel to one of the 2 2 2 coordinate axis
More informationMathematical derivations of inscribed & circumscribed radii for three externally touching circles (Geometry of Circles by HCR)
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter February 15, 2015 Mathematical derivations of inscribed & circumscribed radii for three externally touching circles Geometry of Circles
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationnotes13.1inclass May 01, 2015
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationWhat you will learn today
What you will learn today Conic Sections (in 2D coordinates) Cylinders (3D) Quadric Surfaces (3D) Vectors and the Geometry of Space 1/24 Parabolas ellipses Hyperbolas Shifted Conics Conic sections result
More informationMath 21a Tangent Lines and Planes Fall, What do we know about the gradient f? Tangent Lines to Curves in the Plane.
Math 21a Tangent Lines and Planes Fall, 2016 What do we know about the gradient f? Tangent Lines to Curves in the Plane. 1. For each of the following curves, find the tangent line to the curve at the point
More informationName: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.
. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function
More informationAQA GCSE Further Maths Topic Areas
AQA GCSE Further Maths Topic Areas This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need
More informationTotal. Math 2130 Practice Final (Spring 2017) (1) (2) (3) (4) (5) (6) (7) (8)
Math 130 Practice Final (Spring 017) Before the exam: Do not write anything on this page. Do not open the exam. Turn off your cell phone. Make sure your books, notes, and electronics are not visible during
More informationMath 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.
Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x
More informationLook up partial Decomposition to use for problems #65-67 Do Not solve problems #78,79
Franklin Township Summer Assignment 2017 AP calculus AB Summer assignment Students should use the Mathematics summer assignment to identify subject areas that need attention in preparation for the study
More information8(x 2) + 21(y 1) + 6(z 3) = 0 8x + 21y + 6z = 55.
MATH 24 -Review for Final Exam. Let f(x, y, z) x 2 yz + y 3 z x 2 + z, and a (2,, 3). Note: f (2xyz 2x, x 2 z + 3y 2 z, x 2 y + y 3 + ) f(a) (8, 2, 6) (a) Find all stationary points (if any) of f. et f.
More informationAdvanced Algebra. Equation of a Circle
Advanced Algebra Equation of a Circle Task on Entry Plotting Equations Using the table and axis below, plot the graph for - x 2 + y 2 = 25 x -5-4 -3 0 3 4 5 y 1 4 y 2-4 3 2 + y 2 = 25 9 + y 2 = 25 y 2
More informationP1 REVISION EXERCISE: 1
P1 REVISION EXERCISE: 1 1. Solve the simultaneous equations: x + y = x +y = 11. For what values of p does the equation px +4x +(p 3) = 0 have equal roots? 3. Solve the equation 3 x 1 =7. Give your answer
More informationMath 209, Fall 2009 Homework 3
Math 209, Fall 2009 Homework 3 () Find equations of the tangent plane and the normal line to the given surface at the specified point: x 2 + 2y 2 3z 2 = 3, P (2,, ). Solution Using implicit differentiation
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationGeometry Diagnostic Test
Geometry Diagnostic Test 1. A pentagonal-based prism is sliced horizontally by a plane parallel to the base. Which figure best represents the shape of the cross-section parallel to the slice? rectangle
More information1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:
Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila January 26, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
More informationMathematical Analysis of Tetrahedron (solid angle subtended by any tetrahedron at its vertex)
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter March 29, 2015 Mathematical Analysis of Tetrahedron solid angle subtended by any tetrahedron at its vertex) Harish Chandra Rajpoot Rajpoot,
More informationCoordinate Transformations in Advanced Calculus
Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,
More informationINTRODUCTION TO THREE DIMENSIONAL GEOMETRY
Chapter 1 INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Mathematics is both the queen and the hand-maiden of all sciences E.T. BELL 1.1 Introduction You may recall that to locate the position of a point in
More informationQuadric surface. Ellipsoid
Quadric surface Quadric surfaces are the graphs of any equation that can be put into the general form 11 = a x + a y + a 33z + a1xy + a13xz + a 3yz + a10x + a 0y + a 30z + a 00 where a ij R,i, j = 0,1,,
More informationMath 233. Lagrange Multipliers Basics
Math 233. Lagrange Multipliers Basics Optimization problems of the form to optimize a function f(x, y, z) over a constraint g(x, y, z) = k can often be conveniently solved using the method of Lagrange
More informationMath 213 Exam 2. Each question is followed by a space to write your answer. Please write your answer neatly in the space provided.
Math 213 Exam 2 Name: Section: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used other than a onepage cheat
More informationMATH 261 FALL 2000 FINAL EXAM INSTRUCTIONS. 1. This test booklet has 14 pages including this one. There are 25 questions, each worth 8 points.
MATH 261 FALL 2 FINAL EXAM STUDENT NAME - STUDENT ID - RECITATION HOUR - RECITATION INSTRUCTOR INSTRUCTOR - INSTRUCTIONS 1. This test booklet has 14 pages including this one. There are 25 questions, each
More information2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0.
Midterm 3 Review Short Answer 2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. 3. Compute the Riemann sum for the double integral where for the given grid
More information