Marywood University Department of Mathematics Course Information for Math 204 by Dr. J.

Math 204 Calculus IV

The fourth course in the four-semester Calculus sequence. It covers graphing in three-dimensional space, vectors, vector-valued functions, partial derivatives, and multiple integrals. The course outline and sample test problems are given below. The textbook is "Calculus" by Howard Anton published by Wiley. The chapters from this text usually covered in class are Chapters 13, 14, 15, and 16. The course grade is determined by 3 tests, 3 homework sets, 2 Maple computer labs, and the final examination. Useful websites for the course are listed at the end of this page. Prerequisiste: Math 203.

f(x,y) = sin x + cos y

Course Outline

Three-Dimensional Space and Vectors	Vector-Valued Functions	Partial Derivatives	Multiple Integrals; Topics in Vector Calculus
Rectangular Coordinates in 3-Space	Vector-valued Functions	Functions of 2 or more Variables, Level curves	Double Integrals
Vectors	Calculus of Vector-Valued Fcns.	Limits and Continuity	Area, volume, center of mass
Dot Product ; Projections	Change of Parameter ; Arclength	Partial Derivatives	Vector Fields
Cross Product	Unit Tangent, Normal, and Binormal Vectors	Differentiability and Chain Rules	Line Integrals
Parametric Eqns of Lines	Curvature	Tangent Planes; Total Differentials	Independence of Path, Exactness
Planes in 3-Space	Motion Along a Curve	Directional Derivatives, The Gradient
Quadric Surfaces	Kepler's Laws of Planetary Motion	Dir.Deriv. & Grad. II	Multiple Integrals
3-D Vectors	Vector-valued Fns.	Partial Derivatives	Vector Calculus

3-D Space and Vectors

All bold symbols represent vectors. The "o" symbol represents the dot product of two vectors.

• True or false.
  • For any non-zero real a, b, the vectors ai + bj and bi - aj are orthogonal.
  • If u o v = u o w, then v = w, for any vectors u, v, w
  • If a o b = a o c and a x b = a x c , then b = c.
  • k x ( i x j ) = 0
  • u x v = v x u
  • (u x v) x w = u x (v x w)
  • If u and v are orthogonal, then u x v = 0.
  • (u x v) o w = u o (v x w)
  • A o (A x B) = 0
  • Projw(u) = 0 if and only if u and w are orthogonal. (u, w non-zero)
  • | u o v | <= || u || || v ||
  • i x (i x j) = k
  • If u x v = 0 , then u and v must have the same direction.
  • a o (b + c) = a o b + a o c
• Let u = 3i - 5j, v = -2i + j.
  • Find cosq, where q is the angle between u and v.
  • Find Projv(u)
• If v = xi + yj, and vo = 2i - 7j , write an equation in Cartesian coordinates to describe the set of all points (x, y) for which || v - vo || = 6. What is the name of this curve?
• Find a unit vector in the direction of PQ, where P = (3,-1,2) and Q = (-4,1,7).
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• Find a vector of magnitude 12 having direction cosines cos a = sqrt(3)/2, cos b= - sqrt(2)/3, cos g = 1/6
• Let v = 2i - 2j + k, w = i + 3j - 4k. Then determine the following:
  • i) The length of v
  • ii) The direction of v + w
  • iii) Projw(v)
  • iv) The angle between v and w.
• A vector v has a direction angle of 60 degrees with the positive x-axis and 45 degrees with the positive y-axis.
  • i) What angle does it make with the positive z-axis? (There are two answers.)
  • ii) If the magnitude of v is ||v|| = 4, find v for each of the angles found in (i).
• Given any vector v = ai + bj + ck with direction angles a, b, g, show that cos² a+ cos² b + cos² g = 1.
• A shell is fired from ground level with a speed of 320 ft/sec and elevation angle of 60 degrees. Find:
  • i) parametric equations for the shell's trajectory.
  • ii) the maximum height reached by the shell.
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• Find the equation of the plane containing the point (-1,1,6) that is perpendicular to the line of intersection of the planes 2x - 3y - z = 6 and -4x + y - 2z = 9.
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• Find the plane through (2, 1, -1) perpendicular to the line of intersection of the planes 2x + y - z = 3 and x + 2y + z = 2.
• Find the cosine of the angle between the two planes 3x - 6y - 2z = 7 and 2x + y - 2z = 5. Also find a vector parallel to their line of intersection.
• Determine whether the line given by x = -1 + 2t, y = 4 + t, z = 1 - t and the plane 4x + 2y - 2z = 7 are perpendicular. Give a reason for your answer.
• Find the equation of the plane that contains the point (2,1,5) and the line x = -1 + 3t , y = -2t , z = 2 + 4t.
• Show that the lines given by: x = -2 + t, y = 3 + 2t, z = 4 - t and x = 3 - t, y = 4 - 2t, z = t are parallel and find an equation of the plane they determine.

Vector-Valued Functions

All bold symbols represent vectors.

• Find the domain of r(t) = (cos pt)i - (ln t)j + sqrt(t - 2)k .
• Use the chain rule to find dF/dt if F(x) = (xe^x)i + (ln 3x)j and x = ln t. Remember to express the result as a function of t alone.
• If r(t) = (t + 5)i + (t² )j + (1/3 t³)k is the position vector of a moving particle at time t, find the velocity and acceleration at t = 2.
• Consider the position vector r(t) = e^t(cost i + sint j). If the curvature for this curve is given by k = 1/sqrt(2)e^-t , find the following :
  • (i) v (ii) T (iii) N (iv) a_T (v) a_N (vi) a
• Find the curvature k(x) for the parabola y = (x -1)² . Also find the equation of the osculating circle at the point (1,0), assuming its center is on the axis of symmetry of the parabola.
• Find the length of the curve given by the position vector r(t) = (6 sin2t)i + (6 cos2t)j + 5t k between t = 0 and t = p.
• The force that acts on a particle P of mass m is F = (mt)i + (mt² )j + mk. Find the velocity v at any time t if the particle has original velocity v(0)= 2i + 3j.
• Find the curvature of the helix r(t) = (2 cos t)i + (2 sin t)j + t k.
• A mortar shell is fired from ground level with an initial speed of 320 ft/sec and an elevation angle of 60 degrees. (g = 32 ft/sec²) Find:
  • (i) a vector function for the shell's trajectory
  • (ii) the maximum altitude attained.
• Let T, N, and B be the unit tangent, normal, and binormal respectively. Then B = T x N, T = N x B , and N = B x T. If T, N, and B are given in terms of arclength s, then it is true that dB/ds = - tau N where tau is known as the torsian. Using properties of vector differentiation, show that dN/ds = - kappa T + tau B, where kappa is the curvature.
• A particle moves in the xy-plane in such a manner that the derivative of the position vector is always perpendicular to the position vector. Show that the particle moves on a circle with center at the origin.
• The eccentricity of an orbital path of an object m in motion around a mass M (with M at the origin of the coordinate system) is given by e = r₀v₀²/GM - 1, where r₀ and v₀ are the initial position and speed of m. Show that the speed v of m in a circular orbit is given by v = sqrt(GM/r₀).

Partial Derivatives

• Sketch the domain of f(x,y) = sqrt(x² + y² - 4).
• If T(x,y) is the temperature at a point (x,y) on a thin metal plate in the xy-plane, then the level curves of T are called isothermal curves. Suppose that a plate occupies the first quadrant and T(x,y) = xy.
  • i) Sketch the isothermal curves on which T = 1, T = 2, T = 3.
  • ii) An ant, initially at (1,4), wants to walk on the plate so that the temperature along its path remains constant. What path should the ant take? ( Write the correct equation.)
• Find parametric equations for the tangent line at (1, 3, 3) to the curve of intersection of the surface z = x²y and the plane y = 3.
• Given z = xy² - y sinx, determine all first and second partial derivatives.
• The volume V of a right circular cylinder is given by V = ¼r² h, where r is the radius and h is the height. Suppose h has a constant value of 4 in., but r varies. Find the rate of change of V with respect to r at the instant when r = 6 in.
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• Let w = xz + xy + yz. Use the chain rule to find dw/dr if x = r cosq , y = sinq, z = z. (Cylindrical coordinates).
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• Let z = x lny + y lnx, x = r cosq, y = r sinq . Use the chain rule to find dz/dr and dz/dq .
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• Find the equation of the tangent plane and the normal line to the surface sin xy - 2cos yz = 0 at the point (p/2, 1, p/3).
• The derivative of f(x,y,z) at a given point P is greatest in the direction of the vector v = i + j - k. In this direction the value of the derivative is 2 sqrt(3) .
  • (i) Find the gradient vector of f at P.
  • (ii) Find the derivative of f at P in the direction of the vector i + j.
  •  
• Two surfaces are said to be orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that the sphere x² + y² + z² = a² and the cone z² = x² + y² are orthogonal at every point of intersection.
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• Given the function f(x,y,z) = e^-(x² + y² + z² ) , in what direction does f increase most rapidly at (1,1,1) and what is the value of the directional derivative in that direction?
• Let f(x,y,z) = x² + xy + y² . Find the directional derivative at (1,-1) in the direction of 3i + 4j. Also, what is the maximum value of the directional derivative at this point?
•  
• How many critical points does the function f(x,y) = x e^y have? Why?
• Let f(x,y) = 4x² - 4xy + y² + 5 . How many critical points does f have? Is it possible to determine the nature of the critical points by the second derivative test? Why or why not?
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• One leg of a right triangle increases from 3 cm to 3.2 cm, while the other leg decreases from 4 cm to 3.96 cm. Use a total differential to approximate the change in the length of the hypotenuse.
• Determine the points (x,y) and the directions for which w = 3x² + y² has its largest directional derivative, if (x,y) must be on the circle x² + y² = 1.
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• LAGRANGE MULTIPLIERS
• Use the method of Lagrange multipliers to find a vector in 3-space whose length is 5 and whose components have the largest possible sum.
• Find positive numbers x, y, z such that x + y + z = 24 and xyz² is as large as possible.
• Use the method of LaGrange multipliers to find the minimum distance from the origin to the plane x - y + 3z = 7.

Multiple Integrals

• Find the x-coordinate of the centroid of the area in the plane bounded by the lines q = 0° and q = 45° and by the circles r = 1 and r = 2.
• Find the volume enclosed by the cylinders z = 5 - x² , z = 4x² and the planes y = 0, x + y = 1.
• Find the volume in the first octant bounded by the cylinder x = 4 - y² and the planes z = y , x = 0 , z = 0.
• Sketch the region bounded by x = y - y² and x + y = 0 and use a double integral to find the area.
• Evaluate Int(-inf,+inf) (e^-(x² +y² ) dx dy) by converting to an equivalent double integral in polar coordinates.
• Use a double integral to find the total area enclosed by the lemniscate r² = 2a² cos 2q.
• Find the volume that is bounded above by the paraboloid z = 9 - x² - y² , below by the xy-plane, and that lies outside the cylinder x² + y² = 1.
• A lamina with density d(x,y) = xy is in the first quadrant and is bounded by the circle x² + y² = a² and the coordinate axes. Set up but do not evaluate the expressions needed to find the center of gravity.
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• For any area in the xy-plane, show that its polar moment of inertia Io about an axis through the origin perpendicular to the xy-plane is equal to Ix + Iy . Let d(x,y) stand for the density of the area at (x,y).
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• THE FOLLOWING ARE NOT APPLICABLE in 2005:
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• Consider the solid of constant density d bounded by two concentric spheres of radii a and b (a < b). Set up but do not evaluate a triple integral representing the moment of inertia of this solid about a diameter. Use spherical coordinates.
• Set up but do not evaluate an integral to find the volume of the solid inside the sphere x² + y² + z² = 4 and outside the cone z² = x² + y² . (Hint: use spherical coordinates.)
• Find the area of the surface cut from the plane z = cx, (c constant), by the cylinder x² + y² = a² .
• Find the moment of inertia about the x-axis of the area bounded by x = y² and x = 2y - y² if the density d(x,y) = y + 1.
• Find the area of the surface of that portion of the sphere x² + y² + z² = 4a² that lies inside the cylinder x² + y² = 2ax.
• Find the surface area of the portion of the cone z² = 4x² + 4y² that is above the region in the first quadrant bounded by the line y = x and the parabola y = x² .

Vector Calculus

All bold symbols represent vectors. LInt means line integral.

• Verify that the vector field F(x,y) = -(siny)i + (y - x cosy)j is conservative and find all functions f such that grad f = F.
• Find the value of LInt(C) (y dx + (x - y) dy) along the parabola x = t , y = t² from (0,0) to (1,1).
• Which of the following vector fields is conservative?
  • F(x,y) = -siny i + (y - xcosy) j
  • F(x,y) = 2xy i + (1/y² - x² ) j
• Show that LInt [(x²+ y² )dx + (2xy)dy] along a curve from (0,0) to (1,2) is independent of path and determine its value.
• Show that the force F = (2xy³ )i + (1 + 3x² y² )j is conservative and find the potential function. Use this function to find the work done as F acts on a particle moving from (-2, 3) to (1, 2).
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• THE FOLLOWING WILL NOT BE PART OF THE 2005 FINAL.
• A function f is harmonic in G if it satisfies Laplace's equation d² f/dx² + d² f/dy² + d² f/dz² = 0 throughout a region G in 3-space.
  • Show that the function f(x,y,z) = e^4x e^3y cos 5z is harmonic everywhere.
  • Let f be any function harmonic in a region G in 3-space. Let s be the boundary of G, n the outward unit normal on s, and df/dn = (grad f) o n, the directional derivative of f in the direction of n. Show that DInt(s) (df/dn dS) = 0. Quote any theorem you may need by name.
• Prove the identity curl (grad f) = 0.
• Show that curl(F + grad f) = curl F
• Use Stoke's Theorem to evaluate LInt (C)(z² dx + 2y dy - y³ dz) over the circle C: x² + y² = 1 in the xy-plane traversed counterclockwise looking down the positive z-axis. (Hint: Look for a convenient surface whose boundary is C.)
• Show that Green's Theorem is a special case of Stoke's Theorem in the case of F(x,y) = f(x,y)i + g(x,y)j defined just in the xy-plane.

Useful Websites

http://www.math.psu.edu/dna/graphics.html (Great graphics for learning Calculus from Penn State)

http://www.ping.be/math (Tutorials about most of the main topics in Calculus and Linear Algebra)

http://www.calculus.net (Very interesting practice problems and projects in Calculus and Diff. Equations)

http://www.math.duke.edu/modules/materials/index.html (Modules and projects created by Duke University for enhancing the learning of Calculus, Diff. Equations, Linear Algebra, and Engineering Mathematics)

http://www.maplesoft.com/apps/powertools/ (Maple 6 Resource Center)

http://www.maa.org (Mathematical Association of America)

http://archives.math.utk.edu (Math Archives)

http://forum.swarthmore.edu (Math Forum)

Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update: August 30, 2001

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