1 1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) We would like to make the length 6. The only vectors in the same direction as v are those of the form λv where λ is a positive number. Now λv λ v λ v. So we let λ..(6pts) Which line below is an equation of the line of intersection of the two planes x + 3y 4z 10 and 3x 4y + z. Note ( 1, 0, ) is a point on both planes. (a) t 10, 16, , 0, (b) t, 3, 4 + 1, 0, (c) t 10, 16, 16 +, 3, 4 (d) t 3, 4, + 1, 0, (e) t 10, 16, 17 Actually ( 1, 0, ) is a point on neither plane since ( 1) + 3(0) 4() 10 and 3( 1) 4(0) + () 1. However if you do what you should you should never notice. The two normal vectors are, 3, 4 and 3, 4,. Hence a vector parallel to the line is , 16, Hence t 10, 16, , 0, is an equation of the line. 3.(6pts) Given two lines t 1, 3, + 1, 1, 1 and t 3,, 1 + 7, 8, 8, find the point of intersection, if any? (a) (4, 10, 7) (b) (3, 7, 5) (c) (4, 10, 1) (d) (4, 7, 10) (e) There are no points of intersection. The equations are t + 1 3s + 7 3t + 1 s + 8 t + 1 s + 8 Then t 3s + 6 so (3s + 6) + 1 s + 8 and 6s + 13 s + 8 or 5s 5 or s 1. Then t so t 3 and the point is (4, 10, 7). 4.(6pts) Compute the curvature of the curve r(t) t sin(t), 1 cos(t), t at t π 3. (a) κ 10 8 (b) κ (c) κ 5 4 (d) κ 5 (e) κ 1
2 r 1 cos(t), sin(t), 1 and r sin(t), cos(t), 0 so κ r r r 3 cos(t), sin(t), cos(t) 1 1 cos(t), sin(t), (cos(t) 1) ( 3 cos(t) ) 3 At t π/3, cos(t) 1/ and so κ(π/3) 5/ (6pts) Given three vectors u, v and w in R 3, which of the following statements is not necessarily true? (a) If u v 0, then we must have either u 0 or v 0. (b) u v v u. (c) (u + v) w u w + v w. (d) u u 0. (e) u v is perpendicular to u and v. All the listed properties are true, except for u v implying u 0 or v 0. We could have, e.g., u v 1, 0, 0. 6.(6pts) Suppose that two vectors u and v R 3 are such that u 7, v 3 and the angle between them is θ 90. Compute u v. (a) 1. (b) 3 7. (c) 7. (d) 1π. 3 (e) There is not enough information. Using the identity u v u v sin(θ), we get u v 1. 7.(6pts) If f(x, y) x + e xy and x s + t, y sin(t), compute f at (s, t) (1, 0). (a) 3 (b) 0 (c) 1 (d) (e) 1
3 The chain rule says that f f x x + f y y. In this case, f (x + yexy ) 1 + xe xy cos(t) At (s, t) (1, 0), we have x 1, y 0, so that f (1, 0) OR f x + ye xy, xe xy and if r(t) s + t, sin(t) and r 1, cos(t). When s 1, t 0, x 1, y 0 so f(1, 0) r (1, 0), 1 1, (6pts) The plane S contains the points (0, 1, 3), (,, ), and (3,, 1). Which of the following is an equation for S? (a) x y + z (b) x + y + z 6 (c) x + 3y + z 6 (d) y + z 4 (e) x 4y + 3z 5 There are two ways to solve this problem. You can calculate the normal vector as in the worksheet and find a plane with that normal and one of those points and get x y + z. Alternately, you can test each equation to see if all three given points satisfy it, and the only one which works for all three is x y + z. 9.(6pts) Let f(x, y, z) e xy + z y. Find the sum of partial derivatives f x + f y + f z. (a) (x + y)e xy + yz + z (b) e y + e x + z + zy (c) ye xy (d) e xy + z + zy (e) e x + e y + z + 1 f x + f y + f z ye xy + xe xy + z + zy. 10.(6pts) Which one of the following functions has level curves as concentric circles? (a) f(x, y) e (x +y ) (d) f(x, y) e 4x y (b) f(x, y) 8 x y (c) f(x, y) x y (e) f(x, y) sin(x + y) The contour at level c is given by f(x, y) c. If f(x, y) e (x +y ) the contour at level c is x + y ln c, which is a circle centered at (0, 0). (Note c (0, 1]) For f(x, y) 8 x y the level curves are x+y 8 c, which are lines. For f(x, y) x y the level curves are x y c, which are hyperbolas and two lines. For f(x, y) e 4x y the level curves are 4x y ln c, which are parabolas. For f(x, y) sin(x + y) the level curves are x + y arcsin c which are lines. 11.(10pts) Find the area of the triangle with vertices P (1, 4, 6), Q(, 5, 1) and R(1, 1, 1).
4 We know that the area of the parallelogram determined by the vectors P Q and QR equals, P Q QR so the area of the triangle is just half this quantity. From here we just compute P Q 3, 1, 7 QR 3, 6, P Q QR , 15, , 3, and then Area of P Q QR (10pts) Let z z(x, y) be the function of x, y given implicitly by the equation Find and y. x + y 3 + z 4 + xyz 1 Let F (x, y) x + y 3 + z 4 + xyz. The function G(x, y) given by G(x, y) F (x, y, z(x, y)) is constant equal to 1, hence both G Rule for G, we may solve for : F F Using the Chain Rule for G, we may solve for y y : y F y F (x + yz) 4z 3 + xy (3y + xz) 4z 3 + xy and G y are zero. Using the Chain 13.(10pts) The line L passes through the point (, 7, 8) and is perpendicular to the plane R whose equation is 3x y + z 14. Find the point where L intersects R. Since L is perpendicular to R, its direction vector is R s normal, 3,, 1. Since we have a direction and a point, we can write an equation for L: r(t) 3,, 1 t +, 7, 8 3t +, t + 7, t + 8 To find the intersection of L and R, we substitute these coordinates into the equation for R. 3(3t + ) ( t + 7) + (t + 8) 14 9t t 14 + t t 14 t 1 Plugging this back into L, we get that the intersection is at (5, 5, 9). A quick check confirms that this point is indeed on R.
5 14.(10pts) A space curve is described by the equation 3 r(t) t, t cos(t) sin(t), t sin(t) + cos(t) for (0, ). (a) Write an equation for the unit tangent vector to the curve at time t. (b) Write an equation for the binormal to the curve at time t. (c) For what values of t (0, π) is the osculating plane at t of the curve perpendicular to the line with equations x 1 z 3 y? 1 3 r 3t, t sin(t), t cos(t), r t, T 3/, sin(t)/, cos(t)/, T 0, cos(t)/, sin(t)/, T 1/, N 0, cos(t), sin(t), B T N 1/, 3 sin(t)/, 3 cos(t)/. OR r (t) 3t, t sin(t), t cos(t) r (t) 3, sin(t) t cos(t), cos(t) t sin(t) r r 3t t sin(t) t cos(t) 3 sin(t) t cos(t) cos(t) t sin(t) t, 3t sin(t), 3t cos(t). and a parallel vector is 1, 3 sin(t), 3 cos(t). Hence a formula for the binormal is B 1/, 3 sin(t)/, 3 cos(t)/. OR A vector parallel to r (t) 3t, t sin(t), t cos(t), is p(t) 3, sin(t), cos(t). p (t) 0, cos(t), sin(t) and p p 3 sin(t) cos(t) 0 cos(t) sin(t) 1, 3 sin(t), 3 cos(t). Hence a formula for the binormal is B 1/, 3 sin(t)/, 3 cos(t)/. The binormal has to be parallel to u 1, 0, 3, the vector of the line. When sin(t) 0 and cos(t) 1, B 1 1, 0, 3 1 u. But in (0, π) the only value(s) of t for which sin(t) 0, cos(t) 1 is t π.
Chapter 4 Approximating functions by Taylor Polynomials. 4.1 Linear Approximations We have already seen how to approximate a function using its tangent line. This was the key idea in Euler s method. If
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the
Problem Set II: budget set, convexity Paolo Crosetto email@example.com Exercises will be solved in class on January 25th, 2010 Recap: Walrasian Budget set, definition Definition (Walrasian budget
Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) =
I. Vectors and Geometry in Two and Three Dimensions I.1 Points and Vectors Each point in two dimensions may be labeled by two coordinates (a,b) which specify the position of the point in some units with
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
4. The rent control agency of New York City has found that aggregate demand is Q D = 100-5P. Quantity is measured in tens of thousands of apartments. Price, the average monthly rental rate, is measured
WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? DAVIDE P. CERVONE, WILLIAM V. GEHRLEIN, AND WILLIAM S. ZWICKER Abstract. Consider an election in which each of the n voters casts a vote consisting of
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
Find-The-Number 1 Find-The-Number With Comps Consider the following two-person game, which we call Find-The-Number with Comps. Player A (for answerer) has a number x between 1 and 1000. Player Q (for questioner)
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
Mathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review