jf 00 (x)j ds (x) = [1 + (f 0 (x)) 2 ] 3=2 (t) = jjr0 (t) r 00 (t)jj jjr 0 (t)jj 3
|
|
- Randolf Simmons
- 6 years ago
- Views:
Transcription
1 M73Q Multivariable Calculus Fall 7 Review Problems for Exam The formulas in the box will be rovided on the exam. (s) dt jf (x)j ds (x) [ + (f (x)) ] 3 (t) jjt (t)jj jjr (t)jj (t) jjr (t) r (t)jj jjr (t)jj 3. True or False? Circle ONE answer for each. Hint: For eective study, exlain why if `true' and give a counterexamle if `false.' (a) T or F: If a?b and b?c; then a?c: Solution: False. Let a i; b j; and c < ; ; > : Then a b i j and b c < ; ; > < ; ; > : However, a c < ; ; > < ; ; > : (b) T or F: If a b ; then jja bjj jjajjjjbjj: Solution: True. Assume both a and b are nonero. (If either is ero the result is obvious). If is the angle between a and b; then since a and b are orthogonal. jja bjj jjajjjjbjj sin jjajjjjbjj sin jjajjjjbjj: (c) T or F: For any vectors u; v in R 3 ; jju vjj jjv ujj: Solution: True. If is the angle between u and v; then jju vjj jjujjjjvjj sin jjvjjjjujj sin jjv ujj: (Or, jju vjj jj v ujj j jjjv ujj jjv ujj:) (d) T or F: The vector < 3; ; > is arallel to the lane 6x y + 4 : Solution: False. A normal vector to the lane is n < 6; ; 4 > : Because < 3; ; > n; the vector is arallel to n and hence erendicular to the lane. (e) T or F: If u v ; then u or v : Solution: False. For examle i j but i 6 and j 6 : (f) T or F: If u v ; then u or v : Solution: False. For examle i i but i 6 : (g) T or F: If u v and u v ; then u or v : Solution: True. If both u and v are nonero, then u v imlies u and v are orthogonal. But u v imlies that u and v are arallel. Two nonero vectors can't be both arallel and orthogonal, so at least one of them must be : (h) T or F: The curve r(t) ; t ; 4t is a arabola. Solution: True. Parametric equations for the curve are x ; y t ; 4t; and since t 4 we have y t or y ; x : This is an equation of a arabola in the y lane. 4 6
2 M73Q Multivariable Calculus Review Problems for Exam - Page of 9 (i) T or F: If (t) for all t; the curve is a straight line. Solution: True. Notice that (t) () jjt (t)jj () T (t) for all t: But then T(t) C; a constant vector, which is true only for a straight line. (j) T or F: Dierent arameteriations of the same curve result in identical tangent vectors at a given oint on the curve. Solution: False. For examle, r (t) < t; t > and r (t) < t; t > both reresent the same lane curve (the line y x), but the tangent vector r (t) < ; > for all t; while r (t) < ; > : In fact, dierent arametriations give arallel tangent vectors at a oint, but their magnitudes may dier.. Which of the following are vectors? (a) Vector Scalar Nonsense [(a b)c] a (b) Vector Scalar Nonsense c [(a b) c] (c) Vector Scalar Nonsense c [(a b)c] (d) Vector Scalar Nonsense (a b) c 3. Which of the following are meaningful? (a) Meaningful Nonsense jjwjj(u v) (b) Meaningful Nonsense (u v) w (c) Meaningful Nonsense u (v w) 4. Find the values of x such that the vectors < 3; ; x > and < x; 4; x > are orthogonal. Solution: For the two vectors to be orthogonal, we need < 3; ; x > < x; 4; x > : That is, 3(x) + (4) + x(x) ; or x or x 4: 5. Find the decomosition a a kb + a?b of a < ; ; > along b < ; ; 3 > : Solution: a kb a b b b b ; 7 ; : ; 7 ; : Finally, a?b a a 7 kb < ; ; > 3 7 ; 7 ; 7 6. Let a ; ; b ; ; and u < 3; > : (a) Show that a and b are orthogonal unit vectors. Solution: a b and jjajj + ; and jjbjj +( ) : (b) Find the decomosition of u along a: Solution: u jja (u a)a 3 ; 3 ; 3 : 3 3 ; : Thus, u?a < 3; > 3 ; 3 (c) Find the decomosition of u along b:
3 M73Q Multivariable Calculus Review Problems for Exam - Page 3 of 9 Solution: Similarly, u jjb (u b)b < 3; > ; Thus, u?b < 3; 3 > ; 3 3 ; 3 u?a : ; 3 3 ; : 7. (a) Find an equation of the shere that asses through the oint (6, -, 3) and has center (-,,). Solution: Use the distance formula to nd the distance between (6, -, 3) and (-,,). Then, the equation for the circle is (x + ) + (y ) + ( ) 69: (b) Find the curve in which this shere intersects the y-lane. Solution: The intersection of this shere with the y lane is the set of oints on the shere whose x coordinate is. Putting x in to the equation, (y ) + ( ) 68; which reresents a circle in the y lance with center (; ; ) and radius 68: 8. For each of the following quantities (cos ; sin ; x; y; ; and w) in the icture below, ll in the blank with the number of the exression, taken from the list to the right, to which it is equal. a y w θ x b cos 5. sin 4. x 3. y 3 4. a b jjajj a b jjbjj jja bjj jjbjj jja bjj jjajjjjbjj 7 5. ja bj jjajjjjbjj w 6 6. jjb ajj 7. (b a) b jjbjj 9. Find an equation for the line through (4; ; ) and (; ; 5): Solution: The line has direction v < 3; ; 3 > : Letting P (4; ; ); arametric equations are x 4 3t; y + t; + 3t:
4 M73Q Multivariable Calculus Review Problems for Exam - Page 4 of 9. Find an equation for the line through ( ; ; 4) and erendicular to the lane x y + 5 : Solution: A direction vector for the line is a normal vector for the lane, n < ; ; 5 >; and arametric equations for the line are x + t; y t; 4 + 5t:. Find an equation of the lane through (; ; ) and arallel to x + 4y 3 : Solution: Since the two lanes are arallel, they will have the same normal vectors. Then we can take n < ; 4; 3 > and an equation of the lane is (x ) + 4(y ) 3( ) :. Find an equation of the lane that asses through the oint ( ; 3; ) and contains the line x(t) t; y(t) 4t; (t) + t: Solution: To nd an equation of the lane we must rst nd two nonarallel vectors in the lane, then their cross roduct will be a normal vector to the lane. But since the given line lies in the lane, its direction vector a < ; 4; > is one vector arallel to the lane. To nd another vector b which lies in the lane ick any oint on the line [say ( ; ; ); found by setting t ] and let b be the vector connecting this oint to the given oint in the lane ( ; 3; ): (But beware; we should rst check that the given oint is not on the given line. If it were on the line, the lane wouldnt be uniquely determined. What would n then be when we set n a b?) Thus b < ; 3; > and n a b < + 3; ; 6 >< 3; ; 6 > and an equation of the lane is 3(x + ) + 6( ) or x + 3: 3. Find the oint at which the line x(t) t; y t; (t) + t and the lane x + y intersect. Solution: Substitute the line into the equation of the lane. x+y ) +t ( t)+t ) t ) the oint of intersection is (; ; ): 4. (a) Find an equation of the lane that asses through the oints A(; ; ); B( ; ; ); and C(; 3; 4:)! Solution: The vector AB < 3; ; 9 > and! AC < ; ; 5 > lie in the lane, so n! AB! AC < 8; 4; 8 > or equivalently, < ; 3; > is a normal vector to the lane. The oint A(; ; ) lies on the lane so an equation for the lane is (x ) + 3(y ) + ( ) : (b) A second lane asses through (; ; 4) and has normal vector < ; 4; 3 > : Find an equation for the line of intersection of the two lanes. Solution: The oint (,,4) lies on the second lane, but the oint also satises the equation of the rst lane, so the oint lies on the line of intersection of the lanes. A vector v in the direction of this intersecting line is erendicular to the normal vectors of both lanes, so take v < ; 3; > < ; 4; 3 >< 5; 5; > or just use < ; ; > : Parametric equations for the line are x + t; y t; 4 + t: 5. Provide a clear sketch of the following traces for the quadratic surface y x + + in the given lanes. Label your work aroriately. x ; x ; y ; y ; :
5 M73Q Multivariable Calculus Review Problems for Exam - Page 5 of 9 y x -axis x y -axis -axis -axis Solution: 6. Match the equations with their grahs. Give reasons for your choices. (a) II 8x + y + 3 I sin x + cos y (b) (c) IV sin (d) III ey + x + y 7. Find a vector function that reresents the curve of intersection of the cylinder x + y 6 and the lane x + 5: Solution: The rojection of the curve C of intersection onto the xy lane is the circle x + y 6; : So we can write x 4 cos t; y 4 sin t; t : From the equation of the lane, we have 5 x 5 4 cos t; so arametric equations for C are x 4 cos t; y 4 sin t; 5 4 cos t; t ; and the corresonding vector function is r(t) < 4 cos t; 4 sin t; 5 4 cos t >; t : 8. Find an equation for the tangent line to the curve x sin t; y sin t; and sin 3t at the oint (; 3; ):
6 M73Q Multivariable Calculus Review Problems for Exam - Page 6 of 9 Solution: The curve is given by r(t) < sin t; sin t; sin 3t >; so r (t) < cos t; 4 cos t; 6 cos 3t > : The oint (; 3; ) corresonds to t 6; so the tangent vector there is r (6) < 3; ; > : Then the tangent line has direction vector < 3; ; > and includes the oint (; 3; ); so arametric equations are x + 3t; y 3 + t; : 9. A helix circles the axis, going from (; ; ) to (; ; 6) in one turn. (a) Parameterie this helix. Solution: r(t) < cos t; sin t; 3t > : (Note that revolution is ; so (3) 6:) (b) Calculate the length of a single turn. Solution: For t ; jjr (t)jj : Thus s (c) Find the curvature of this helix. Z 3dt 3(): Solution: The unit tangent vector is T(t) 3 < sin t; cos t; 3 >; so T (t) 3 < cos t; sin t; >. Thus, jjt (t)jj 3 : Since jjr (t)jj 3; 3. (a) Sketch the curve with vector function r(t) ht; cos t; sin ti ; t : Solution: The corresonding arametric equations for the curve are x t; y cos t; sin t: Since y + ; the curve is contained in a circular cylinder with axis the x axis. Since x t; the curve is a helix. (b) Find r (t) and r (t): Solution: Since r(t) ht; cos t; sin ti ; r (t) h; sin t; cos ti ; and r (t) ; i cos t; sin t :. Which curve below is traced out by r(t) sin t; cos 4 t; t ; t : Solution: Grah. Note that r() < ; ; > :. Find a oint on the curve r(t) t + ; t ; 5 where the tangent line is arallel to the lane x + y 4 5:
7 M73Q Multivariable Calculus Review Problems for Exam - Page 7 of 9 Solution: The lane has normal vector n < ; ; 4 > : Since r (t) < ; 4t; >; we want < ; 4t; > < ; ; 4 > : That is, + 8t ; and r( ) 8 < 7 8 ; 63 4 ; 5 > : 3. Let r(t) t; (e t )t; ln(t + ) : (a) Find the domain of r: Solution: The exressions t; (e t )t; and ln(t + ) are all dened when t : Thus, t ; t 6 ; and t + > ) t > : Finally, the domain of r is ( ; ) [ (; ]: (b) Find lim r(t): t! Solution: lim r(t) t! D ; ; E : Note that in the y comonent we use l'hosital's Rule. (c) Find r (t): D Solution: r (t) E ; te t e t + t t ; : t+ 4. Suose that an object has velocity v(t) 3 + t; sin(t); 6e 3t at time t; and osition r(t) < ; ; > at time t : Find the osition, r(t); of the object at time t: Z Solution: r(t) v(t)dt Z 3 + tdt; Z sin(t)dt; Z 6e 3t dt D( + t) 3 + c ; cos(t) + c ; e 3t + c 3 E : Thus, r() < + c ; + c ; + c 3 >< ; ; >) c ; c ; c 3 : So, r(t) D( + t) 3 ; cos(t) + ; e 3t E : 5. If r(t) t ; t cos t; sin t ; evaluate Z r(t)dt: Solution: Z r(t)dt Z Z Z t dt; t cos tdt; sin tdt 3 ; ; : 6. Find the length of the curve: x cos(t); y t 3 ; and sin(t); t : Solution: s 7 : Z s dx + dt dy dt + d dt dt Z q 6 sin (t) + 6 cos (t) + 9t dt Z 6 + 9tdt 7. Rearameterie the curve r(t) < e t ; e t sin t; e t cos t > with resect to arc length measured from the oint (; ; ) in the direction of increasing t: Solution: The arametric value corresonding to the oint (; ; ) is t : Since r (t) < e t ; e t (cos t+ sin t); e t (cos t sin t) >; jjr (t)jj e t + (cos t + sin t) + (cos t sin t) 3e t ; and s(t) R t eu 3du D E 3(e t ) ) t ln + s : 3 Therefore, r(t(s)) + s; 3 + s 3 sin ln + s ; 3 + s 3 cos ln + s : 3
8 M73Q Multivariable Calculus Review Problems for Exam - Page 8 of 9 8. Find the tangent line to the curve of intersection of the cylinder x + y 5 and the lane x at the oint (3; 4; 3): Solution: Let x(t) 5 cos t; y(t) 5 sin t; and (t) 5 cos t; so that r(t) < 5 cos t; 5 sin t; 5 cos t > and r (t) < 5 sin t; 5 cos t; 5 sin t > : When (x; y; ) (3; 4; 3); x(t) (t) 5 cos t 3; and y(t) 5 sin t 4; so r (t ) < 4; 3; 4 > is a direction vector for the line. The tangent line has arametric equations x(t) 3 4t; y(t) 4 + 3t; and (t) 3 4t: ANOTHER SOLUTION: Let x(t) t; y 5 x 5 t ; and (t) x(t) t: Then r(t) D E case, r t (t) ; 5 t ; : When x 3; t 3; and r (3) < ; 3 arametric equations x(t) 3 + t; y(t) 4 t; and (t) 3 + t: 4 D t; (5 t) ; t E : In this 3 ; > : So the tangent line has 4 9. For the curve given by r(t) 3 t3 ; t ; t ; nd (a) the unit tangent vector ; t; > : Solution: T(t) r (t) jjr (t)jj < t ; t; > t 4 + t + < t (t + ) (b) the unit normal vector Solution: T (t) < 4t; 4 t ; 4t > (t + ) ) jjt (t)jj 4(t4 + 4t + 4) (t + ) t + < t; t ; t > and N(t) t + (c) the curvature Solution: (t) jjt (t)jj jjr (t)jj (t + ) 3. A article moves with osition function r(t) < t ln t; t; e t >. Find the velocity, seed, and acceleration of the article. Solution: v(t) r (t) < + ln t; ; e t > : (t) jjv(t)jj ( + ln t) + + ( e t ) + ln t + (ln t) + e t : a(t) v (t) < t ; ; e t > : 3. A article starts at the origin with initial velocity < ; ; 3 > and its acceleration is a(t) < 6t; t ; 6t > : Find its osition function. Z Z Z Z Solution: v(t) a(t)dt 6t dt; t dt; 6t dt 3t ; 4t 3 ; 3t +C; but < ; ; 3 > v()+c; so C < ; ; 3 > and v(t) 3t + ; 4t 3 ; 3t + 3 Z : r(t) v(t)dt t 3 + t; t 4 t; 3t t 3 D: But r() ; so D ; and r(t) t 3 + t; t 4 t; 3t t 3 : 3. A ying squirrel has osition r(t) t + t ; t; + t at time t: Comute the following at time t : (a) The velocity at time t ; v() < ; ; >.
9 M73Q Multivariable Calculus Review Problems for Exam - Page 9 of 9 Solution: v(t) r (t) < + t; ; t > : (b) The seed at time t ; () 3. Solution: jjv(t)jj: jjv()jj : 33. Consider the vector valued function r(t) describing the curve shown below. Put the curvature of r at A; B and C in order from smallest to largest. Draw the osculating circles at those oints. Solution: B, A, C.
AP Calculus Testbank (Chapter 10) (Mr. Surowski)
AP Calculus Testbank (Chater 1) (Mr. Surowski) Part I. Multile-Choice Questions 1. The grah in the xy-lane reresented by x = 3 sin t and y = cost is (A) a circle (B) an ellise (C) a hyerbola (D) a arabola
More informationMath 273 Solutions to Review Problems for Exam 1
Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More informationFind the equation of a plane perpendicular to the line x = 2t + 1, y = 3t + 4, z = t 1 and passing through the point (2, 1, 3).
CME 100 Midterm Solutions - Fall 004 1 CME 100 - Midterm Solutions - Fall 004 Problem 1 Find the equation of a lane erendicular to the line x = t + 1, y = 3t + 4, z = t 1 and assing through the oint (,
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationMA 351 Fall 2007 Exam #1 Review Solutions 1
MA 35 Fall 27 Exam # Review Solutions THERE MAY BE TYPOS in these solutions. Please let me know if you find any.. Consider the two surfaces ρ 3 csc θ in spherical coordinates and r 3 in cylindrical coordinates.
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More informationSection Vector Functions and Space Curves
Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of
More information3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.
Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out bythe vector-valued function r!!t # f!t, g!t, h!t $. The vector T!!t r! % r! %!t!t is the unit tangent vector
More informationLecture 8: The First Fundamental Form Table of contents
Math 38 Fall 2016 Lecture 8: The First Fundamental Form Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this
More informationShort Solutions to Practice Material for Test #2 MATH 2421
Short Solutions to Practice Material for Test # MATH 4 Kawai (#) Describe recisely the D surfaces listed here (a) + y + z z = Shere ( ) + (y ) + (z ) = 4 = The center is located at C (; ; ) and the radius
More informationSOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003
SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of
More informationAPPM 2350, Summer 2018: Exam 1 June 15, 2018
APPM 2350, Summer 2018: Exam 1 June 15, 2018 Instructions: Please show all of your work and make your methods and reasoning clear. Answers out of the blue with no supporting work will receive no credit
More informationChapter 14: Vector Calculus
Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity
More informationMATH 280 Multivariate Calculus Fall Integrating a vector field over a curve
MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.
More informationMathematics 2203, Test 1 - Solutions
Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationCurves I: Curvature and Torsion. Table of contents
Math 48 Fall 07 Curves I: Curvature and Torsion Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this lecture
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 1 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH SOME SOLUTIONS TO EXAM 1 Fall 018 Version A refers to the regular exam and Version B to the make-up 1. Version A. Find the center
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More information0, such that. all. for all. 0, there exists. Name: Continuity. Limits and. calculus. the definitionn. satisfying. limit. However, is the limit of its
L Marizzaa A Bailey Multivariable andd Vector Calculus Name: Limits and Continuity Limits and Continuity We have previously defined limit in for single variable functions, but how do we generalize this
More informationVANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions
VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the
More informationPractice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
More informationVectors, dot product, and cross product
MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point
More informationM273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3
M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly
More informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
More informationParametric Functions and Vector Functions (BC Only)
Parametric Functions and Vector Functions (BC Only) Parametric Functions Parametric functions are another way of viewing functions. This time, the values of x and y are both dependent on another independent
More information0.1 Practical Guide - Surface Integrals. C (0,0,c) A (0,b,0) A (a,0,0)
. Practical Guide - urface Integrals urface integral,means to integrate over a surface. We begin with the stud of surfaces. The easiest wa is to give as man familiar eamles as ossible ) a lane surface
More informationNo calculators, books, notebooks or any other written materials are allowed. Question Points Score Total: 40
Be sure this exam has 6 ages including the cover The University of British Columbia MATH 317, Section 11, Instructor Tai-Peng Tsai Midterm Exam 1 October 16 Family Name Student Number Given Name Signature
More information= ( 2) = p 5.
MATH 0 Exam (Version ) Solutions Setember, 00 S. F. Ellermeyer Name Instructions. Your work on this exam will be raded accordin to two criteria: mathematical correctness clarity of resentation. In other
More informationMAT 272 Test 1 Review. 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same
11.1 Vectors in the Plane 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same direction as. QP a. u =< 1, 2 > b. u =< 1 5, 2 5 > c. u =< 1, 2 > d. u =< 1 5, 2 5 > 2. If u has magnitude
More informationCHAPTER 4 DIFFERENTIAL VECTOR CALCULUS
CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS 4.1 Vector Functions 4.2 Calculus of Vector Functions 4.3 Tangents REVIEW: Vectors Scalar a quantity only with its magnitude Example: temperature, speed, mass, volume
More informationMATH 32A: MIDTERM 1 REVIEW. 1. Vectors. v v = 1 22
MATH 3A: MIDTERM 1 REVIEW JOE HUGHES 1. Let v = 3,, 3. a. Find e v. Solution: v = 9 + 4 + 9 =, so 1. Vectors e v = 1 v v = 1 3,, 3 b. Find the vectors parallel to v which lie on the sphere of radius two
More informationAP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
More informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
NAURAL SCIENCES RIPOS Part IA Wednesday 5 June 2005 9 to 2 MAHEMAICS (2) Before you begin read these instructions carefully: You may submit answers to no more than six questions. All questions carry the
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationCHAPTER 11 Vector-Valued Functions
CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................
More information7. The olynomial P (x) =x, 9x 4 +9x, 4x +4x, 4 can be written in the form P (x) =(x, ) n Q(x), where n is a ositive integer and Q(x) is not divisible
. Which of the following intervals contains all of the real zeros of the function g(t) =t, t + t? (a) (,; ) (b) ( ; ) (c) (,; ) (d) (; ) (e) (,; ) 4 4. The exression ((a), + b), is equivalent to which
More informationMTHE 227 Problem Set 2 Solutions
MTHE 7 Problem Set Solutions 1 (Great Circles). The intersection of a sphere with a plane passing through its center is called a great circle. Let Γ be the great circle that is the intersection of the
More informationCentral Force Motion Challenge Problems
Central Force Motion Challenge Problems Problem 1: Ellitic Orbit A satellite of mass m s is in an ellitical orbit around a lanet of mass m which is located at one focus of the ellise. The satellite has
More informationLecture for Week 6 (Secs ) Derivative Miscellany I
Lecture for Week 6 (Secs. 3.6 9) Derivative Miscellany I 1 Implicit differentiation We want to answer questions like this: 1. What is the derivative of tan 1 x? 2. What is dy dx if x 3 + y 3 + xy 2 + x
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 14.1
OHSx XM5 Multivariable Differential Calculus: Homework Solutions 4. (8) Describe the graph of the equation. r = i + tj + (t )k. Solution: Let y(t) = t, so that z(t) = t = y. In the yz-plane, this is just
More informationMath 114, Section 003 Fall 2011 Practice Exam 1 with Solutions
Math 11, Section 003 Fall 2011 Practice Exam 1 with Solutions Contents 1 Problems 2 2 Solution key 8 3 Solutions 9 1 1 Problems Question 1: Let L be the line tangent to the curve r (t) t 2 + 3t + 2, e
More informationComplex Analysis Homework 1
Comlex Analysis Homework 1 Steve Clanton Sarah Crimi January 27, 2009 Problem Claim. If two integers can be exressed as the sum of two squares, then so can their roduct. Proof. Call the two squares that
More informationCalculus Vector Principia Mathematica. Lynne Ryan Associate Professor Mathematics Blue Ridge Community College
Calculus Vector Principia Mathematica Lynne Ryan Associate Professor Mathematics Blue Ridge Community College Defining a vector Vectors in the plane A scalar is a quantity that can be represented by a
More informationAPPM 2350 Section Exam points Wednesday September 26, 6:00pm 7:30pm, 2018
APPM 2350 Section Exam 1 140 points Wednesday September 26, 6:00pm 7:30pm, 2018 ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3) lecture section/time (4) your instructor
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: 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. You may use a graphing
More informationMathematics Engineering Calculus III Fall 13 Test #1
Mathematics 2153-02 Engineering Calculus III Fall 13 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Which of the following statements is always true? (a) If x = f(t), y = g(t) and f (1) = 0, then dy/dx(1)
More information1. (a) The volume of a piece of cake, with radius r, height h and angle θ, is given by the formula: [Yes! It s a piece of cake.]
1. (a The volume of a piece of cake, with radius r, height h and angle θ, is given by the formula: / 3 V (r,h,θ = 1 r θh. Calculate V r, V h and V θ. [Yes! It s a piece of cake.] V r = 1 r θh = rθh V h
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationGeometry and Motion Selected answers to Sections A and C Dwight Barkley 2016
MA34 Geometry and Motion Selected answers to Sections A and C Dwight Barkley 26 Example Sheet d n+ = d n cot θ n r θ n r = Θθ n i. 2. 3. 4. Possible answers include: and with opposite orientation: 5..
More informationSection 14.1 Vector Functions and Space Curves
Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationMATH 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2
MATH 3A: MIDTERM REVIEW JOE HUGHES 1. Curvature 1. Consider the curve r(t) = x(t), y(t), z(t), where x(t) = t Find the curvature κ(t). 0 cos(u) sin(u) du y(t) = Solution: The formula for curvature is t
More information1 Vectors and 3-Dimensional Geometry
Calculus III (part ): Vectors and 3-Dimensional Geometry (by Evan Dummit, 07, v..55) Contents Vectors and 3-Dimensional Geometry. Functions of Several Variables and 3-Space..................................
More information2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4).
Exam 3 Study Guide Math 223 Section 12 Fall 2015 Instructor: Dr. Gilbert 1. Which of the following vector fields are conservative? If you determine that a vector field is conservative, find a valid potential
More informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
More informationEXERCISES Practice and Problem Solving
EXERCISES Practice and Problem Solving For more ractice, see Extra Practice. A Practice by Examle Examles 1 and (ages 71 and 71) Write each measure in. Exress the answer in terms of π and as a decimal
More information(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)
eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your
More information= =5 (0:4) 4 10 = = = = = 2:005 32:4 2: :
MATH LEC SECOND EXAM THU NOV 0 PROBLEM Part (a) ( 5 oints ) Aroximate 5 :4 using a suitable dierential. Show your work carrying at least 6 decimal digits. A mere calculator answer will receive zero credit.
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationCalculus of Vector-Valued Functions
Chapter 3 Calculus of Vector-Valued Functions Useful Tip: If you are reading the electronic version of this publication formatted as a Mathematica Notebook, then it is possible to view 3-D plots generated
More informationPractice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b.
Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More information16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21
16.2 Line Integrals Lukas Geyer Montana State University M273, Fall 211 Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall 211 1 / 21 Scalar Line Integrals Definition f (x) ds = lim { s i } N f (P i ) s
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationMultiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous.
Multiple Choice. Circle the best answer. No work needed. No partial credit available. + +. Evaluate lim + (a (b (c (d 0 (e None of the above.. Evaluate lim (a (b (c (d 0 (e + + None of the above.. Find
More informationPractice Midterm Exam 1. Instructions. You have 60 minutes. No calculators allowed. Show all your work in order to receive full credit.
MATH202X-F01/UX1 Spring 2015 Practice Midterm Exam 1 Name: Answer Key Instructions You have 60 minutes No calculators allowed Show all your work in order to receive full credit 1 Consider the points P
More informationMath 153 Calculus III Notes
Math 153 Calculus III Notes 10.1 Parametric Functions A parametric function is a where x and y are described by a function in terms of the parameter t: Example 1 (x, y) = {x(t), y(t)}, or x = f(t); y =
More information1 4 (1 cos(4θ))dθ = θ 4 sin(4θ)
M48M Final Exam Solutions, December 9, 5 ) A polar curve Let C be the portion of the cloverleaf curve r = sin(θ) that lies in the first quadrant a) Draw a rough sketch of C This looks like one quarter
More informationTopic 2-2: Derivatives of Vector Functions. Textbook: Section 13.2, 13.4
Topic 2-2: Derivatives of Vector Functions Textbook: Section 13.2, 13.4 Warm-Up: Parametrization of Circles Each of the following vector functions describe the position of an object traveling around the
More informationThe Calculus of Vec- tors
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 3 1 The Calculus of Vec- Summary: tors 1. Calculus of Vectors: Limits and Derivatives 2. Parametric representation of Curves r(t) = [x(t), y(t),
More information13.3 Arc Length and Curvature
13 Vector Functions 13.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. We have defined the length of a plane curve with parametric equations x = f(t),
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: 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. You may use a graphing
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
More informationCalculus and Parametric Equations
Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how
More informationExam 2 Review Solutions
Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationNORTHEASTERN UNIVERSITY Department of Mathematics
NORTHEASTERN UNIVERSITY Department of Mathematics MATH 1342 (Calculus 2 for Engineering and Science) Final Exam Spring 2010 Do not write in these boxes: pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 Total (100 points)
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
More informationTopic 30 Notes Jeremy Orloff
Toic 30 Notes Jeremy Orloff 30 Alications to oulation biology 30.1 Modeling examles 30.1.1 Volterra redator-rey model The Volterra redator-rey system models the oulations of two secies with a redatorrey
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationExaminer: D. Burbulla. Aids permitted: Formula Sheet, and Casio FX-991 or Sharp EL-520 calculator.
University of Toronto Faculty of Applied Science and Engineering Solutions to Final Examination, June 216 Duration: 2 and 1/2 hrs First Year - CHE, CIV, CPE, ELE, ENG, IND, LME, MEC, MMS MAT187H1F - Calculus
More informationMA 126 CALCULUS II Wednesday, December 14, 2016 FINAL EXAM. Closed book - Calculators and One Index Card are allowed! PART I
CALCULUS II, FINAL EXAM 1 MA 126 CALCULUS II Wednesday, December 14, 2016 Name (Print last name first):................................................ Student Signature:.........................................................
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More informationSolutions to Problem Set 5
Solutions to Problem Set Problem 4.6. f () ( )( 4) For this simle rational function, we see immediately that the function has simle oles at the two indicated oints, and so we use equation (6.) to nd the
More information1 Exam 1 Spring 2007.
Exam Spring 2007.. An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = 5. 2. Use the
More informationMTH 234 Exam 1 February 20th, Without fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More information