# Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector

Size: px
Start display at page:

Download "Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector"

Transcription

1 Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a = 3, 4, 0 and b = 1, 4,. Find a, a + 3 b, 3 a b. Find the normalization of a. (b) Let a = 1, 0, 1 and b = 1, 1, 0. Find a b. Find the projection of b onto a. (c) Let a = 5, 3, 7 and b = 6, 8,. Find a b and a b. Determine if the vectors are parallel, perpendicular or neither. 3. Lines and Planes. (a) Find an equation of the line through the point (, 4, 10) and parallel to the vector 3, 1, 8. (b) Find an equation of the line through the point (1, 0, 6) and perpendicular to the plane x + 3y + z = 5. (c) Find an equation of the line through the points (3, 1, 1) and (3,, 6). (d) Find an equation of the plane through the point (6, 3, ) and perpendicular to the vector, 1, 5. (e) Find an equation of the plane through the point (4,, 3) and parallel to the plane 3x 7z = 1. (f) Find an equation of the plane through the points (0, 1, 1), (1, 0, 1) and (1, 1, 0). (g) Find an equation of the line of the intersection of the planes x+y z = 0 and x 5y z = Curves in Space. (a) Consider the curve x = cos t y = sin t z = t. Find an equation of the tangent line to the curve at the point where t = 0. Find the length of the curve from t = 0 to t = 1. (b) Let C be the curve of intersection of the cylinder x + y = 1 with the plane y + z =. Find parametric equations of this curve and an equation of the tangent line to the curve at the point where t = 0 in parametrization. Using the calculator, estimate the length of the curve from t = 0 to t = π/. (c) Consider the curve C which is the intersection of the surfaces x + y = 9 and z = 1 y. i) Find the parametric equations that represent the curve C. ii) Find the equation of the tangent line to the curve C at point (0, 3, 8). iii) Find the length of the curve from (3, 0, 1) to (0, 3, 8). You can use the calculator for the integral that you are going to get. 1

2 (d) Consider the curve C which is the intersection of the surfaces y + z = 16 and x = 8 y z i) Find the parametric equations that represent the curve C. ii) Find the equation of the tangent line to the curve C at point ( 8, 4, 0). iii) Find the length of the curve from (4, 0, 4) to ( 8, 4, 0). Use the calculator to evaluate the integral that you are going to get. (e) Find the length of the boundary of the part of the paraboloid z = 4 x y in the first octant. 5. Partial Derivatives. Find the indicated derivatives. (a) z = 3x + xy 5y ; z x, z y, z xx, z xy, z yx and z yy. (b) z = e x sin y; z x, z y, z xx, z xy, z yx and z yy. (c) z = ax e x xy where a is a constant; z x, z y, z xx, z xy and z yy. (d) z = x ln(xy ); z x, z y, z xx, z xy and z yy. (e) xy + yz + zx = 3; z x and z y at (1, 1, 1). (f) x yz = cos(x + y + z); z x and z y at (0, 1, 1). (g) z = 3x + xy 5y, x = + t, y = 1 t 3 ; z (t) when t = 0. (h) z = x ln(x + y), x = cos t, y = sin t; z (t) when t = 0. (i) z = e x y + xy, x = st, y = s + t ; z s and z t at (1, 1). (j) z = x + xy, x = e t cos s, y = e t sin s; z s and z t at (π, 0). 6. Tangent plane. Find an equation of the tangent plane to a given surface at a specified point. (a) z = y x, at ( 4, 5, 9) (b) z = e x ln y, at (3, 1, 0) (c) x + y + 3z = 1, at (4, 1, 1) (d) xy + yz + zx = 3; at (1, 1, 1). (e) x yz = cos(x + y + z); at (0, 1, 1). 7. Linear Approximation. (a) If f(, 3) = 5, f x (, 3) = 4 and f y (, 3) = 3, approximate f(.0, 3.1). (b) If f(1, ) = 3, f x (1, ) = 1 and f y (1, ) =, approximate f(.9, 1.99). (c) Find the linear approximation of z = 0 x 7y at (, 1) and use it to approximate the value at (1.95, 1.08). (d) Find the linear approximation of z = ln(x 3y) at (7, ) and use it to approximate the value at (6.9,.06). 8. Applications. (a) The pressure of 1 mole of an ideal gas is increasing at a rate of 0.05 kpa/s and the temperature is rising at a rate of 0.15 K/s. The pressure P, volume V and temperature T are related by the equation P V = 8.31T. Find the rate of change of the volume when the pressure is 0 kpa and temperature 30 K.

3 (b) The number N of bacteria in a culture depends on temperature T and pressure P which depend on time t in minutes. Assume that 3 minutes after the experiment started, the pressure is increasing at a rate of 0.1 kpa/min and the temperature at a rate of 0.5 K/min. The number of bacteria changes at the rates of 3 bacteria per kpa and 5 bacteria per Kelvin. i) Find the rate at which the number of bacteria is increasing 3 minutes after the experiment started. ii) Assume that the rates of 3 bacteria per kpa and 5 bacteria per Kelvin are constant. If there is 300 bacteria initially when T = 305 K and P = 10 kpa, estimate the number of bacteria when T = 309 K and P = 100 kpa. (c) The number of flowers N in a closed environment depends on the amount of sunlight S that the flowers receive and the temperature T of the environment. Assume that the number of flowers changes at the rates N S = and N T = 4. i) If there are 100 flowers when S = 50 and T = 70, estimate the number of flowers when S = 5 and T = 73. ii) If the temperature depends on time as T (t) = 85 8 and the amount of sunlight 1+t decreases on time as S(t) = 1, find the rate of change of the flower population at time t t = days. (d) The temperature at a point (x, y) is T (x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = 1 + t, y = + 1 t where x and y are 3 measured in centimeters. The temperature function satisfies T x (, 3) = 4 and T y (, 3) = 3. Determine how fast the temperature rises on the bug s path after 3 seconds. (e) An object moves in the space away from its initial position so that after t hours it is at x = t and y = t 1 and z = 3t + 1 miles from its initial position. Find the speed of that object 5 hours after it started moving. (Recall that the speed is the length of the tangent vector at a point.) 3

4 Solutions More detailed solutions of the problems can be found on the class handouts. 1. Surfaces. (a) Cylinder, base is a circle x + y = 9 in xy-plane. (b) Sphere, center at origin, radius. (c) Horizontal plane, passes (0, 0, 1). (d) Plane, passes (3, 0, 0), (0,, 0) and (0, 0, 6). (e) Cone, obtained by rotating a line z = y in zy-plane about z-axis. obtained by rotating a parabola z = y in zy-plane about z-axis.. Review of Vectors. (a) a = 5, a + 3 b = 3, 0, 6, 3 a b = 11, 4, 4. a a = 3 5, 4 5, 0. (b) a b = 1 The projection of b onto a is 1, 0, 1. (f) Paraboloid, (c) a b = 40 0 so the vectors are not perpendicular, a b = 6, 5, 0, 0, 0 so the vectors are not parallel. 3. Lines and Planes. (a) x = + 3t y = 4 + t z = 10 8t (b) x = 1 + t y = 3t z = 6 + t (c) x = 3 y = 1 + t z = 1 5t (d) x + y + 5z = 1 (e) 3x 7z = 9 (f) x + y + z = (g) x = 1 + 6t y = t z = 1 + 7t 4. Curves in Space. (a) Tangent: x = 1, y = t, z = t. Length: Length: 1.91 (b) Tangent: x = 1, y = t, z = t. (c) i) x = 3 cos t, y = 3 sin t, z = 1 y = 1 9 sin t. ii) (0, 3, 8) corresponds to t = π/. Plugging π/ in derivative gives you 3, 0, 0. Tangent line: x = 3t y = 3 z = 8. iii) (3, 0, 1) corresponds to t = 0 and (0, 3, 8) to t = π/. So, the bounds of integration are 0 to π/. The length is (d) i) y = 4 cos t, z = 4 sin t, x = 8 y z = 8 16 cos t 4 sin t. ii) ( 8, 4, 0) corresponds to t = π. Plugging π in derivative gives you 4, 0, 4. Tangent line: x = 4t 8 y = 4 z = 4t. iii) (4, 0, 4) corresponds to t = π/ and ( 8, 4, 0) to t = π. So, the bounds of integration are π/ to π. The length is (e) A set of parametric equations for the three curves in the intersection is x = cos t, y = sin t, z = 0 with 0 t π, x = t, y = 0, z = 4 t with 0 t, and x = 0, y = t, z = 4 t with 0 t. The three derivative vectors and length elements are x = sin t, y = cos t, z = 0 ds = 4 sin t + 4 cos tdt = 4dt = dt x = 1, y = 0, z = t ds = 1 + 4t dt, and x = 0, y = 1, z = t ds = 1 + 4t dt. The total length is π/ 0 dt t dt t dt = π

5 5. Partial Derivatives. (a) z x = 6x + y, z y = x 10y, z xx = 6, z xy = z yx =, z yy = 10 (b) z x = e x sin y, z y = e x cos y, z xx = e x sin y, z xy = z yx = e x cos y, z yy = e x sin y (c) z x = axe x xy + ax e x xy (x y) = a(x + x 3 x y)e x xy, z y = ax e x xy ( x) = ax 3 e x xy. Then z xx = a(+6x xy)e x xy +a(x+x 3 x y)e x xy (x y) and z yy = ax 3 e x xy ( x) = ax 4 e x xy. Differentiating z x with respect to y get z xy = a( x )e x xy + a(x+x 3 x y)e x xy ( x) = a( x x x 4 +x 3 y)e x xy = a( 3x x 4 +x 3 y)e x xy. Alternatively, differentiating z y with respect to x get z xy = 3ax e x xy ax 3 e x xy (x y) = a( 3x x 4 + x 3 y)e x xy (d) z x = ln(xy ) + 1, z y = x/y, z xx = 1/x, z xy = z yx = /y, z yy = x/y (e) z x = (y + xz)/(yz + x ), z y = (xy + z )/(yz + x ). At (1, 1, 1), z x = 1, z y = 1. (f) z x = (1 + sin(x + y + z))/( y + sin(x + y + z)) and z y = ( z + sin(x + y + z))/( y + sin(x + y + z)). At (0, 1, 1), z x = 1 and z y = 1. (g) z (t) = (6x + y)(t) + (x 10y)( 3t ); z (0) = 0 (h) z (t) = (ln(x + y) + x/(x + y))( sin t) + (x/(x + y))(cos t); z (0) =. (i) z s = (e x y + y )t + (e x + xy)s; z t = (e x y + y )s + (e x + xy)t; z s (1, 1) = 4e + 1 and z t (1, 1) = 4e + 1 (j) z s = (x + y)( e t sin s) + x(e t cos s), z t = (x + y)(e t cos s) + x(e t sin s), z s (π, 0) = 1 and z t (π, 0) = 6. Tangent plane. (a) 8x + 10y z = 9 (b) e 3 y z = e 3 (c) F x = x, F y = 4y, F z = 6z. At (4, 1, 1) this produces vector 8, 4, 6. The tangent plane is 4x y + 3z = 1. (d) F x = y + xz, F y = xy + z, F z = yz + x. At (1, 1, 1) this produces vector 3, 3, 3. The tangent plane x + y + z = 3. (e) F x = 1 + sin(x + y + z), F y = z + sin(x + y + z), F z = y + sin(x + y + z). At (0, 1, 1) this produces vector 1, 1, 1. The tangent plane is x + y z =. 7. Linear Approximation. (a) f(.0, 3.1) 5.38 (b) f(.9, 1.99).9 (c) f(1.95, 1.08).847 (d) f(6.9,.06) Applications. (a) -.7 liter per second (b) i) Since P (3) = 0.1, T (3) = 0.5, N P = 3, and N T = 5, and N (t) = N T T +N P P We have that N (3) = =.8 bacteria/minute. ii) Using linear approximation formula, N(T, P ) N(T 0, P 0 ) + N T (T T 0 ) + N P (P P 0 ) N(309, 100) N(305, 10) + 5( ) + 3(100 10) = (4) + 3( ) = 314 bacteria. (c) i) (5 50) + 4(73 70) = 116 flowers. ii) N = N S S + N T T = = flowers/day. (d) degrees Celsius per second (e) Speed = 0.1 miles per hour. 5

### Review for the Final Exam

Calculus 3 Lia Vas Review for the Final Exam. Sequences. Determine whether the following sequences are convergent or divergent. If they are convergent, find their limits. (a) a n = ( 2 ) n (b) a n = n+

### Tangent Plane. Linear Approximation. The Gradient

Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,

### Practice 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)

### M273Q 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

### (a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

### Practice Problems for the Final Exam

Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of

### Math 10C - Fall Final Exam

Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient

### Vectors, 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

### Calculus 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.

### Math 222 Spring 2013 Exam 3 Review Problem Answers

. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w

### Mathematics 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

### Page Points Score Total: 210. No more than 200 points may be earned on the exam.

Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200

### Exam 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.

### MA FINAL EXAM Form B December 13, 2016

MA 6100 FINAL EXAM Form B December 1, 016 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME 1. You must use a # pencil on the scantron. a. If the cover of your exam is GREEN, write 01 in the TEST/QUIZ NUMBER

### Exercises 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

### DEPARTMENT 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

### (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

### a 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

### e x3 dx dy. 0 y x 2, 0 x 1.

Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

### Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a

### SCORE. 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

### Review for the Final Exam

Calculus Lia Vas. Integrals. Evaluate the following integrals. (a) ( x 4 x 2 ) dx (b) (2 3 x + x2 4 ) dx (c) (3x + 5) 6 dx (d) x 2 dx x 3 + (e) x 9x 2 dx (f) x dx x 2 (g) xe x2 + dx (h) 2 3x+ dx (i) x

### Math 2163, Practice Exam II, Solution

Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so

### SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253

SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the

### Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Fall 2014 MTH 234 FINAL EXAM December 8, 2014 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 2 5 1 3 5 4 5 5 5 6 5 7 5 2 8 5 9 5 10

### MATH 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

### 1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.

Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is

### Department 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.

### SCORE. 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

### MIDTERM EXAMINATION. Spring MTH301- Calculus II (Session - 3)

ASSALAM O ALAIKUM All Dear fellows ALL IN ONE MTH3 Calculus II Midterm solved papers Created BY Ali Shah From Sahiwal BSCS th semester alaoudin.bukhari@gmail.com Remember me in your prayers MIDTERM EXAMINATION

### Final Review Worksheet

Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;

### G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv

1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N

### Final Exam. Math 3 December 7, 2010

Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.

### Solutions 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

### x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the

1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle

### 1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r

### 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

### Spring 2015 Sample Final Exam

Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than

### LB 220 Homework 4 Solutions

LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class

### z 2 = 1 4 (x 2) + 1 (y 6)

MA 5 Fall 007 Exam # Review Solutions. Consider the function fx, y y x. a Sketch the domain of f. For the domain, need y x 0, i.e., y x. - - - 0 0 - - - b Sketch the level curves fx, y k for k 0,,,. The

### Bi. lkent Calculus II Exams

Bi. lkent Calculus II Exams 988-208 Spring 208 Midterm I........ Spring 208 Midterm II....... 2 Spring 207 Midterm I........ 4 Spring 207 Midterm II....... 5 Spring 207 Final........... 7 Spring 206 Midterm

### Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions 1969-010 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

### Review Sheet for the Final

Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence

### Math 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

### Parametric Curves. Calculus 2 Lia Vas

Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider

### Math 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will

Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The

### MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane

MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.

### MA 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.

### 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.

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

### Final Examination 201-NYA-05 May 18, 2018

. ( points) Evaluate each of the following limits. 3x x + (a) lim x x 3 8 x + sin(5x) (b) lim x sin(x) (c) lim x π/3 + sec x ( (d) x x + 5x ) (e) lim x 5 x lim x 5 + x 6. (3 points) What value of c makes

### AP Calculus Free-Response Questions 1969-present AB

AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions

### Math Review for Exam 3

1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)

### There are some trigonometric identities given on the last page.

MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during

### MATH 52 FINAL EXAM SOLUTIONS

MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

### 1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6

Contents 1 Functions of Several Variables 1 1.1 Some Examples.................................. 2 1.2 Level Curves / Contours............................. 4 1.3 Functions of More Variables...........................

### 7a3 2. (c) πa 3 (d) πa 3 (e) πa3

1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

### Calculus III (MAC )

Calculus III (MAC2-) Test (25/9/7) Name (PRINT): Please show your work. An answer with no work receives no credit. You may use the back of a page if you need more space for a problem. You may not use any

### MATH 151, SPRING 2018

MATH 151, SPRING 2018 COMMON EXAM II - VERSIONBKEY LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF

### Vector Calculus handout

Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f

### Review problems for the final exam Calculus III Fall 2003

Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)

### 4 Partial Differentiation

4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which

### MA CALCULUS II Friday, December 09, 2011 FINAL EXAM. Closed Book - No calculators! PART I Each question is worth 4 points.

CALCULUS II, FINAL EXAM 1 MA 126 - CALCULUS II Friday, December 09, 2011 Name (Print last name first):...................................................... Signature:........................................................................

### Study Guide/Practice Exam 2 Solution. This study guide/practice exam is longer and harder than the actual exam. Problem A: Power Series. x 2i /i!

Study Guide/Practice Exam 2 Solution This study guide/practice exam is longer and harder than the actual exam Problem A: Power Series (1) Find a series representation of f(x) = e x2 Explain why the series

### Math Review for Exam Compute the second degree Taylor polynomials about (0, 0) of the following functions: (a) f(x, y) = e 2x 3y.

Math 35 - Review for Exam 1. Compute the second degree Taylor polynomial of f e x+3y about (, ). Solution. A computation shows that f x(, ), f y(, ) 3, f xx(, ) 4, f yy(, ) 9, f xy(, ) 6. The second degree

### MATH 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,

### Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

### No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.

Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear

### DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle

### Calculus with Analytic Geometry 3 Fall 2018

alculus with Analytic Geometry 3 Fall 218 Practice Exercises for the Final Exam I present here a number of exercises that could serve as practice for the final exam. Some are easy and straightforward;

### A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#\$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y \$ Z % Y Y x x } / % «] «] # z» & Y X»

### Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

### Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

### MTH 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

### Brief answers to assigned even numbered problems that were not to be turned in

Brief answers to assigned even numbered problems that were not to be turned in Section 2.2 2. At point (x 0, x 2 0) on the curve the slope is 2x 0. The point-slope equation of the tangent line to the curve

### Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

### 3 Applications of partial differentiation

Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives

### ARNOLD PIZER rochester problib from CVS Summer 2003

WeBWorK assignment VmultivariableFunctions due 3/3/08 at 2:00 AM.( pt) setvmultivariablefunctions/ur VC 5.pg Match the surfaces with the verbal description of the level curves by placing the letter of

### (b) x = (d) x = (b) x = e (d) x = e4 2 ln(3) 2 x x. is. (b) 2 x, x 0. (d) x 2, x 0

1. Solve the equation 3 4x+5 = 6 for x. ln(6)/ ln(3) 5 (a) x = 4 ln(3) ln(6)/ ln(3) 5 (c) x = 4 ln(3)/ ln(6) 5 (e) x = 4. Solve the equation e x 1 = 1 for x. (b) x = (d) x = ln(5)/ ln(3) 6 4 ln(6) 5/ ln(3)

### In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute

alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In

### WORKSHEET #13 MATH 1260 FALL 2014

WORKSHEET #3 MATH 26 FALL 24 NOT DUE. Short answer: (a) Find the equation of the tangent plane to z = x 2 + y 2 at the point,, 2. z x (, ) = 2x = 2, z y (, ) = 2y = 2. So then the tangent plane equation

### APPM 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

### 1 + 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

### LSU AP Calculus Practice Test Day

LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3

### Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

### Answer sheet: Final exam for Math 2339, Dec 10, 2010

Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,

### Fundamental Theorems of Vector

Chapter 17 Analysis Fundamental Theorems of Vector 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

### Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

### MA 113 Calculus I Fall 2015 Exam 1 Tuesday, 22 September Multiple Choice Answers. Question

MA 113 Calculus I Fall 2015 Exam 1 Tuesday, 22 September 2015 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions

### Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

### 4/5/2012: Second midterm practice A

Math 1A: introduction to functions and calculus Oliver Knill, Spring 212 4/5/212: Second midterm practice A Your Name: Problem 1) TF questions (2 points) No justifications are needed. 1) T F The formula

### MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014

MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014 Dr. E. Jacobs The main texts for this course are Calculus by James Stewart and Fundamentals of Differential Equations by Nagle, Saff

### 4/8/2014: Second midterm practice C

Math 1A: introduction to functions and calculus Oliver Knill, Spring 214 4/8/214: Second midterm practice C Your Name: Start by writing your name in the above box. Try to answer each question on the same

### Exam 1 Review SOLUTIONS

1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make

### MAT 132 Midterm 1 Spring 2017

MAT Midterm Spring 7 Name: ID: Problem 5 6 7 8 Total ( pts) ( pts) ( pts) ( pts) ( pts) ( pts) (5 pts) (5 pts) ( pts) Score Instructions: () Fill in your name and Stony Brook ID number at the top of this

### Sample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed.

Sample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed. Part A: (SHORT ANSWER QUESTIONS) Do the following problems. Write the answer in the space provided. Only the answers

### Name: 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

### Mathematical Analysis II, 2018/19 First semester

Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically

### Without fully opening the exam, check that you have pages 1 through 12.

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 12. Show all your work on the standard