Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Size: px
Start display at page:

Download "Practice Final Math 122 Spring 12 Instructor: Jeff Lang"

Transcription

1 Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6 n. A) ln 6 B) C) e 6 D) does not exist 3. Find the limit of the sequence a n = n! 6 n +8 n. A) B) C) e 8 D) does not exist. Find a formula for the nth partial sum of the series it to find the series sum if the series converges. 6 9n +n+ n= and use A) 6n 5(3n+5) ; 5 B) 6n 5n+5 ; 5 C) 6n 5(n+) ; 6 5 D) 6n 5n+ ; Find the sum of the telescoping series e n e n 3, if it exists. n= A) e 5 B) e 3 C) e D) does not exist 6. Find the sum of the telescoping series ( ln n= n+ ) ln ( n+ A) ln B) ln 3 C) D) does not exist ), if it exists. 7. Find the sum of the geometric series n= A) 9 7 B) 7 7 C) 8 D) ( ) n 9 8 n. 8. Use the integral test to determine whether the series n e /n converges or diverges. A) converge B) diverge n= 9) Determine whether the series ln ( e / n) converges or diverges. n=

2 A) converge B) diverge. Use the ratio test to determine if the series n= 5(n!) (n)! converges or diverges. A) converge B) diverge. Use the comparison test to determine if the series ( n n 5n+) converges or diverges. A) converge B) diverge n=. Use the limit comparison test to determine whether the series 9 converges or diverges. n 3 ln n+ n= A) converge B) diverge 3. Determine whether the alternating series ( ( ) n ln diverges. n= A) converge B) diverge 6n+ 6n+3. Determine whether the alternating series ( ) n ( n n + ) converges absolutely, conditionally, or diverges. n= ) converges or A) converge absolutely B) diverges C) converges conditionally 5. Estimate the magnitude of the error involved in using the sum of the first four terms to estimate the sum of the entire series n= ( ) n+ (.) n+ n+. A) B) 3. C).5 8 D) Find the radius of convergence of the power series A) 8 B) C) 9 D) n= 7. Find the radius of convergence of the power series n= (x 5) n 9 n n. (x 5) n 9 n n!.

3 A) 8 B) C) D) 9 8. Find the interval of convergence of the power series n= ( ) n (x 5) n (n+)6 n. A) x B) < x C) x 6 D) x < 6 9. Find the first for tems of the McLaurin series for f (x) = ln ( + x). A) x x! + x3 3! x x! + B) x + + x3 3 + x C) x x + x3 3 x. Find the Taylor series sin x at x = π. + x + D) x +! + x3 3! + x! + A) ( ( ) ( ) x π + x π ( ) ) 6 x π 3 + B) ( + ( ) ( ) x π x π ( ) ) 3 x π 3 + C) ( ( ) ( ) + x π 6 x π ( ) ) x π 3 + D) ( + ( ) ( ) x π x π ( ) ) 6 x π 3 + E) None of the above. Find the angle between the two planes x + 3y 3z = and x + y 5z =. A).9 B).3 C).57 D).8. Which of the following vectors is parallel to the two planes x + 3y z = and x + y + z = 8? A) i + 3 j 5 k B) i 3 j 5 k C) i + 3 j 6 k D) i 3 j 6 k 3. Find parametric equations of the line of intersection of the two planes x + y + z = and x y + z =. A) x = t +, y = t, z = 3t B) x = t +, y = t, z = 3t C) x = t, y = 3 + t, z = 5 3t D) x = + t, y = t, z = 3 + 3t. Find the point of intersection of the line x x + y + z =. = y+ = z and the plane 3

4 A) (,, ) B) (,, ) C) ( 9,, ) D) (,, ) 5. Find symmetric equations of the line perpendicular to the plane 5x + y + z = and passing through the point (, 3, ). A) x+ = y + 3 = z+ 5 B) x+ 5 = y + 3 = z+ C) x 5 = y 3 = z D) x 5 = y 3 = z 6. Which of the vectors below is tangent to the curve r (t) = cos t i + sin t j + e t k at t =? A) j k B) j + k D) All of the above C) j k 7. Calculate the arc length of the curve r (t) = t i + t sin t j + t cos t k ; t. A) 5 B) C) D) 6 8. Find the acceleration at t = for r (t) = ( t t 3) i + 5t j + ( t 3 ln t ) k. A) i + k B) i + k C) 6 i + 6 k D) i + 5 j + k x 9. Find lim +y + (x,y)(,) x 6xy+7. A) B) 7 C) 7 D) No limit 3. Find lim x+y. (x,y)(,) x +y A) B) No limit C) 7 D) - 3. At what points is the function f (x, y) = xy x continuous? A) All (x, y) such that xy > and x B) All (x, y) such that x i y and x > C) All (x, y) such that x D) All (x, y) 3. Find all first order partial derivatives of the function f (x, y) = x ln (xy).

5 A) f f x = + ln (xy) ; y = x y C) f f x = x + ln (xy) ; y = x y B) f f x = ln (xy) ; y = x y D) f x = x + x y ; f y = y 33. Find all second order partial derivatives of the function f (x, y) = xye y. A) f x B) f x C) f x D) f x = ye y ; f y = xye y ( y 6 ) ; = ; f y = xye y ( y 3 ) ; = ye y ; f y = xye y ( y ) ; = ; f y = xye y ( y 3 ) ; 3. Find a chain rule formula for w z = k(r, s, t). f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y if w = f(x, y, z), x = g(r, s), y = h(t), A) w C) w = w dx dt + w z = w dy y dt + w z z B) w D) w = w z z = dy dt + z 35. Compute the gradient of f(x, y, z) = ln ( x 5y + 7z ) at ( 5, 5, 5). A) 5 i + 3 j B) 5 i + 3 j 5 C) i + 3 j 5 k D) i + 3 j k k k 36. Find the derivative of f (x, y, z) = 3xy 3 z at the point ( 3, 7, 9) in the direction of v = i + j k. A) 53, B) 885, 735 C) 78, 588 D) 35, Find the direction in which the function f (x, y) = xe y ln x is decreasing most rapidly at the point (, ). A) 7 i + 7 j B) 7 i 7 j C) 7 i 7 j D) 7 i 38. Find the derivative of the function f (x, y) = arctan ( y x) at the point ( 8, 8) in the direction in which the function increases most rapidly. A) 3 B) 6 C) D) 3 6 5

6 39. A simple electrical circuit consists of a resistor connected between the terminals of a battery. The voltage V (in volts) is dropping at the rate of. volts per second as the battery wears out. At the same time the resistance R (in ohms) is increasing at the rate of ohms per second as the resistor heats up. The power P (in watts) disippated by the circuit is given by P = V R. How much is the power changing when R = 5 and V =? A).3 watts B).6 watts C).3 watts D).6 watts. Find an equation for the tangent plane to the surface ln z = 8x + 3y at the point (,, ). A) 6x + 6y z = 9 B) 6x + 6y + z = C) 6x 6y + z = 3 D) 6x 6y z =. Find all extreme values of the function f(x, y) = 5x y + 7xy and identify each as a local maximum, local minimum, or saddle point. A) f ( 7, ) 5 = 7 5, local minimum B) f (35, 35) = 5, 5, local max C) f ( 5, ) 7 = 35, local minimum D) f (, ) =, saddle point. Find all extreme values of the function f(x, y) = x 3 + y 3 3x 8y 8 and identify each as a local maximum, local minimum, or saddle point. A) f (, 6) =, local minimum; f (, 6) = 576, saddle point; f (, 6) = 56, saddle point; f (, 6) =, local maximum B) f (, 6) = 576, saddle point; f (, 6) = 56, saddle point C) f (, 6) =, local maximum D) f (, 6) =, local minimum; f (, 6) =, local maximum 3. Find the absolute maximum and minimum of the function f (x, y) = 7x + y on the trapezoidal region with vertices (, ) (, ), (, ), (, ). A) Absolute maximum: at (, ); absolute minimum: 7 at (, ) B) Absolute maximum: 8 at (, ); absolute minimum: at (, ) C) Absolute maximum: at (, ); absolute minimum: at (, ) D) Absolute maximum: at (, ); absolute minimum: 7 at (, ). Use Lagrange multipliers to find the maximum and minimum values of f (x, y, z) = x + y z subject to the constraint x + y + z = 9. A) Maximum: 8 at (,, ); minimum -8 at (,, ) B) Maximum: 9 at (,, ); minimum -9 at (,, ) C) Maximum: at (,, ); minimum - at (,, ) D) Maximum: at (,, 3); minimum - at (,, 3) 6

7 5. Find the point on the plane x + y z = that is nearest the origin. A) (,, ) B) (, 8, ) C) (,, ) D) (,, ) 6. Evaluate π 5π (sin x + cos y) dxdy A) π B) 9π C) 5π D) 8π 7. Write an integral equivalent to ln 8 ln integration reversed. A) 8 ln x ln 8 ln 7y dy dx B) 8 C) 8 ln x ln 7y dy dx D) 8 ln 8 e y 7y dx dy but with the order of ln x ln 8 7y dy dx ln x 7y dy dx ln 8 8. Write an integral equivalent to 6 integration reversed. 36 8y dx dy but with the order of y A) 36 x 8y dy dx B) 36 x 8y dy dx 6 C) 6 x 8y dy dx D) 6 x 6 8y dy dx 9. Express the area of the region bounded by x = y and the line y = x as a double integral. A) y+8 dx dy B) y C) 5. Evaluate ln y+8 dx dy D) e y dx dy. e y y+8 y dx dy y+8 dx dy A) 8 B) C) 8 D) 5. Integrate f (x, y) = ln x over the region bounded by the x axis, the line x = 3, and the curve y = ln x. A) B) C) 3 D) 5. Find the volume of the region bounded by the coordinate planes, the parabolic cylinder z = x, and the plane y = 5. 7

8 A) 8 3 B) 3 C) 8 D) Evaluate ln 8 ln 8 dx dy by reversing the order of integration. y/ ex A) 36 B) 35 C) D) 5 5. Change 6 dy dx to polar coordinates and evaluate. 36 x + x +y A) π(6+ln 7) π(6 ln 7) B) 55. Find the area of the region enclosed by r = 9 sin θ. π(6 ln 7) π(6+ln 7) C) D) A) 7 π B) 8 π C) 8 8 π D) 8 π 56. Write a triple integral in the order dz dy dx for the volume of the solid enclosed by the parabaloids z = x y and z = x + y. A) x x y dz dy dx B) x x x +y x y dz dy dx x x +y C) x x x y x +y dz dy dx D) 57. Write cylindrical coordinates. A) π C) π x x y x x +y dz dy dx as an equivalent integral in r r dz dr dθ B) π r 5 r r dz dr dθ D) π/ r x x y dz dy dx x x +y r r dz dr dθ r r r r dz dr dθ 58. Set up a triple integral for the volume of the solid in the first octant inside the cone ϕ = π 3 and between the spheres ρ = and ρ = 7. A) π/ π/3 C) π π/ 7 ρ sin ϕ dρ dϕ dθ B) π/ π/ π/3 7 ρ sin ϕ dρ dϕ dθ D) π/ π/3 7 π/ π/3 ρ sin ϕ dρ dϕ dθ 7 ρ sin ϕ dρ dϕ dθ 59. Use cylindrical coordinates to find the volume of the solid bounded below by the xy-plane, laterally by the cylinder r = sin θ, and above by the plane z = 7 x. A) 9π B) 9 π C) 7 π D) 7π 8

9 6. Find the mass of the solid in the first octant between the spheres x + y + z = 6 and x + y + z = if the density at any point is inversely proportional to the distance from the origin. A) 68kπ B) 8kπ C) k D) kπ Answer Key. C. B 3. D. A 5. B 6. D 7. C 8. A 9. B. A. A. B 3. A. C 5. D 6. C 7. D 8. B 9. C. D. A. B 3. C. D 5. D 6. D 7. C 8. C 9. B 3. B 3. A 3. A 33. D 3. C 35. B 36. A 37. C 9

10 38. B 39. A. D. D. A 3. C. B 5. B 6. D 7. C 8. A 9. B 5. C 5. D 5. A 53. B 5. A 55. D 56. C 57. A 58. B 59. D 6. C

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0, Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

More information

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

1 3 4 = 8i + 20j 13k. x + w. y + w

1 3 4 = 8i + 20j 13k. x + w. y + w ) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations

More information

Section 12.6: Directional Derivatives and the Gradient Vector

Section 12.6: Directional Derivatives and the Gradient Vector Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

More information

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5 1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

More information

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series 1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

More information

Figure 1: Volume between z = f(x, y) and the region R.

Figure 1: Volume between z = f(x, y) and the region R. 3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

More information

MATH FINAL EXAMINATION - 3/22/2012

MATH FINAL EXAMINATION - 3/22/2012 MATH 22 - FINAL EXAMINATION - /22/22 Name: Section number: About this exam: Partial credit will be given on the free response questions. To get full credit you must show all of your work. This is a closed

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent. CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n

More information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

More information

Name: ID: Discussion Section:

Name: ID: Discussion Section: Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

More information

MATH 2300 review problems for Exam 3 ANSWERS

MATH 2300 review problems for Exam 3 ANSWERS MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

More information

Solutions to Vector Calculus Practice Problems

Solutions to Vector Calculus Practice Problems olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

More information

Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

More information

x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c . Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x + 2 + 2 ln x + c (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( +

More information

Solutions to Final Practice Problems

Solutions to Final Practice Problems s to Final Practice Problems Math March 5, Change the Cartesian integral into an equivalent polar integral and evaluate: I 5 x 5 5 x ( x + y ) dydx The domain of integration for this integral is D {(x,

More information

Multivariable Calculus Practice Midterm 2 Solutions Prof. Fedorchuk

Multivariable Calculus Practice Midterm 2 Solutions Prof. Fedorchuk Multivariable Calculus Practice Midterm Solutions Prof. Fedorchuk. ( points) Let f(x, y, z) xz + e y x. a. (4 pts) Compute the gradient f. b. ( pts) Find the directional derivative D,, f(,, ). c. ( pts)

More information

Surface Normals and Tangent Planes

Surface Normals and Tangent Planes Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to

More information

Math 21a Review Session for Exam 2 Solutions to Selected Problems

Math 21a Review Session for Exam 2 Solutions to Selected Problems Math 1a Review Session for Exam Solutions to Selected Problems John Hall April 5, 9 Note: Problems which do not have solutions were done in the review session. 1. Suppose that the temperature distribution

More information

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3) MATH 255 FINAL NAME: Instructions: You must include all the steps in your derivations/answers. Reduce answers as much as possible, but use exact arithmetic. Write neatly, please, and show all steps. Scientists

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

Assignment 5 Math 101 Spring 2009

Assignment 5 Math 101 Spring 2009 Assignment 5 Math 11 Spring 9 1. Find an equation of the tangent line(s) to the given curve at the given point. (a) x 6 sin t, y t + t, (, ). (b) x cos t + cos t, y sin t + sin t, ( 1, 1). Solution. (a)

More information

Section 3.1 Calculus of Vector-Functions

Section 3.1 Calculus of Vector-Functions Section 3.1 Calculus of Vector-Functions De nition. A vector-valued function is a rule that assigns a vector to each member in a subset of R 1. In other words, a vector-valued function is an ordered triple

More information

FINAL EXAM SOLUTIONS Math 21a, Spring 03

FINAL EXAM SOLUTIONS Math 21a, Spring 03 INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

Extra Problems for Midterm 2

Extra Problems for Midterm 2 Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

More information

Math 263 Assignment 7 SOLUTIONS

Math 263 Assignment 7 SOLUTIONS Problems to turn in: Math 6 Assignment 7 SOLUTIONS In each case sketch the region and then compute the volume of the solid region. a The ice-cream cone region which is bounded above by the hemisphere z

More information

MULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then

MULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.

More information

Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems

Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Objectives In this lecture you will learn the following Define different coordinate systems like spherical polar and cylindrical coordinates

More information

AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss

AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =

More information

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those 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) + + 9 3 We would like to make

More information

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

More information

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

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom Free Response Questions 1969-005 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

More information

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

More information

6. In cylindrical coordinate system, the differential normal area along a z is calculated as: a) ds = ρ d dz b) ds = dρ dz c) ds = ρ d dρ d) ds = dρ d

6. In cylindrical coordinate system, the differential normal area along a z is calculated as: a) ds = ρ d dz b) ds = dρ dz c) ds = ρ d dρ d) ds = dρ d Electrical Engineering Department Electromagnetics I (802323) G1 Dr. Mouaaz Nahas First Term (1436-1437), Second Exam Tuesday 07/02/1437 H االسم: الرقم الجامعي: Start from here Part A CLO 1: Students will

More information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

SAMPLE TEST 1a: SOLUTIONS

SAMPLE TEST 1a: SOLUTIONS LAST (family) NAME: Test # 1 FIRST (given) NAME: Math 2Q04 ID # : TUTORIAL #: Instructions: You must use permanent ink. Tests submitted in pencil will not be considered later for remarking. This exam consists

More information

VECTOR CALCULUS Stokes Theorem. In this section, we will learn about: The Stokes Theorem and using it to evaluate integrals.

VECTOR CALCULUS Stokes Theorem. In this section, we will learn about: The Stokes Theorem and using it to evaluate integrals. VECTOR CALCULU 16.8 tokes Theorem In this section, we will learn about: The tokes Theorem and using it to evaluate integrals. TOKE V. GREEN THEOREM tokes Theorem can be regarded as a higher-dimensional

More information

AP Calculus BC 2004 Free-Response Questions

AP Calculus BC 2004 Free-Response Questions AP Calculus BC 004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be

More information

Applications of Integration to Geometry

Applications of Integration to Geometry Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular cross-sections by revolving regions in the plane along a line, giving a solid of revolution. Note that

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9. .(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

More information

AP Calculus BC 2004 Scoring Guidelines

AP Calculus BC 2004 Scoring Guidelines AP Calculus BC Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from

More information

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 2 6 2004 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

More information

Solutions to Practice Problems for Test 4

Solutions to Practice Problems for Test 4 olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

More information

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000) Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

More information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

More information

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0 . Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.

More information

Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

More information

The Gradient and Level Sets

The Gradient and Level Sets The Gradient and Level Sets. Let f(x, y) = x + y. (a) Find the gradient f. Solution. f(x, y) = x, y. (b) Pick your favorite positive number k, and let C be the curve f(x, y) = k. Draw the curve on the

More information

AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

More information

Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates

Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in

More information

Chapter 1 Vectors, lines, and planes

Chapter 1 Vectors, lines, and planes Simplify the following vector expressions: 1. a (a + b). (a + b) (a b) 3. (a b) (a + b) Chapter 1 Vectors, lines, planes 1. Recall that cross product distributes over addition, so a (a + b) = a a + a b.

More information

4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin

4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin 4.2. LINE INTEGRALS 1 4.2 Line Integrals MATH 294 FALL 1982 FINAL # 7 294FA82FQ7.tex 4.2.1 Consider the curve given parametrically by x = cos t t ; y = sin 2 2 ; z = t a) Determine the work done by the

More information

Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

More information

AP Calculus AB 2003 Scoring Guidelines Form B

AP Calculus AB 2003 Scoring Guidelines Form B AP Calculus AB Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the

More information

MA FINAL EXAM Form A (Test/Quiz Number 01) May 8, You must use a #2 pencil on the mark sense sheet (answer sheet).

MA FINAL EXAM Form A (Test/Quiz Number 01) May 8, You must use a #2 pencil on the mark sense sheet (answer sheet). MA 600 FINAL EXAM Form A (Test/Quiz Number 0) May 8, 05 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME. You must use a # pencil on the mark sense sheet (answer sheet).. If the cover of your question

More information

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

More information

Student Performance Q&A:

Student Performance Q&A: Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

More information

The Mathematics Diagnostic Test

The Mathematics Diagnostic Test The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether

More information

Chapter 17. Review. 1. Vector Fields (Section 17.1)

Chapter 17. Review. 1. Vector Fields (Section 17.1) hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your

More information

x = y + 2, and the line

x = y + 2, and the line WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first

More information

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

More information

II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3.

II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3. Math 34 April I. It is estimated that t days from now a farmer s crop will be increasing at the rate of.5t +.4t + bushels per day. By how much will the value of the crop increase during the next 5 days

More information

RARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III

RARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III RARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III I. Basic Course Information A. Course Number and Title: MATH 251 Calculus III B. New or Modified Course: Modified Course C.

More information

The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better! The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

More information

Section 2.1 Rectangular Coordinate Systems

Section 2.1 Rectangular Coordinate Systems P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

More information

2008 AP Calculus AB Multiple Choice Exam

2008 AP Calculus AB Multiple Choice Exam 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

Figure 1: u2 (x,y) D u 1 (x,y) yρ(x, y, z)dv. xρ(x, y, z)dv. M yz = E

Figure 1: u2 (x,y) D u 1 (x,y) yρ(x, y, z)dv. xρ(x, y, z)dv. M yz = E Contiune on.7 Triple Integrals Figure 1: [ ] u2 (x,y) f(x, y, z)dv = f(x, y, z)dz da u 1 (x,y) Applications of Triple Integrals Let be a solid region with a density function ρ(x, y, z). Volume: V () =

More information

Notes on Elastic and Inelastic Collisions

Notes on Elastic and Inelastic Collisions Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

More information

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule. Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.

More information

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z). Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field

More information

Math 1B, lecture 5: area and volume

Math 1B, lecture 5: area and volume Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in

More information

Chapter 1: Essentials of Geometry

Chapter 1: Essentials of Geometry Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

More information

Practice Problems for Midterm 2

Practice Problems for Midterm 2 Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,

More information

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

Catholic Schools Trial Examination 2004 Mathematics

Catholic Schools Trial Examination 2004 Mathematics 0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and

More information

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation: Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region

More information

Derive 5: The Easiest... Just Got Better!

Derive 5: The Easiest... Just Got Better! Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering

More information

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50 Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

More information

ELECTROSTATICS. Ans: It is a fundamental property of matter which is responsible for all electrical effects

ELECTROSTATICS. Ans: It is a fundamental property of matter which is responsible for all electrical effects ELECTROSTATICS One Marks Questions with Answers: 1.What is an electric charge? Ans: It is a fundamental property of matter which is responsible for all electrical effects 2. Write the SI unit of charge?

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

Math 32B Practice Midterm 1

Math 32B Practice Midterm 1 Math B Practice Midterm. Let be the region bounded b the curves { ( e)x + and e x. Note that these curves intersect at the points (, ) and (, e) and that e

More information

1. AREA BETWEEN the CURVES

1. AREA BETWEEN the CURVES 1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da

More information

Sphere centered at the origin.

Sphere centered at the origin. A Quadratic surfaces In this appendix we will study several families of so-called quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz

More information

Review for Final - Geometry B

Review for Final - Geometry B Review for Final - Geometry B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A model is made of a car. The car is 4 meters long and the model is 7 centimeters

More information

Solutions - Homework sections 17.7-17.9

Solutions - Homework sections 17.7-17.9 olutions - Homework sections 7.7-7.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points - and therefore the triangle between them - are on the plane x +

More information

Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

More information

Definition: Let S be a solid with cross-sectional area A(x) perpendicular to the x-axis at each point x [a, b]. The volume of S is V = A(x)dx.

Definition: Let S be a solid with cross-sectional area A(x) perpendicular to the x-axis at each point x [a, b]. The volume of S is V = A(x)dx. Section 7.: Volume Let S be a solid and suppose that the area of the cross-section of S in the plane P x perpendicular to the x-axis passing through x is A(x) for a x b. Consider slicing the solid into

More information