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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 . 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 ln x + c

2 (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( + x 2 ) 2 dx 3 2( + x 2 ) + c (f) 2x arctanxdx x 2 arctan x x + arctanx = π 2

3 2. Determine the area bounded by the curves f(x) = 4 x and g(x) = 3x +. (3x + 4 x ) dx = 3 2 x2 + x ln 4 4x = ln 4 3. Determine the area bounded by the x-axis, the y-axis, the line y = 3 and the curve y = x 2. 3 (y 2 + 2) dx = 3 3 y3 + 2y = 5

4 4. Calculate the volume obtained by rotating the region in the first quadrant bounded by f(x) = x 3 and g(x) = 2x x 2 about the x-axis. π(2x x 2 x 6 ) dx = ( 4 π(4x 2 4x 3 +x 4 x 6 ) dx = π = 7) 4π 5 5. Calculate the volume obtained by rotating the region in the first quadrant bounded by f(x) = x 3 and g(x) = 2x x 2 about the y-axis. 2π(2x 2 x 3 x 4 ) dx = 2π ( ) = 3π 3

5 6. A 6-lb boulder is suspended over a roof by a 4-ft cable that weighs lb/ft. How much work is required to raise the boulder with the cable over the roof, the distance of 4 ft? 6(4) + 4 xdx = 24 + ( 5x 2) 4 = = 4 ft-lb 7. A trough is filled with water and its vertical ends have the shape of a parabola with top length 8 ft and height 4 ft. Find the hydrostatic force on one end of the trough (4 y)(4 4 y) dy = 62.5 (6 y 4y 3/2 ) dy = 62.5 ( ) ( ) 2 = 62.5(256) lbs 5 5

6 8. Determine the arclength of the curve y = 2 x2 ln x on 2 x 4 4 ( ) 2 dy + = x 2 + dx 2 + ( 6x = x + ) 2 2 4x 4 2 ( x + ) dx = 4x 2 x ln x = ln 2 9. Evaluate the integral if it converges. Show divergence otherwise. (a) 3 2 (x ) 2 dx ( ) 3 2 = diverges x x (b) 2 + x 2 dx 2 arctanx x = π converges

7 . Solve the initial value differential equation explicitly for y(t): dy dt = 2(y )2 y() = 2. dy = 2 dt (y ) 2 y = 2t + c y = 2t + 2. Solve the initial value differential equation explicitly for y(x): dy dx = xy x y() =. dy = xdx y ln y = 2 x2 + c y = + 9e x2 /2

8 2. Solve the initial value first order linear differential equation: y = y + x y() = 2. I.Factor = e dx = e x e x y e x y = xe x (e x y) = xe x e x y = xe x e x + c y = x + 3e x 3. Solve the initial value first order linear differential equation: xy + y = 3x 2 y() = 2. y + x y = 3x I.Factor = e x dx = e ln x = x xy + y = 3x 2 (xy) = 3x 2 xy = x 3 + c y = x 2 + x

9 4. Use Euler s Method to approximate y() if dy dx = y + x with y() = and x = 2. x n y n dy/dx y n + (dy/dx)(δx) + (/2) /2 3/2 2 3/2 + 2(/2) 5/2 5. Use Simpsons Rule to approximate 3 dx with n = 4. x ( ( ( ( (2) 3) 2) 5) ) =

10 6. Solve the initial value second order homogeneous differential equation: y + 2y + y = y() = 2 y () = 4. r 2 + 2r + = r = r 2 =. y = Me x + Nxe x M = 2 y = 2e x + Ne x Nxe x N + 6 y = 2e x + 6xe x 7. Solve the initial value second order homogeneous differential equation: y + 4y = y() = y () = 3. r = r = ±2i y = M cos (2t) + N sin (2t) M = y = 2 sin (2t) + 2N cos (2t) N = 3 2 y = cos (2t) + 3 sin (2t) 2

11 8. Solve the initial value second order nonhomogeneous differential equation using the method of undetermined coefficients. y + 5y + 4y = sin x y() = y () =. y p y p y p = A sin x + B cosx = A cosx B sin x = A sin x B cos x 4A sin x + 4B cosx + 5A cosx 5B sin x A sin x B cosx = sin x 3A 5B = 5A + 3B = r 2 + 5r + 4 = A = 3 6 B = 5 6 y p = 3 6 sin x cosx r = 4, r 2 = therefore y c = Me 4x + Ne x y = Me 4x + Ne x 3 6 sin x cosx M + N = 5 6 4M N = 3 6 M = 2 48 N = 7 48 y = 24 e 4x 7 48 e x 3 6 sin x cos x

12 9. Tell whether the series converges or diverges and justify your answer by showing reason by a valid test. (a) n= ( ) n 3n + Converges; Alternating with b n = 3n+ decreasing and lim n b n =. (b) n= 2 3n 5 n Diverges; Geometric with r = 8 5 > (c) n= 4 n + Diverges; Comparison to 4 dx = lim 4 ln (x + ) = x + x (d) n= 3 n Converges; Comparison to dx = lim x arctanx = 3π x 2

13 2. Determine the given sum: (a) 5 2 n n= 3 n /3 2 3 = (b) ( ) n n= n! e = e (c) n= 2 n 2 + 4n + 3 n= ( n + ) = 3 n (d) n= 4 n! 4e

14 2. Determine the Taylor Series about x = for: (a) f(x) = + x ( ) n x n n= (b) g(x) = ( + x) 2 n= n( ) n x n (c) h(x) = + x 2 ( ) n x 2n n= (d) k(x) = arctan x n= ( ) n 2n + x2n+

15 22. Determine the fourth degree Taylor Polynomial about x = for the function f(x) = + x. + x = c + c x + c 2 x 2 + c 3 x 3 + c 4 x 4 2 ( + x) /2 = c + 2c 2 x + 3c 3 x 2 + 4c 4 x 3 4 ( + x) 3/2 = 2c c 3 x + 4 3x ( + x) 5/2 = 3 2c c 4 x 5 6 ( + x) 7/2 = 4 3 2c 4 + x + 2 x 8 x2 + 6 x x4 23. Determine the fourth degree Taylor Polynomial about x = for the function f(x) = ln x. ln x = c + c (x ) + c 2 (x ) 2 + c 3 (x ) 3 + c 4 (x ) 4 x = c + 2c 2 (x ) + 3c 3 (x ) 2 + 4c 4 (x ) 3 x 2 = 2c c 3 (x ) + 4 3(x ) 2 2 x 3 = 3 2c c 4 (x ) 6 x 4 = 4 3 2c 4 ln x (x ) 2 (x )2 + 3 (x )3 (x )4 4

16 24. Determine the interval and radius of convergence of the given series: n= (x ) n 3 n. 2 < x < 4 R = Determine the interval and radius of convergence of the given series: n= (x ) n. n x < 2 R =

17 26. Given points P(, 4, 6) and Q( 3, 6, 7) and R( 4, 7, 6), (a) determine the angle θ between PQ and PR. PQ = 2, 2, and PR = 3, 3, 2 cosθ = ( 9)( 62) = θ = 9 (b) Determine the equation of the plane which contains the points P, Q, and R. i j k = 27 i 27 j + 2 k 9x + 9y 4z = 3 (c) Determine the volume of the parallelopiped formed by the vectors: and OR where O is the origin (,, ). OP, OQ = ( 36 49) 4(8 + 28) + 6( ) =

18 27. Change coordinates: (a) from rectangular coordinates to cylindrical coordinates. i. P( 3, 3, 6) (3 2, 3π4 ), 6 ii. z = 4x 2 + 4y 2 z = 2r (b) from spherical coordinates to rectangular coordinates. i. P(4, π/3, π/4) ( 2, 6, 2 2) (c) from rectangular to spherical coordinates. i. x 2 + y 2 + z 2 = 9 ρ = 3

### Section 4.7. More on Area

Difference Equations to Differential Equations Section 4.7 More on Area In Section 4. we motivated the definition of the definite integral with the idea of finding the area of a region in the plane. However,

### If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

### 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

### 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

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### Volumes by Cylindrical Shell

Volumes by Cylindrical Shell Problem: Let f be continuous and nonnegative on [a, b], and let R be the region that is bounded above by y = f(x), below by the x - axis, and on the sides by the lines x =

### 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,

### 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

### MATH 52 FINAL EXAM SOLUTIONS (AUTUMN 2003)

MATH 5 FINAL EXAM OLUTION (AUTUMN 3). Evaluate the integral by reversing the order of integration olution. 3 3y e x dxdy = = 3 3y 3 x/3 3 = ex 3 [ e x y e x dxdy. e x dxdy ] y=x/3 y= = e9 dx =. Rewrite

### 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

### Contents. 12 Applications of the Definite Integral Area Solids of Revolution Surface Area...

Contents 12 Applications of the Definite Integral 179 12.1 Area.................................................. 179 12.2 Solids of Revolution......................................... 181 12.3 Surface

### 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

### 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

### 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

### VI. Transcendental Functions. x = ln y. In general, two functions f, g are said to be inverse to each other when the

VI Transcendental Functions 6 Inverse Functions The functions e x and ln x are inverses to each other in the sense that the two statements y = e x, x = ln y are equivalent statements In general, two functions

### 3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13

Contents 1 Limits 2 2 Derivatives 3 2.1 Difference Quotients......................................... 3 2.2 Average Rate of Change...................................... 4 2.3 Derivative Rules...........................................

### 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

### 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,

### 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

### Review Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.

Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u

### Practice Final Math 122 Spring 12 Instructor: Jeff Lang

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

### 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

### AP Calculus BC 2006 Free-Response Questions

AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

### 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.

### Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

### 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

### 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

### 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,

### 1. A convergent, B divergent. 2. both series divergent A divergent, B convergent

Version Homework grandi () This print-out should have 8 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. CalCcExam. points Let h be a continuous,

### 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)

### 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

### 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

### MAC Calculus II Spring Homework #2 SOLUTIONS

MAC 232-593-Calculus II Spring 23 Homework #2 SOLUTIONS Note. Beginning already with this homework, sloppiness costs points. Missing dx (or du, depending on the variable of integration), missing parentheses,

### Calculating Areas Section 6.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined

### 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

### 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

### Time: 1 hour 10 minutes

[C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Non-calculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may

### Math 201 Lecture 23: Power Series Method for Equations with Polynomial

Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in

### Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0

Math -0 Calc Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series We are working with

### Notes from FE Review for mathematics, University of Kentucky

Notes from FE Review for mathematics, University of Kentucky Dr. Alan Demlow March 19, 2012 Introduction These notes are based on reviews for the mathematics portion of the Fundamentals of Engineering

### 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

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

### Example 1. Example 1 Plot the points whose polar coordinates are given by

Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

### Math 121 Final Review

Math Final Review Pre-Calculus. Let f(x) = (x + ) and g(x) = x, find: a. (f + g)(x) b. (f g)(x) c. (f/g)(x) d. (f g)(x) e. (g f)(x). Given f(x) = x + x and g(x) = x 5x 5, find the domain of a. f + g b.

### Review for Exam 2 (The following problems are all from old exams)

Review for Exam 2 (The following problems are all from old exams). Write down, but do not evaluate, an integral which represents the volume when y = e x, x, is rotated about the y axis. Soln: Divide the

### Math 220 October 11 I. Exponential growth and decay A. The half-life of fakeium-20 is 40 years. Suppose we have a 100-mg sample.

Math 220 October 11 I. Exponential growth and decay A. The half-life of fakeium-20 is 40 years. Suppose we have a 100-mg sample. 1. Find the mass that remains after t years. 2. How much of the sample remains

### Test Two Review Cal 2

Name: Class: Date: ID: A Test Two Review Cal 2 Short Answer 1. Set up the definite integral that gives the area of the region bounded by the graph of y 1 x 2 2x 1 and y 2 2x.. Find the area of the region

### RELEASED. Student Booklet. Precalculus. Fall 2015 NC Final Exam. Released Items

Released Items Public Schools of North Carolina State Board of Education epartment of Public Instruction Raleigh, North Carolina 27699-6314 Fall 2015 NC Final Exam Precalculus Student Booklet Copyright

### D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table.

. Find the domains of the following functions: (a) f(x) = ln(x ) We need x > 0, or x >. Thus D f = (, ) (x + )(x 3) (b) g(x) = x We need the thing inside the root to be greater than or equal to 0. So we

### SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

### Chapter 2 Test Review

Name Chapter 2 Test Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If the following is a polynomial function, then state its degree and leading

### MATHEMATICS FOR ENGINEERS & SCIENTISTS 7

MATHEMATICS FOR ENGINEERS & SCIENTISTS 7 We stress that f(x, y, z) is a scalar-valued function and f is a vector-valued function. All of the above works in any number of dimensions. For instance, consider

### 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

### 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

### Student name: Earlham College. Fall 2011 December 15, 2011

Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use

### f (x) = x 2 (x) = (1 + x 2 ) 2 (x) = (1 + x 2 ) 2 + 8x 2 (1 + x 2 ) 3 8x (1 + x 2 ) x (1 + x 2 ) 3 48x3 f(0) = tan 1 (0) = 0 f (0) = 2

Math 5 Exam # Practice Problem Solutions. Find the Maclaurin series for tan (x (feel free just to write out the first few terms. Answer: Let f(x = tan (x. Then the first few derivatives of f are: f (x

### 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

### 1 Mathematical Induction

Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem

### 1 Chapter Chapter Chapter Chapter 4 1

Contents 1 Chapter 1 1 2 Chapter 2 1 3 Chapter 3 1 4 Chapter 4 1 5 Applications of Integrals 2 5.1 Area and Definite Integrals............................ 2 5.1.1 Area of a Region Between Two Curves.................

### 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,

### 11.2. Using a table of derivatives. Introduction. Prerequisites. Learning Style. Learning Outcomes

Using a table of derivatives 11.2 Introduction In Block 1 you were introduced to the idea of a derivative and you calculated some derivatives from first principles. Rather than always calculate the derivative

### 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)

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### 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.

### Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.

CALCULUS WORKSHEET ON 7. Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.. f x x and g x x on x Find the area bounded by the given

### 15.5. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes. Learning Style

Lengths of curves and surfaces of revolution 15.5 Introduction Integration can be used to find the length of a curve and the area of the surface generated when a curve is rotated around an axis. In this

### Worksheet # 1: Review

Worksheet # 1: Review 1. (MA 113 Exam 1, Problem 1, Spring 27). Find the equation of the line that passes through (1, 2) and is parallel to the line 4x + 2y = 11. Put your answer in y = mx + b form. 2.

### 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

### 7.3 Volumes Calculus

7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

### Answers Spring 2014 Math 125 Final Exam

Answers Spring 04 Math 5 Final Exam. (a) 3 (x + )3/ 4(x + ) / + C (b) 6 sec3 (e x ) sec(ex ) + C. (a) 3 6 8 ln (b) π 3 3 + π 3. + ln( + 3) 4. 5. ( ) 3 /3 3 4 6., 00 ft-lb ( 7. (b) 3 + 0 6 + 4 + 6 5 ln

### The line segment can be parametrized as x = 1 + t, y = 1 + 2t, z = 1 + 3t 0 t 1. Then dx = dt, dy = 2dt, dz = 3dt so the integral becomes

MATH rd Midterm Review Solutions. Evaluate xz dx z dy + y dz where is the line segment from (,, ) to (,, ). The line segment can be parametrized as x = + t, y = + t, z = + t where t. Then dx = dt, dy =

### 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

### AP Calculus BC 2012 Free-Response Questions

AP Calculus BC 0 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

### Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find

### Test 3 Review. Jiwen He. Department of Mathematics, University of Houston. math.uh.edu/ jiwenhe/math1431

Test 3 Review Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431 November 25, 2008 1 / Test 3 Test 3: Dec. 4-6 in CASA Material - Through 6.3. November

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### IB Mathematics HL, Year 1 (Paper 1)

Puxi Campus High School Examinations Semester Two June 2010 IB Mathematics HL, Year 1 (Paper 1) Student Name: Thursday, June 3 rd, 2010 12:45-2:00 Time: 1 hour, 15 minutes Mr. Surowski Instructions to

### 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

### 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

### Numerical integration example The probability that someone s IQ falls between a and b is given by the area under the curve

Numerical integration example The probability that someone s IQ falls between a and b is given by the area under the curve D(x) = 5 p 2 e (x 00) 2 (5) 2 2 (this is the normal distribution with average

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### AP Calculus AB 2007 Free-Response Questions

AP Calculus AB 2007 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

### Calculus II MAT 146 Integration Applications: Area Between Curves

Calculus II MAT 46 Integration Applications: Area Between Curves A fundamental application of integration involves determining the area under a curve for some interval on the x- or y-axis. In a previous

### 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

### Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector) valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x) = φ(x 1,

### 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,

### 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)

### 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,

### M1120 Class 5. Dan Barbasch. September 4, Dan Barbasch () M1120 Class 5 September 4, / 16

M1120 Class 5 Dan Barbasch September 4, 2011 Dan Barbasch () M1120 Class 5 September 4, 2011 1 / 16 Course Website http://www.math.cornell.edu/ web1120/index.html Dan Barbasch () M1120 Class 5 September

### Pre-Calculus Review Lesson 1 Polynomials and Rational Functions

If a and b are real numbers and a < b, then Pre-Calculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b

### 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,

### 1. First-order Ordinary Differential Equations

Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential

### Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### National Quali cations 2015

H National Quali cations 05 X77/76/ WEDNESDAY, 0 MAY 9:00 AM 0:0 AM Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only

### 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,