Calculus 1: Sample Questions, Final Exam, Solutions

Calculus 1 Final Exam Review Part 1 | Behind the Scenes with Professor V | How I Write Exams
Calculus 1 Final Exam Review Part 1 | Behind the Scenes with Professor V | How I Write Exams

Calculus 1: Sample Questions, Final Exam, Solutions

  • Elmer Hunter
  • 7 years ago
  • Views:

Transcription

1 Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer. e 3 e Solution: x dx=ln x e3 e = ln e3 ln e = ln(e 3 ) ln(e )=3 =. Evaluate π π the form of an integer. Solution: cosθdθ. Your answer should be in π π sin( π )= ( )=. cosθdθ = sinθ π π = sin( π ) Compute e ln3+ln. Your answer should be in the form of an integer. Solution: e ln3+ln = e ln3 e ln = 3 =6. 5 x 3 Compute dx. Your answer should be in 5 e x + 9 the form of an integer. 5 x 3 Solution: dx= since it is a definite 5 e x + 9 integral over an interval of zero length. (This integral is probably impossible to do otherwise.) Evaluate 3x sin(x 3 + )dx. Solution: Calculate for a substitution u=x 3 +, du=3x dx, and so 3x sin(x 3 +)dx= sinudu= cosu+c= cos(x 3 +)+C. + C (f) Evaluate the derivative D x e x. Solution: e x.

2 (g) Compute the derivative D x ln(x + ). Solution: Use the Chain Rule. D x ln(x + )= x x= x + x +. (h) y 5 Evaluate dy. Your answer should be in y + the form of an integer. Solution:. This is because it is an integral of an odd function over a symmetric interval[, ]. To see the integrand is an odd function, compute ( y) 5 ( y) + = y5 y5 = y + y +. You can also do this the long way. Use long division of polynomials to show y5 y + = y3 y+ y y + and integrate y 5 y + dy = = (y 3 y+ (y 3 y)dy+ y y + )dy y y + dy = ( 4 y4 y ) + ln(y + ) = ( 4 ) ( 4 )+ ln ln=. (This computation uses the fact that y y + dy = ln(y + ), which follows by using the substitution u=y +.) (i) Evaluate 3 z dz. Your answer should be in the form of an integer. Solution: Make the substitution u=z, du= dz, u()=, u()=9 to compute 3 z dz= 9 3 udu=3( 3 u ) 3 9 = (9 3 3 )=(7 )=5.

3 . Let f be a continuous function on the interval[,] which satisfies f(x)dx=5. Given this information, compute the integral Show your work and justify your answer. f(y)dy. Solution: Use the substitution x=y, dx=dy, dy= dx, x()= ()=, x()=()= to find f(y)dy= f(x) dx= f(x)dx= (5)= A population of bacteria undergoes exponential growth. If at noon, there are bacteria, and there are by pm, when does the number of bacteria reach 8? Show your work and simplify your answer. Solution: The doubling time is hours, and so to reach 8 bacteria from, it must double twice (since 8/= 4= ). This means that =4 more hours past pm are required, and so the population will be 8 at 6pm. Alternately, the general formula for exponential growth is y=y e kt, where t is the time in hours past noon and y = is the initial size. We need to solve for the constant k: At t=, we know the population is y=, so we find =y= y()=y e kt = e k and so e k =, k=ln, k= ln. So plug this back in to find the general formula y= y e kt = e t( ln) = e (ln) t = (e ln ) t = t. Therefore, y= 8 when 8= t, and 3 = 8= t, 3= t and so t=6pm. 4. Find the particular solution to the differential equation dy dx = xy+ x which satisfies y= 3 when x=. Show your work.

4 Solution: This equation is separable: dy = xy+ x=x(y+ ), dx dy y+ = xdx, dy y+ = xdx, ln y+ = x + C (substitute u=y+ ) Now the point we are interested in is y= 3, which means that y+ > for our solution. So we can drop the absolute value and find ln(y+ ) = x + C, y+ = e x +C, y = e x +C. For the particular solution, we should plug in y= 3 and x=to compute C. 3 = e ( )+C, 4 = e C, C = ln4, y = e x +ln4 = 4 e x. 5. Consider the following functions. Circle the one(s) which are concave up on an open interval containing x=. No explanation necessary. lnx x cosx x tanx

5 Solution: Only x is concave up near x=: lnx is not even defined at x=. If f(x)=x, then f (x)=> and so f is concave up always. If g(x)=cosx, g (x)= cosx and g ()= cos= <, so g is concave down near x=. If h(x)= x =(x ), h (x)= x(x ) and h (x)= (x ) x( )(x ) 3 (x) and h ()= <. So h is concave down near x=. Finally k(x)=tanx has an inflection point at x=. Compute k (x)= sec x, k (x)=secx(secxtanx)=sec xtanx, and k ()=sec tan=() ()=. We can also see that k (x)> for <x< π (concave up there) and k (x)< for π < x< (concave down there), and so x= is an inflection point. 6. Consider the function g(x)= x+cosx for <x<π. (a) Find all the critical points of g(x) for <x<π. Show your work. Hint: there are two of them. Solution: g (x)= sinx=when sinx=. This is possible only in the first and second quadrants (this is where the sine function is positive); these correspond to <x< π and π < x<π). To find the specific value, note that sin π 6 = for π 6 in the first quadrant. The corresponding solution in the second quadrant is π π 6 = 5π 6. So the only critical points are x= π 6 and x= 5π 6. (b) Classify each of the critical points you found in part (a) as a local maximum or a local minimum (or neither). Justify your answers. Solution: Apply the second derivative test: g (x) = cosx. g ( π 6 )= cos π 6 = 3 <. (You can tell the sign of the solution just by knowing that π 6 is in the first quadrant, and thus its cosine must be positive.) This means x= π 6 is a local maximum. Similarly g ( 5π 6 )= cos(5π 6 )= ( 3 )= 3 > and so x= 5π 6 is a local minimum. 7. Compute the derivative d dx [ x + x 3 (x ) ]. Show your work.

6 Solution: Use logarithmic differentiation. y = x + x 3 (x ), x lny = ln[ + + ) 3lnx ln(x ), x 3 (x ) ]=ln(x d dx (lny) = d dx [ln(x + ) 3lnx ln(x )], dy y dx = x x + 3 x x, dy x = y[ dx x + 3 x x ] x = + x x 3 (x ) [ x + 3 x x ] = 3(x + ) (x + ) x (x ) x 4 (x ) x 3 (x ) 3 = x (x ) 3(x + )(x ) (x + )x x 4 (x ) 3 = 3×3 + x 5x+3 x 4 (x ) 3 8. Consider the function h(x)= ex + e x for <x<. (a) Find the interval(s) on which h(x) is increasing. Show your work. Solution: Compute h (x)= [ex + e x ( )]= (ex e x ). So h (x)= if (ex e x ) =, e x = e x, e x =, ln(e x ) = ln=, x =, x =. So this splits the real line up into two intervals(,) and(, ). Check for each interval h ( )= (e e)< (use your calculator

7 or the fact that e>). So h (x)<on the interval(,). Similarly, plug in h ()= (e e )> and so h (x)> on the interval(, ). Thus h is increasing on the interval(, ). (b) Show that h(x) is always concave up. Solution: Compute h (x)= (ex + e x )=h(x), which is clearly always positive. Therefore h is always concave up. 9. Recall that x represents the greatest integer function ( x is the greatest integer x). Compute the graph.) 4 x dx. Show your work. (Hint: Draw Solution: See the picture below. The integral is equal to the area under the graph, which is +++3=6.. (a) If f(x)=sinx, what is f (x)? Solution: cos x. (b) For f(x)=sinx, write down the formula for f () using the definition of the derivative.

8 Solution: f () = lim h f(+h) f() h = lim h sin(+h) sin h = lim h sinh h. sinh (c) Use parts (a) and (b) to compute the limit lim. Your answer h h should be in the form of an integer. Justify your answer. sinh Solution: lim h h = f ()=cos=.. Consider the function p(r)=r 3 + 6r + 9r 4. (a) Find all the critical points of p(r) for r in the interval[,]. Show your work. Solution: The critical points in this interval are r=,,. First of all, the endpoints r=, are critical points. Compute p (r)=3r +r+9= if 3(r+3)(r+)= if r=, 3. Only r= is in our interval, so this is the only stationary critical point. (b) For which r does p(r) attain its minimum and maximum on the interval[,]? Show your work. Solution: Compute p( ) = ( ) 3 + 6( ) + 9( ) 4= = 6, p( ) = ( ) 3 + 6( ) + 9( ) 4= = 8, p() = 3 + 6( )+9() 4= =46. So r= is the minimum and r= is the maximum. u. Compute the limit lim 9. Show your work. u 3 u 4u+3 Solution: First of all, if we plug in u=3, we get, and so we must do some work: u lim 9 u 3 u 4u+3 = lim (u 3)(u+ 3) u 3(u 3)(u ) = lim u+3 u 3 u = = 3.

9 3. (a) Sketch the graph of a function y=f(x) which satisfies all the following properties: F(x) has domain(, ). F(x) has a vertical asymptote at x=. F(x) is increasing on the interval(, ). F(x) is concave down on the interval(, ). Solution: (b) Give a formula for a function F(x) which satisfies all the properties listed in part (a). Justify your answer. Solution: F(x)=lnx works: It has the correct domain. lim x + lnx= and so there is a vertical asymptote at x=. F (x)= x > for x> and so F is increasing on(, ). Moreover, F (x)= x < for x> and so F is concave down on(, ). There are other possible answers. F(x)= x also works.

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)


Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.


PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

Table of Contents

Homework # 3 Solutions


Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6×2 + x 3 ) = 2 + 6x 2 + x 3 x 8

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:


Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

Inverse Functions and Logarithms


Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that

Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations


Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a

14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style


Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of

Math 120 Final Exam Practice Problems, Form: A


Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

x 2 y 2 +3xy ] = d dx dx [10y] dy dx = 2xy2 +3y


MA7 – Calculus I for thelife Sciences Final Exam Solutions Spring -May-. Consider the function defined implicitly near (,) byx y +xy =y. (a) [7 points] Use implicit differentiation to find the derivative

Microeconomic Theory: Basic Math Concepts


Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

2 Integrating Both Sides


2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

The Derivative. Philippe B. Laval Kennesaw State University


The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition

To differentiate logarithmic functions with bases other than e, use


To ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions with

5.1 Derivatives and Graphs


5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

Differentiation and Integration


This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

Techniques of Integration


CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration

Chapter 7 Outline Math 236 Spring 2001


Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

MATH 381 HOMEWORK 2 SOLUTIONS


MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e

WORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS.


WORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises


CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C


Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

AP CALCULUS AB 2007 SCORING GUIDELINES (Form B)


AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and

100. In general, we can define this as if b x = a then x = log b


Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?


36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

2.2 Separable Equations


2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123


Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

Lecture 3: Derivatives and extremes of functions


Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

Section 2.7 One-to-One Functions and Their Inverses


Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

1 if 1 x 0 1 if 0 x 1


Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

Calculus with Parametric Curves


Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function

Math 432 HW 2.5 Solutions


Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

Algebra 2: Themes for the Big Final Exam


Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity


MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

Euler s Formula Math 220


Euler s Formula Math 0 last change: Sept 3, 05 Complex numbers A complex number is an expression of the form x+iy where x and y are real numbers and i is the imaginary square root of. For example, + 3i

Taylor and Maclaurin Series


Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

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

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA


FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.


MA123 Elem. Calculus Fall 2015 Exam 2 2015-10-22 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during

6 Further differentiation and integration techniques


56 6 Further differentiation and integration techniques Here are three more rules for differentiation and two more integration techniques. 6.1 The product rule for differentiation Textbook: Section 2.7

Practice with Proofs


Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

Constrained optimization.


ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

Linear and quadratic Taylor polynomials for functions of several variables.


ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

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

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

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

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx


Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

Math Placement Test Practice Problems


Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

TOPIC 4: DERIVATIVES


TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

Chapter 11. Techniques of Integration


Chapter Techniques of Integration Chapter 6 introduced the integral. There it was defined numerically, as the limit of approximating Riemann sums. Evaluating integrals by applying this basic definition

Objectives. Materials


Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways

6.4 Logarithmic Equations and Inequalities


6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

4 More Applications of Definite Integrals: Volumes, arclength and other matters


4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the

Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015


Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax

3 Contour integrals and Cauchy s Theorem


3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals


ALGEBRA REVIEW LEARNING SKILLS CENTER The “Review Series in Algebra” is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

Notes and questions to aid A-level Mathematics revision


Notes and questions to aid A-level Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and

Using a table of derivatives


Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.

a cos x + b sin x = R cos(x α)


a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this

Student Performance Q&A:


Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief

*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) MONDAY, 21 MAY 1.00 PM 2.30 PM NATIONAL QUALIFICATIONS 2012


X00//0 NTIONL QULIFITIONS 0 MONY, MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (40 marks) Instructions for completion

5.3 Improper Integrals Involving Rational and Exponential Functions


Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1


Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.


MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

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

1 Lecture: Integration of rational functions by decomposition


Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1


MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on

Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)


Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis


Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010


MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0


College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1


Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

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

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I – ORDINARY DIFFERENTIAL EQUATIONS PART A


RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I – ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

Calculus AB 2014 Scoring Guidelines


P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official

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

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates


Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.


MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x – 6 A) 1, x 6 B) 8x – 1, x 6 x –

Techniques of Integration


8 Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it

TOPPER Sample Paper – I. Class : XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100


TOPPER Sample Paper – I Class : XI MATHEMATICS Questions Time Allowed : 3 Hrs Maximum Marks: 100 1. All questions are compulsory.. The question paper consist of 9 questions divided into three sections

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F


FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

Representation of functions as power series


Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:


Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS


SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise

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.

Functions: Piecewise, Even and Odd.


Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

Exercises and Problems in Calculus. John M. Erdman Portland State University. Version August 1, 2013


Exercises and Problems in Calculus John M. Erdman Portland State University Version August 1, 2013 c 2010 John M. Erdman E-mail address: erdman@pdx.edu Contents Preface ix Part 1. PRELIMINARY MATERIAL

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions


College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

AP Calculus AB 2011 Scoring Guidelines


AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the

6. Differentiating the exponential and logarithm functions


1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

4.5 Chebyshev Polynomials


230 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION 4.5 Chebyshev Polynomials We now turn our attention to polynomial interpolation for f (x) over [ 1, 1] based on the nodes 1 x 0 < x 1 < < x N 1. Both

Calculus. Contents. Paul Sutcliffe. Office: CM212a.


Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical

SUBSTITUTION I.. f(ax + b)


Integrtion SUBSTITUTION I.. f(x + b) Grhm S McDonld nd Silvi C Dll A Tutoril Module for prctising the integrtion of expressions of the form f(x + b) Tble of contents Begin Tutoril c 004 g.s.mcdonld@slford.c.uk

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,

10 Polar Coordinates, Parametric Equations


Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

The Heat Equation. Lectures INF2320 p. 1/88


The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

Objective: Use calculator to comprehend transformations.


math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given

GRE Prep: Precalculus


GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

GRAPHING IN POLAR COORDINATES SYMMETRY


GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry – y-axis,

You are watching: Calculus 1: Sample Questions, Final Exam, Solutions. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts