Ph.D. Katarína Bellová Page 1 Mathematics 1 (10-PHY-BIPMA1) RETAKE EXAM, 4 April 2018, 10:00 12:00
|
|
- Bennett Powell
- 5 years ago
- Views:
Transcription
1 Ph.D. Katarína Bellová Page Mathematics (0-PHY-BIPMA) RETAKE EXAM, 4 April 08, 0:00 :00 Problem [4 points]: Prove that for any positive integer n, the following equality holds: (3n ) = n(3n ). Solution: We will prove the equality by mathematical induction. For n =, the left-hand side equals, while the right hand side equals (3 ) = and the equality is true. Assume now that the equality is true for some n = k N, we will prove it for n = k +. Using the induction hypothesis in the first line, we have (3k ) + (3(k + ) ) = k(3k ) + (3(k + ) ) = 3k k + (3k + ) = 3k + 5k + (k + )(3k + ) = (k + )(3(k + ) ) =. Since this is eactly the equality for n = k +, the second induction step is proved and so is the statement. Problem [4 points]: Is the sequence (a n ) n, where a n = cos(n π), convergent? If yes, compute its it. If no, compute sup n a n and inf n a n. Solution: For even n, n is even, and cos(n π) =. For odd n, n is also odd, and hence cos(n π) =. Thus, for any n N we have cos(n π) = ( ) n, which is not convergent, sup n a n = and inf n a n =. Problem 3 [4 points]: Prove that there eists at least one solution R of the equation tan + =. Solution: We want to use intermediate value theorem. For that, we need to make sure to work on an interval [a, b] where the function f() = tan + is continuous, for eample on some closed subinterval of ( π, π ). Checking some values of the function, we find out that for eample interval [0, π 3 ] works: f(0) = tan = 0; f( π 3 ) = tan π 3 + ( π 3 ) = 3 + ( π 3 ) > + = ; f() is continuous on [0, π 3 ]. Hence, as the value lies between the values f(0) and f( π 3 ), by intermediate value theorem there must eist some (0, π 3 ) such that f() =. Problem 4 [4 points]: Compute the it ( e ).
2 Ph.D. Katarína Bellová Page Mathematics (0-PHY-BIPMA) e Solution: First, = > 0, so the epression is well defined and the it is of type (which can be anything). Due to continuity of the eponential function, we will have ( e ) = e ln ( e ) = e ln ( e ), if the it in the eponent eists. Let us compute it: by using l Hospital rule twice (indicating the type each time; second use applies only to the second it in the epression) and algebraic properties of its, ln ( ) e 0 0 = = (e ) (e )() ( e ) e 0 0 = = e + e e = e =. e e + (e ) + e () (e ) + () The calculation is justified backwards: since the its and epressions (we do not divide by 0) on the right eist, also the its on the left eist and equal the right-hand sides. Getiing back to our it, we get ( e ( ) ) = e / = e. Problem 5 [5 points]: For which values of a R is the following function f continuous at = 0? For which a R is it differentiable at = 0? { sin(a) f() = for 0, for = 0. Solution: For continuity at 0, we must have f() = f(0). Here sin(a) f() = = { sin(a) a a = a = a if a 0, sin(0 ) = 0 = 0 = a if a = 0. In the first case we used substitution y = a: y 0 as 0 and y 0 if 0; then sin y y 0 y =. In any case, we got f() = a. Hence, the function will be continuous at 0 if and only if a = f(0), i.e. a =. If f is differentiable at 0, it must be continuous at 0. Hence, f can not be differentiable for
3 Ph.D. Katarína Bellová Page 3 Mathematics (0-PHY-BIPMA) any a. Let us check from definition whether f is differentiable at 0 for a = : f f() f(0) (0) = 0 = sin() = sin() 0 0 = cos() cos() = 0 0 = sin() = 0. Here we used l Hospital rule twice for its of type 0 0, the calculations are justified backwards as the right hand sides eist. We see that for a =, f (0) eists, and f is differentiable at 0 if and only if a =. Problem 6 [4 points]: Give the definition domain of the function f() = sin(ln ) and compute its first and second derivative. Solution: Logarithm is defined for > 0, sin y is defined for any y R. Hence, the definition domain of f is (0, + ). For (0, + ), by chain rule and then quotient and chain rule, f () = cos(ln ) cos(ln ) =, f () = (cos(ln )) cos(ln ) () = ( sin(ln )) cos(ln ) sin(ln ) cos(ln ) =. Problem 7 [4 points]: What is the maimal value of a > 0 such that the function f() = ln is invertible on (0, a]? Solution: For > 0, f () = = is positive if and only if < =, and negative if and only if > =. We see that f is continuous on (0, + ), while it is ( ] [ ( ] increasing on 0, and decreasing on )., + Thus, f is invertible on 0,, but not on any interval (0, a] for any a >, and the maimal a we are looking for is a =. Problem 8 [5 points]: Find all local maima of the function f() = cos + on R. Is any of them a global maimum? Solution: As the function is differentiable, in any local maimum we must have f (0) = 0, i.e. sin + = 0, sin = /, = { π 6 5π 6 + kπ, k Z, + kπ, k Z. Furthermore, as f is twice differentiable, in any local maimum we must have f () 0. We
4 Ph.D. Katarína Bellová Page 4 Mathematics (0-PHY-BIPMA) have f ( () = cos, f π ) 6 + kπ = 3 < 0, f ( 5π 6 + kπ ) = 3 > 0. We see that the only possible local maima occur at = π 6 +kπ, k Z; as f () = 0, f () < 0 is also a sufficient condition for a local maimum, these are indeed local maima. Since f() = + (as cos is bounded and = + ), function f does not have any global maimum. Problem 9 [4 points]: Compute the indefinite integral arctan d. Solution: Integrating by parts, arctan d = arctan + d = arctan + d = arctan ( ) d + = arctan ( arctan ) + C = (( + ) arctan ) + C. Problem 0 [4 points]: Compute the definite integral 3 π 0 sin( 3 ) d. Solution: Substituting y = 3, dy = 3 d, we get 3 π 0 sin( 3 ) d = π 0 3 sin y dy = 3 cos y π Problem [4 points]: Does the following series converge? ( ) tan k k=0 0 = 3 ( ) = 3. Solution: We compare the series with the series + k= (the convergence does not depend k 3/ on any finitely many first terms of the series, so it does not matter that our series starts with
5 Ph.D. Katarína Bellová Page 5 Mathematics (0-PHY-BIPMA) k = 0 and the second series with k = ): ( ) tan k + = k 3/ k + = k + ( ) tan tan ( tan y = y 0 + y k + = =. ) k 3/ k k 3/ k + k k 3 Since the it is finite and series + k= ( ) k=0 tan k k 3/ converges (as 3/ > ), so does the series Problem [3 points]: Write the comple number i 3 i in the algebraic form. Solution: i 3 i = i 3 i 3 + i 3 + i = 3 + i 3i + = 4 i = i. Problem 3 [5 points]: Prove that 3 vectors in space are linearly dependent if and only if they are coplanar. Solution: First assume that vectors v, v, v 3 R 3 are linearly dependent. Then there eist coefficients α, α, α 3, not all of them zero, such that α v + α v + α 3 v 3 = 0. Without loss of generality, assume that α 0 (otherwise, just rename the vectors). Then v = α α v α 3 α v 3. This means that if vectors v and v 3 span a plane, then v lies in this plane. If v and v 3 lie in one line, then v also lies in this line. If v = v 3 = 0, then also v = 0. In any case, vectors v, v, v 3 are coplanar (if v and v 3 are colinear, we just have many options to choose the plane). For the reverse implication, assume that v, v, v 3 R 3 are coplanar. Assume first that the three vectors span a plane P. As P is a -dimensional subspace of R 3, there must eist two of the vectors v, v, v 3 which span P - assume these are v, v 3 (otherwise rename the vectors). Then (v, v 3 ) form a basis of P, and we can epress v P in this basis: v α v α 3 v 3 = 0. v = α v + α 3 v 3, This is a non-trivial linear combination of v, v, v 3 which gives 0 (the first coefficient 0), so v, v, v 3 are linearly dependent. If v, v, v 3 do not span a plane, they all lie in a single line. If at least one of the vectors v, v, v 3 is non-zero, e.g. v 3 0, then vectors v, v are scalar multiples of v 3, and v = αv 3, v αv 3 = 0.
6 Ph.D. Katarína Bellová Page 6 Mathematics (0-PHY-BIPMA) This is again a non-trivial linear combination of v, v, v 3 which gives 0. The remaining case when v = v = v 3 = 0 is trivial: then e.g. v = 0 is non-trivial linear combination of v, v, v 3 which gives 0. Problem 4 [3 points]: Consider the set V of all polynomials with real coefficients of degree at most 3. With the usual addition and multiplication by real numbers, V forms a vector space over R. What is its dimension? Justify your answer! Solution: We have V = {α α + α + α 0 : α 0, α, α, α 3 R}. We see that polynomials,,, 3 form a basis of V (they span V by the above equality, and they are linearly independent, as α α + α + α 0 = 0 as a polynomial implies α 0 = α = α = α 3 = 0). Hence, the dimension of V is the number of its basis vectors, i.e. 4.
Ph.D. Katarína Bellová Page 1 Mathematics 1 (10-PHY-BIPMA1) EXAM SOLUTIONS, 20 February 2018
Ph.D. Katarína Bellová Page Mathematics 0-PHY-BIPMA) EXAM SOLUTIONS, 0 February 08 Problem [4 points]: For which positive integers n does the following inequality hold? n! 3 n Solution: Trying first few
More informationNational University of Singapore Department of Mathematics
National University of Singapore Department of Mathematics Semester I, 2002/2003 MA505 Math I Suggested Solutions to T. 2. Let f() be a real function defined as follows: sin(a) if
More informationDifferential Calculus
Differential Calculus. Compute the derivatives of the following functions a() = 4 3 7 + 4 + 5 b() = 3 + + c() = 3 + d() = sin cos e() = sin f() = log g() = tan h() = 3 6e 5 4 i() = + tan 3 j() = e k()
More informationDifferentiation of Logarithmic Functions
Differentiation of Logarithmic Functions The rule for finding the derivative of a logarithmic function is given as: If y log a then dy or y. d a ( ln This rule can be proven by rewriting the logarithmic
More informationReview of elements of Calculus (functions in one variable)
Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints
More informationSolutions Exam 4 (Applications of Differentiation) 1. a. Applying the Quotient Rule we compute the derivative function of f as follows:
MAT 4 Solutions Eam 4 (Applications of Differentiation) a Applying the Quotient Rule we compute the derivative function of f as follows: f () = 43 e 4 e (e ) = 43 4 e = 3 (4 ) e Hence f '( ) 0 for = 0
More informationSolutions to Homework 8, Mathematics 1
Solutions to Homework 8, Mathematics Problem [6 points]: Do the detailed graphing definition domain, intersections with the aes, its at ±, monotonicity, local and global etrema, conveity/concavity, inflection
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationCore Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description
Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation;
More informationR3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
More information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More informationMath 180, Exam 2, Spring 2013 Problem 1 Solution
Math 80, Eam, Spring 0 Problem Solution. Find the derivative of each function below. You do not need to simplify your answers. (a) tan ( + cos ) (b) / (logarithmic differentiation may be useful) (c) +
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.6 Derivatives of Logarithmic Functions In this section, we: use implicit differentiation to find the derivatives of the logarithmic functions and, in particular,
More informationCalculus 1 (AP, Honors, Academic) Summer Assignment 2018
Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have
More informationEconomics 205 Exercises
Economics 05 Eercises Prof. Watson, Fall 006 (Includes eaminations through Fall 003) Part 1: Basic Analysis 1. Using ε and δ, write in formal terms the meaning of lim a f() = c, where f : R R.. Write the
More informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
More informationUnit 5 MC and FR Practice
Name: Date:. If y = e, then y = ( )e ( )e ( )e. If y = e /, then y = e/ e / e / e /. If y = e cos, then dy d = e cos sin e cos sin e cos e cos sin. curve is defined by y = e sin. Find dy d. e sin cos sin
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationA.P. Calculus Summer Packet
A.P. Calculus Summer Packet Going into AP calculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationMath 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationGiven the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.
Selected problems from the tetbook J. Neustupa, S. Kračmar: Sbírka příkladů z Matematiky I Problems in Mathematics I I. LINEAR ALGEBRA I.. Vectors, vector spaces Given the vectors u, v, w and real numbers
More informationIntermediate Algebra Section 9.3 Logarithmic Functions
Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where
More informationMath 1000 Final Exam Review Solutions. (x + 3)(x 2) = lim. = lim x 2 = 3 2 = 5. (x + 1) 1 x( x ) = lim. = lim. f f(1 + h) f(1) (1) = lim
Math Final Eam Review Solutions { + 3 if < Consider f() Find the following limits: (a) lim f() + + (b) lim f() + 3 3 (c) lim f() does not eist Find each of the following limits: + 6 (a) lim 3 + 3 (b) lim
More informationy x is symmetric with respect to which of the following?
AP Calculus Summer Assignment Name: Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number. Part : Multiple Choice Solve
More informationLimits, Continuity, and Differentiability Solutions
Limits, Continuity, and Differentiability Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
More informationSTEP Support Programme. Pure STEP 3 Solutions
STEP Support Programme Pure STEP 3 Solutions S3 Q6 Preparation Completing the square on gives + + y, so the centre is at, and the radius is. First draw a sketch of y 4 3. This has roots at and, and you
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationSolutions to Math 41 Final Exam December 9, 2013
Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain
More informationName. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
REVIEW Eam #3 : 3.2-3.6, 4.1-4.5, 5.1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Leading Coefficient Test to determine the end behavior
More informationTechnical Calculus I Homework. Instructions
Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationOutline. 1 Integration by Substitution: The Technique. 2 Integration by Substitution: Worked Examples. 3 Integration by Parts: The Technique
MS2: IT Mathematics Integration Two Techniques of Integration John Carroll School of Mathematical Sciences Dublin City University Integration by Substitution: The Technique Integration by Substitution:
More informationExact Equations. M(x,y) + N(x,y) y = 0, M(x,y) dx + N(x,y) dy = 0. M(x,y) + N(x,y) y = 0
Eact Equations An eact equation is a first order differential equation that can be written in the form M(, + N(,, provided that there eists a function ψ(, such that = M (, and N(, = Note : Often the equation
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationMath 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationIt s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]
It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)
More informationThe Fundamental Theorem of Calculus Part 3
The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
More informationTHEOREM: THE CONSTANT RULE
MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:
More informationSome commonly encountered sets and their notations
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their
More informationSolutions to Math 41 Exam 2 November 10, 2011
Solutions to Math 41 Eam November 10, 011 1. (1 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it is or.
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationC) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.
. The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +
More informationMATH 1325 Business Calculus Guided Notes
MATH 135 Business Calculus Guided Notes LSC North Harris By Isabella Fisher Section.1 Functions and Theirs Graphs A is a rule that assigns to each element in one and only one element in. Set A Set B Set
More informationMath 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below.
Math Final Eam - Practice Problem Solutions. A function f is graphed below. f() 8 7 7 8 (a) Find f(), f( ), f(), and f() f() = ;f( ).;f() is undefined; f() = (b) Find the domain and range of f Domain:
More information(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.
Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive
More informationMethods of Integration
Methods of Integration Essential Formulas k d = k +C sind = cos +C n d = n+ n + +C cosd = sin +C e d = e +C tand = ln sec +C d = ln +C cotd = ln sin +C + d = tan +C lnd = ln +C secd = ln sec + tan +C cscd
More information10.4: WORKING WITH TAYLOR SERIES
04: WORKING WITH TAYLOR SERIES Contributed by OpenSta Mathematics at OpenSta CNX In the preceding section we defined Taylor series and showed how to find the Taylor series for several common functions
More information1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x)
APPM 5 Final Eam (5 pts) Fall. The following problems are not related: (a) (5 pts, 5 pts ea.) Find the following limits or show that they do not eist: (i) lim e (ii) lim arcsin() (b) (5 pts) Find and classify
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationProblem Set 9 Solutions
8.4 Problem Set 9 Solutions Total: 4 points Problem : Integrate (a) (b) d. ( 4 + 4)( 4 + 5) d 4. Solution (4 points) (a) We use the method of partial fractions to write A B (C + D) = + +. ( ) ( 4 + 5)
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationMath 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.
. Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.
More informationInverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4
Inverse Functions Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the
More informationAvon High School Name AP Calculus AB Summer Review Packet Score Period
Avon High School Name AP Calculus AB Summer Review Packet Score Period f 4, find:.) If a.) f 4 f 4 b.) Topic A: Functions f c.) f h f h 4 V r r a.) V 4.) If, find: b.) V r V r c.) V r V r.) If f and g
More informationCALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg.
CALCULUS: Graphical,Numerical,Algebraic b Finne,Demana,Watts and Kenned Chapter : Derivatives.: Derivative of a function pg. 116-16 What ou'll Learn About How to find the derivative of: Functions with
More informationReview sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston
Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More informationUnit 5 Applications of Antidifferentiation
Warmup 1) If f ( ) cos(ln ) for > 0, then f () (a) sin(ln ) (b) sin(ln ) (c) sin(ln ) (d) sin(ln ) (e) ln sin 2) If f ( ) 2, then f () (a) 2 ( ln 2) (b) 2 (1 ln 2) (c) 2 ln 2 (d) 2 (1 ln 2) (e) 2 (1 ln
More informationWork the following on notebook paper. No calculator. Find the derivative. Do not leave negative exponents or complex fractions in your answers.
ALULUS B WORKSHEET ON 8. & REVIEW Find the derivative. Do not leave negative eponents or comple fractions in your answers. sec. f 8 7. f e. y ln tan. y cos tan. f 7. f cos. y 7 8. y log 7 Evaluate the
More informationAdding and Subtracting Rational Expressions
Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
More information4.3 - How Derivatives Affect the Shape of a Graph
4.3 - How Derivatives Affect the Shape of a Graph 1. Increasing and Decreasing Functions Definition: A function f is (strictly) increasing on an interval I if for every 1, in I with 1, f 1 f. A function
More informationMA Midterm Exam 1 Spring 2012
MA Miterm Eam Spring Hoffman. (7 points) Differentiate g() = sin( ) ln(). Solution: We use the quotient rule: g () = ln() (sin( )) sin( ) (ln()) (ln()) = ln()(cos( ) ( )) sin( )( ()) (ln()) = ln() cos(
More informationAP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student
AP Calculus AB SUMMER ASSIGNMENT Dear future Calculus AB student We are ecited to work with you net year in Calculus AB. In order to help you be prepared for this class, please complete the summer assignment.
More informationCore Mathematics 2 Unit C2 AS
Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences
More informationCourse. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.
Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More information8.3 Zero, Negative, and Fractional Exponents
www.ck2.org Chapter 8. Eponents and Polynomials 8.3 Zero, Negative, and Fractional Eponents Learning Objectives Simplify epressions with zero eponents. Simplify epressions with negative eponents. Simplify
More informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationNotation. 0,1,2,, 1 with addition and multiplication modulo
Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More informationCalculus B Exam III (Page 1) May 11, 2012
Calculus B Eam III (Page ) May, 0 Name: Instructions: Provide all steps necessary to solve the problem. Unless otherwise stated, your answer must be eact and reasonably simplified. Additionally, clearly
More informationIndeterminate Forms and L Hospital s Rule
APPLICATIONS OF DIFFERENTIATION Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at certain points. INDETERMINATE FORM TYPE
More informationPre-Calculus and Trigonometry Capacity Matrix
Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions
More informationMath Midterm Solutions
Math 50 - Midterm Solutions November 4, 009. a) If f ) > 0 for all in a, b), then the graph of f is concave upward on a, b). If f ) < 0 for all in a, b), then the graph of f is downward on a, b). This
More informationChapter 3: Topics in Differentiation
Chapter 3: Topics in Differentiation Summary: Having investigated the derivatives of common functions in Chapter (i.e., polynomials, rational functions, trigonometric functions, and their combinations),
More informationM151B Practice Problems for Exam 1
M151B Practice Problems for Eam 1 Calculators will not be allowed on the eam. Unjustified answers will not receive credit. 1. Compute each of the following its: 1a. 1b. 1c. 1d. 1e. 1 3 4. 3. sin 7 0. +
More information(ii) y = ln 1 ] t 3 t x x2 9
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
More informationMathematical Induction Assignments
1 Mathematical Induction Assignments Prove the Following using Principle of Mathematical induction 1) Prove that for any positive integer number n, n 3 + 2 n is divisible by 3 2) Prove that 1 3 + 2 3 +
More informationAP CALCULUS AB - Name: Summer Work requirement due on the first day of class
AP CALCULUS AB - Name: Summer Work For students to successfully complete the objectives of the AP Calculus curriculum, the student must demonstrate a high level of independence, capability, dedication,
More informationMath 1160 Final Review (Sponsored by The Learning Center) cos xcsc tan. 2 x. . Make the trigonometric substitution into
Math 60 Final Review (Sponsored by The Learning Center). Simplify cot csc csc. Prove the following identities: cos csc csc sin. Let 7sin simplify.. Prove: tan y csc y cos y sec y cos y cos sin y cos csc
More informationSummer Packet Honors PreCalculus
Summer Packet Honors PreCalculus Honors Pre-Calculus is a demanding course that relies heavily upon a student s algebra, geometry, and trigonometry skills. You are epected to know these topics before entering
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationPage 1. These are all fairly simple functions in that wherever the variable appears it is by itself. What about functions like the following, ( ) ( )
Chain Rule Page We ve taken a lot of derivatives over the course of the last few sections. However, if you look back they have all been functions similar to the following kinds of functions. 0 w ( ( tan
More informationIn Praise of y = x α sin 1x
In Praise of y = α sin H. Turgay Kaptanoğlu Countereamples have great educational value, because they serve to illustrate the limits of mathematical facts. Every mathematics course should include countereamples
More informationSummer AP Assignment Coversheet Falls Church High School
Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose:
More informationTest 3, Linear Algebra
Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all
More information0.1. Linear transformations
Suggestions for midterm review #3 The repetitoria are usually not complete; I am merely bringing up the points that many people didn t now on the recitations Linear transformations The following mostly
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationMathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman
03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction
More information3.7 Indeterminate Forms - l Hôpital s Rule
3.7. INDETERMINATE FORMS - L HÔPITAL S RULE 4 3.7 Indeterminate Forms - l Hôpital s Rule 3.7. Introduction An indeterminate form is a form for which the answer is not predictable. From the chapter on lits,
More informationMidterm 1 Solutions. Monday, 10/24/2011
Midterm Solutions Monday, 0/24/20. (0 points) Consider the function y = f() = e + 2e. (a) (2 points) What is the domain of f? Epress your answer using interval notation. Solution: We must eclude the possibility
More informationCalculus Problem Sheet Prof Paul Sutcliffe. 2. State the domain and range of each of the following functions
f( 8 6 4 8 6-3 - - 3 4 5 6 f(.9.8.7.6.5.4.3.. -4-3 - - 3 f( 7 6 5 4 3-3 - - Calculus Problem Sheet Prof Paul Sutcliffe. By applying the vertical line test, or otherwise, determine whether each of the following
More information