math hands Calculus: HW set 150c02s04
|
|
|
- Kelley Porter
- 5 years ago
- Views:
Transcription
1 Calculus: HW st 150c02s04 For all problms, assum inicat rivativs xists, ar finit an whn thy appar on nominator ar non-zro. 1. Comput th following rivativ: x 2 +5x 5 x 2 +5x 5 5 x 2 +5x 4 x 2 +5x 5 x 2 +5x 4 2x+5 2. Comput th following rivativ: x x x 2 +5x 4 x x x 2 3. Comput th following rivativ: sin(x)5 sin(x)5 5sin(x) 4 sin(x) 5sin(x) 4 cos(x) pg. 1 c MathHans.com v.1007
2 4. Comput th following rivativ: Comput th following rivativ: tan(x)5 tan(x)5 5tan(x) 4 tan(x) 5tan(x) 4 sc 2 (x) 6. Comput th following rivativ: g5 g5 5g 4 g 5g 4 g 7. Comput th following rivativ: z5 pg. 2 c MathHans.com v.1007
3 z5 5z 4 z 5z 4 z 8. Comput th following rivativ: 5 ď 5 4 ď 5 ď ď 4 ď 5 ď 9. Comput th following rivativ: y5 y5 5y 4 y 5y 4 y 10. supr famous Comput th following rivativ: yz5 pg. 3 c MathHans.com v.1007
4 yz5 5yz 4 yz 5yz 4 z y +y 5yz 4 z y +yz z 11. Comput th following rivativ: ˇ ) 8 ˇ ) 8 8ˇ ) 7 ˇ ) 8ˇ ) 7 ˇ ) 12. Comput th following rivativ: ( ) 8 1 tan sin 2 3x+5 hmmm fun Comput th following rivativ: 3x+4 3x+4 3x+4 3x+4 3x+4 3 pg. 4 c MathHans.com v.1007
5 14. Comput th following rivativ: x5 x5 x5 x 5 x5 4x Comput th following rivativ: cos(x) cos(x) cos(x) cos(x) cos(x) ( sin(x)) 16. Comput th following rivativ: ( ) pg. 5 c MathHans.com v.1007
6 17. Comput th following rivativ: ln(x) ln(x) ln(x) ln(x) ln(x) 1 x 18. Comput th following rivativ: xtan 1 (x) xtan 1 (x) xtan 1 (x) xtan 1 (x) xtan 1 (x) ( 1 tan 1 (x)+x ) 1 x Comput th following rivativ: sc(x) sc(x) sc(x) sc(x) sc(x) (sc(x)tan(x)) pg. 6 c MathHans.com v.1007
7 20. Comput th following rivativ: cot(x) cot(x) cot(x) cot(x) cot(x) ( csc 2 (x) ) 21. Comput th following rivativ: xlnx xlnx xlnx xlnx xlnx (lnx+1) 22. fin xx hint: s prvious qustion 23. Comput th following rivativ: xln5 xln5 xln5 xln5 xln5 (ln5) pg. 7 c MathHans.com v.1007
8 24. fin 5x hint: s prvious qustion 25. Comput th following rivativ: xln7 xln7 xln7 xln7 xln7 (ln7) 26. fin 7x hint: s prvious qustion 27. Comput th following rivativ: ( ln 5x 2 ) ( ln 5x 2 ) 1 5x 2 1 5x 2 10x 5x 2 5x 2 (10x) pg. 8 c MathHans.com v.1007
9 28. Comput th following rivativ: ( ln sin 2 (x) ) ( ln sin 2 (x) ) 1 sin 2 sin 2 (x) (x) 1 sin 2 (x) cos(x)sin(x) sin 2 (x) ( cos(x) sin(x)) 29. Comput th following rivativ: ln(ln(x)) 1 ln(ln(x)) ln(x) ln(x) 1 (1/x) ln(x) 1/x ln(x) 30. Comput th following rivativ: ( ) ln ď ( ) 1 ln ď ď 1 ď ď ď pg. 9 c MathHans.com v.1007
10 31. Comput th following rivativ: ln(xyz) 1 ln(xyz) xyz 1 xyz xyz ( xy z +xzy +yz xyz y +xz +yz xyz ) 32. (ALTrnativly) Comput th following rivativ: ln(xyz) 1 ln(xyz) xyz 1 xyz xyz ( xy z +xzy +yz xy z +xzy +yz xyz ) 33. Comput th following rivativ: ln(xx sin(x)) pg. 10 c MathHans.com v.1007
11 ln(xx sin(x)) 1 x x sin(x) xx sin(x) 1 x x sin(x) ( x sin(x)+x x sin(x)+x x cos(x)) x sin(x)+x x sin(x)+x x cos(x) x x sin(x) 34. littl famous fin 1 2 (x x ) 1 2 (x + x ) not: th hyprbolic function sinh(x) is fin as sinh(x) 1 2 (x x ) not: th hyprbolic function cosh(x) is fin as cosh(x) 1 2 (x + x ) 35. littl famous fin 1 2 (x + x ) 1 2 (x x ) not: th hyprbolic function sinh(x) is fin as sinh(x) 1 2 (x x ) not: th hyprbolic function cosh(x) is fin as cosh(x) 1 2 (x + x ) pg. 11 c MathHans.com v.1007
12 36. supr famous fin f g 1 not th us of -1 hr is as xponnt, not composition invrs f g 1 f g 1 +f g 1 f g 1 +( 1)f g 2 g f g 1 f g 2g 1 f g f g g 2 1 g f f g 2g f g fg g 2 f g g 2 fg g 2 f g fg g 2 (yippi kay yy) 37. supr famous fin f g hint: s prvious qustion 38. Comput th following rivativ: x 3 cos(x) x 3 (cos(x)) cos(x) x 3 (x 3 ) cos(x) (cos(x)) 2 (cos(x))(3x2 ) (x 3 )( sin(x)) (cos(x)) Comput th following rivativ: lnx 5x 3 +πx+xln7 pg. 12 c MathHans.com v.1007
13 lnx 5x 3 +πx+xln7 (5x3 +πx+xln7) lnx (lnx) 5x 3 +πx+xln7 (5x 3 +πx+xln7) 2 (5x3 +πx+xln7)( 1 x ) (lnx)(15x2 +π+ln7) (5x 3 +πx+xln7) supr famous Comput th following rivativ: sin(x) cos(x) sin(x) (cos(x)) cos(x) sin(x) (sin(x)) cos(x) (cos(x)) 2 (cos(x))(cos(x)) (sin(x))( sin(x)) (cos(x)) supr famous Comput th following rivativ: cos(x) sin(x) cos(x) (sin(x)) sin(x) cos(x) (cos(x)) sin(x) (sin(x)) 2 (sin(x))( sin(x)) (cos(x))(cos(x)) (sin(x)) supr famous Comput th following rivativ: 1 sin(x) pg. 13 c MathHans.com v.1007
14 1 sin(x) (sin(x)) 1 (1) sin(x) (sin(x)) 2 (sin(x))(0) (1)(cos(x)) (sin(x)) supr famous Comput th following rivativ: 1 cos(x) 1 cos(x) (cos(x)) 1 (1) cos(x) (cos(x)) 2 (cos(x))(0) (1)( sin(x)) (cos(x)) Bas on th iagram: (c,f(c)) c (c+h,f(c+h)) h c+h (a) comput th scant slop from pt (c,f(c)) to point (c+h,f(c+h) OR sctant slop f(c+h) f(c) c+h c sctant slop f(c+h) f(c) h pg. 14 c MathHans.com v.1007
15 (b) turn that scant into a tangnt slop (c,f(c)) to point (c+h,f(c+h) tangnt slop at c f (c) lim h 0 f(c+h) f(c) h (c) Us part B to prov if f(x) x 2 thn f (c) 2c f f(c+h) f(c) (c) lim h 0 h (c+h) 2 c 2 lim h 0 h c 2 +2ch+h 2 c 2 lim h 0 h 2ch+h 2 lim h 0 h lim 2c+h h 0 pim 2c 45. Prov vrything on th first two pags of th CH1 & CH2 Summary Sht. or at last 40 of thos statmnts pg. 15 c MathHans.com v.1007
Math 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
Review Problems for Test 1
Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
University Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.
MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,
5, tan = 4. csc = Simplify: 3. Simplify: 4. Factor and simplify: cos x sin x cos x
Precalculus Final Review 1. Given the following values, evaluate (if possible) the other four trigonometric functions using the fundamental trigonometric identities or triangles csc = - 3 5, tan = 4 3.
Math 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).
Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y).
MA Practice Exam #2 Solutions
MA 123 - Practice Exam #2 Solutions Name: Instructions: For some of the questions, you must show all your work as indicated. No calculators, books or notes of any form are allowed. Note that the questions
Math 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.)
(1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are
Calculus I Announcements
Slide 1 Calculus I Announcements Read sections 3.9-3.10 Do all the homework for section 3.9 and problems 1,3,5,7 from section 3.10. The exam is in Thursday, October 22nd. The exam will cover sections 3.2-3.10,
Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018
Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement
MSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
f(g(x)) g (x) dx = f(u) du.
1. Techniques of Integration Section 8-IT 1.1. Basic integration formulas. Integration is more difficult than derivation. The derivative of every rational function or trigonometric function is another
PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
Math 2163, Practice Exam II, Solution
Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so
MAT137 Calculus! Lecture 9
MAT137 Calculus! Lecture 9 Today we will study: Limits at infinity. L Hôpital s Rule. Mean Value Theorem. (11.5,11.6, 4.1) PS3 is due this Friday June 16. Next class: Applications of the Mean Value Theorem.
1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
Solution to Review Problems for Midterm II
Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions
Things you should have learned in Calculus II
Things you should have learned in Calculus II 1 Vectors Given vectors v = v 1, v 2, v 3, u = u 1, u 2, u 3 1.1 Common Operations Operations Notation How is it calculated Other Notation Dot Product v u
INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1.
INVERSE FUNCTIONS DERIVATIVES Recall the steps for computing y implicitly: (1) Take of both sies, treating y like a function. (2) Expan, a, subtract to get the y terms on one sie an everything else on
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15
A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 The scoring for this section is determined by the formula [C (0.25 I)] 1.8 where C is the
3/1/2012: First hourly Practice A
Math 1A: Introduction to functions and calculus Oliver Knill, Spring 2012 3/1/2012: First hourly Practice A Your Name: Start by writing your name in the above box. Try to answer each question on the same
Calculus I Announcements
Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.
AP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.
AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)
MATH 31BH Homework 5 Solutions
MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [
sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...
![sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...](/thumbs/92/110801024.jpg "sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...")
Sec. 01 notes Some Observations Not Linear Doubling an angles does NOT usually double the ratio. Compare sin10 with sin 20, for example. generally,... sin(2x) 2 sinx sin IS NOT a number If x, y and z are
Final Exam. Math 3 December 7, 2010
Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.
Antiderivatives. DEFINITION: A function F is called an antiderivative of f on an (open) interval I if F (x) = f(x) for all x in I EXAMPLES:
Antiderivatives 00 Kiryl Tsishchanka DEFINITION: A function F is called an antiderivative of f on an (open) interval I if F (x) = f(x) for all x in I EXAMPLES:. If f(x) = x, then F(x) = 3 x3, since ( )
Honors Calculus II [ ] Midterm II
![Honors Calculus II [ ] Midterm II](/thumbs/90/103243723.jpg "Honors Calculus II [ ] Midterm II")
Honors Calculus II [3-00] Midterm II PRINT NAME: SOLUTIONS Q]...[0 points] Evaluate the following expressions and its. Since you don t have a calculator, square roots, fractions etc. are allowed in your
Calculus & Analytic Geometry I
TQS 124 Autumn 2008 Quinn Calculus & Analytic Geometry I The Derivative: Analytic Viewpoint Derivative of a Constant Function. For c a constant, the derivative of f(x) = c equals f (x) = Derivative of
AP Calculus Multiple-Choice Question Collection
AP Calculus Multipl-Coic Qustion Collction 985 998 . f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b
Partial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt
Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses
TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
Prof. Israel N. Nwaguru MATH 11 CHAPTER,,, AND - REVIEW WORKOUT EACH PROBLEM NEATLY AND ORDERLY ON SEPARATE SHEET THEN CHOSE THE BEST ANSWER TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
Chapter 13: General Solutions to Homogeneous Linear Differential Equations
Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)
Solved Examples. (Highest power of x in numerator and denominator is ½. Dividing numerator and denominator by x)
Solved Examples Example 1: (i) (ii) lim x (x 4 + 2x 3 +3) / (2x 4 -x+2) lim x x ( (x+c)- x) (iii) lim n (1-2+3-4+...(2n-1)-2n)/ (n 2 +1) (iv) lim x 0 ((1+x) 5-1)/3x+5x 2 (v) lim x 2 ( (x+7)-3 (2x-3))/((x+6)
Differential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
Institute of Computer Science
Institute of Computer Science Academy of Sciences of the Czech Republic Calculus Digest Jiří Rohn http://uivtx.cs.cas.cz/~rohn Technical report No. V-54 02.02.202 Pod Vodárenskou věží 2, 82 07 Prague 8,
1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
AP Calculus BC AP Exam Problems Chapters 1 3
AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum
Hyperbolics. Scott Morgan. Further Mathematics Support Programme - WJEC A-Level Further Mathematics 31st March scott3142.
Hyperbolics Scott Morgan Further Mathematics Support Programme - WJEC A-Level Further Mathematics 3st March 208 scott342.com @Scott342 Topics Hyperbolic Identities Calculus with Hyperbolics - Differentiation
AP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
UNIT 3: DERIVATIVES STUDY GUIDE
Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average
MA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
Math 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
Algebra 2/Trig AIIT.17 Trig Identities Notes. Name: Date: Block:
Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Name: Date: Block: Trigonometric Identities When two trig expressions can be proven to be equal to each other, the statement is called a trig identity
dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
![dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)](/thumbs/75/71716329.jpg "dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)")
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
M152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
Chapter 3 Differentiation Rules (continued)
Chapter 3 Differentiation Rules (continued) Sec 3.5: Implicit Differentiation (continued) Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph
Math 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
MAS113 CALCULUS II SPRING 2008, QUIZ 5 SOLUTIONS. x 2 dx = 3y + y 3 = x 3 + c. It can be easily verified that the differential equation is exact, as
MAS113 CALCULUS II SPRING 008, QUIZ 5 SOLUTIONS Quiz 5a Solutions (1) Solve the differential equation y = x 1 + y. (1 + y )y = x = (1 + y ) = x = 3y + y 3 = x 3 + c. () Solve the differential equation
a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
Question Instructions Read today's Notes and Learning Goals.
63 Proucts an Quotients (13051836) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Instructions Rea toay's Notes an Learning Goals. 1. Question Details fa15 62 chain 1 [3420817] Fin all
(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
PRACTICE PROBLEMS FOR MIDTERM I
Problem. Find the limits or explain why they do not exist (i) lim x,y 0 x +y 6 x 6 +y ; (ii) lim x,y,z 0 x 6 +y 6 +z 6 x +y +z. (iii) lim x,y 0 sin(x +y ) x +y Problem. PRACTICE PROBLEMS FOR MIDTERM I
Calculus II. George Voutsadakis 1. LSSU Math 152. Lake Superior State University. 1 Mathematics and Computer Science
Calculus II George Voutsadakis Mathematics and Computer Science Lake Superior State University LSSU Math 52 George Voutsadakis (LSSU) Calculus II February 205 / 88 Outline Techniques of Integration Integration
School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.
School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm
DRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
Math 1A: Homework 6 Solutions
Math A: Homework Solutions July 30. Sketch the graphs of the following functions. Ensure that your work includes: Domain Intercepts Symmetry Asymptotes and End-behaviour Intervals of Increase/Decrease/Local
Formulas to remember
Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal
WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II.
WORKBOOK. MATH 32. CALCULUS AND ANALYTIC GEOMETRY II. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributors: U. N. Iyer, P. Laul, I. Petrovic. (Many problems have been irectly taken from Single Variable
Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
M GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H
M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester 38-39 H 3 Q. Let A =, B = and C = 3 Compute (if possible) : A+B and BC A+B is impossible. ( ) BC = 3
SET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
Welcome to AP Calculus!!!
Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you
Solving Equations. Pure Math 30: Explained! 255
Solving Equations Pure Math : Explained! www.puremath.com 55 Part One - Graphically Solving Equations Solving trigonometric equations graphically: When a question asks you to solve a system of trigonometric
JUST THE MATHS UNIT NUMBER PARTIAL DIFFERENTIATION 1 (Partial derivatives of the first order) A.J.Hobson
JUST THE MATHS UNIT NUMBER 14.1 PARTIAL DIFFERENTIATION 1 (Partial derivatives of the first order) by A.J.Hobson 14.1.1 Functions of several variables 14.1.2 The definition of a partial derivative 14.1.3
MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points
MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points Dr. Sarah Mitchell Autumn 2014 An important limit To calculate the limits of basic trigonometric functions
Math 229 Mock Final Exam Solution
Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it
TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
PRODUCT & QUOTIENT RULES CALCULUS 2. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Product & quotient rules 1/13 Adrian Jannetta
PRODUCT & QUOTIENT RULES CALCULUS 2 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Proct & quotient rules 1/13 Adrian Jannetta Objectives In this presentation we ll continue learning how to differentiate
5/8/2012: Practice final A
Math 1A: introduction to functions and calculus Oliver Knill, Spring 2012 Problem 1) TF questions (20 points) No justifications are needed. 5/8/2012: Practice final A 1) T F The quantum exponential function
Math 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
Math 251, Spring 2005: Exam #2 Preview Problems
Math 5, Spring 005: Exam # Preview Problems. Using the definition of derivative find the derivative of the following functions: a) fx) = e x e h. Use the following lim =, e x+h = e x e h.) h b) fx) = x
Table of Contents Derivatives of Logarithms
Derivatives of Logarithms- Table of Contents Derivatives of Logarithms Arithmetic Properties of Logarithms Derivatives of Logarithms Example Example 2 Example 3 Example 4 Logarithmic Differentiation Example
1.7 Sums of series Example 1.7.1: 2. Real functions of one variable Example 1.7.2: 2.1 General definitions Example 2.1.3: Example 2.1.
7 Sums of series We often want to sum a series of terms, for example when we look at polynomials As we already saw, we abbreviate a sum of the form For example and u + u + + u r by r u i i= a n x n + a
Test one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental
MATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
Find all of the real numbers x that satisfy the algebraic equation:
Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when
Homework Problem Answers
Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C.
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Tim: 3hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A, B and C. SECTION -A Vry Short Answr Typ Qustions. 0 X = 0. Find th condition
Differentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
Calculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
Final exam for MATH 1272: Calculus II, Spring 2015
Final exam for MATH 1272: Calculus II, Spring 2015 Name: ID #: Signature: Section Number: Teaching Assistant: General Instructions: Please don t turn over this page until you are directed to begin. There
Integration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
cosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L =
Integrals Volume: Suppose A(x) is the cross-sectional area of the solid S perpendicular to the x-axis, then the volume of S is given by V = b a A(x) dx Work: Suppose f(x) is a force function. The work
Friday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector
Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a
Math 206 Practice Test 3
Class: Date: Math 06 Practice Test. The function f (x) = x x + 6 satisfies the hypotheses of the Mean Value Theorem on the interval [ 9, 5]. Find all values of c that satisfy the conclusion of the theorem.
8/28/2017 Assignment Previewer
Proucts an Quotients (10862446) Due: Mon Sep 25 2017 07:31 AM MDT Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Instructions Rea toay's Notes an Learning Goals. 1. Question Details
Inverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
and lim lim 6. The Squeeze Theorem
Limits (day 3) Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar
CHM320: MATH REVIEW. I. Ordinary Derivative:
CHM30: MATH REVIEW I. Ordinary Derivative: Figure : Secant line between two points on a function Ordinary derivatives describe how functions of a single variable change in response to variation of the
MATH 104 Practice Problems for Exam 2
. Find the area between: MATH 4 Practice Problems for Exam (a) x =, y = / + x, y = x/ Answer: ln( + ) 4 (b) y = e x, y = xe x, x = Answer: e6 4 7 4 (c) y = x and the x axis, for x 4. x Answer: ln 5. Calculate