Chapter 3 Differentiation Rules (continued)
|
|
|
- Josephine Floyd
- 5 years ago
- Views:
Transcription
1 Chapter 3 Differentiation Rules (continued)
2 Sec 3.5: Implicit Differentiation (continued)
3 Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph of a function? x 2 + y 2 = 25 at (-3, 4)
4 Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph of a function? x 3 + y 3 = 6xy at (3, 3) Can x 3 + y 3 = 6xy be solved for y in the vicinity of (3, 3)?
5 Implicit Differentiation When an equation in x and y is solved for y, i.e. is written as y = stuff with x s only (e.g. y = x 3 + sinx ), then y is defined explicitly as a function of x. When an equation in x and y is not solved for y, (e.g. ysin(x) = x or sin xy = x + e y2 ), then y is defined implicitly as a function of x.
6 Note: = d dx Implicit Differentiation
7 Implicit Differentiation Problems Ex 3: For the curve defined by x 3 + y 3 = 6xy, a) Find dy dx b) Find the equation of the tangent line to the curve at (3, 3) c) Find the points on the curve that have a horizontal tangent line d) Find the points on the curve that have a vertical tangent line e) What s going on at the point (0, 0)? f) Find d2 y dx 2
8 Implicit Differentiation Problems Ex 4: For the curve defined by e x/y + y 2 tan(y 2 ) = sin xy + 7x y + 1, find dy dx.
9 Derivatives of Elementary Functions
10 Derivatives of Exponential Functions (Sec. 3.4)
11 Derivatives of Exponential Functions Ex 5: Derive the formula for d dx 2x
12 Derivatives of Exponential Functions Rule for the derivative of an exponential function: d dx ax = a x lna
13 Derivatives of Exponential Functions Quickies Ex 6: Find a) d dx 7x = 7 x ln (7) b) 10 x = 10 x ln (10) c) d dt t = 3 t ln 3
14 Derivatives of Inverse Trig. Functions (Sec. 3.5)
15 Derivatives of Inverse Trig Functions Ex 7: Derive the formula for d dx sin 1 x
16 Derivatives of Inverse Trig Functions Ex 8: Derive the formula for d dx tan 1 x
17 Derivatives of Inverse Trig Functions Formulas d dx sin 1 x = 1 d 1 x 2 dx cos 1 x = 1 1 x 2 d dx tan 1 x = 1 d 1 + x 2 dx cot 1 x = x 2 d dx sec 1 x = 1 x x 2 1 d dx csc 1 x = 1 x x 2 1
18 Derivatives of Log Functions (Sec. 3.6)
19 Derivatives of Log Functions Ex 9: Derive the formula for d dx log 2 x
20 Derivatives of Log Functions Rule(s) for the derivative of an log function: In particular d dx log a x = 1 xlna d dx ln x = 1 x and d dx ln x = 1 x
21 Derivatives of Log Functions Quickies Ex 10: Find a) d dx log 12 x = 1 xln12 b) log x = 1 xln10 c) d dx ln x = 1 x d) ln x = 1 x
22 Derivatives of Hyperbolic Functions (Sec. 3.11)
23 Some Things to Know About Hyperbolic Functions Recall the definitions of the hyperbolic functions: sinh x = ex e x 2 tanh x = coth x = cosh x = ex + e x sinh x cosh x = ex e x e x + e x cosh x sinh x = ex + e x e x e x 2
24 Some Things to Know About Hyperbolic Functions Recall the definitions of the hyperbolic functions: sinh x = ex e x 2 cosh x = ex + e x 2 sech x = 1 cosh x = 2 e x e x csch x = 1 sinh x = 2 e x + e x
25 Some Things to Know About Hyperbolic Functions Graphs: f x = sinh x Domain =, Range =, Resitricted Domain for sinh 1 x =,
26 Some Things to Know About Hyperbolic Functions Graphs: f x = cosh x Domain =, Range = 1, ) Resitricted Domain for cosh 1 x = 0, )
27 Some Things to Know About Hyperbolic Functions Graphs: f x = tanh x Domain =, Range = 1, 1 Resitricted Domain for tanh 1 x =,
28 Some Things to Know About Hyperbolic Functions Identities: sinh x = sinh(x) cosh x = cosh (x) cosh 2 x sinh 2 x = 1 1 tanh 2 x = sech 2 x sinh x + y = sinh x cosh y + cosh x sinh y cosh x + y = cosh x cosh y + sinh x sinh y
29 Derivatives of Hyperbolic Functions Ex 11: Derive the formula for d sinh x dx
30 Derivatives of Hyperbolic Functions Ex 12: Derive the formula for d tanh x dx
31 Derivatives of Hyperbolic Functions Formulas d dx sinh x = cosh x d dx cosh x = sinh x d dx tanh x = sech2 x d dx coth x = csch2 x d dx sech x = sech x tanh x d csch x = csch x coth x dx
32 Derivatives of Inverse Hyperbolic Functions (Sec. 3.11)
33 Some Things to Know About Inverse Hyperbolic Functions Recall the formulas for the inverse hyperbolic functions: sinh 1 x = ln x + x 2 Domain = + 1, cosh 1 x = ln x + x 2 1 Domain = 1, ) tanh 1 x = 1 2 ln 1 + x 1 x Domain = 1, 1
34 Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = sinh 1 x Domain =, Range =,
35 Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = cosh 1 x Domain = 1, ) Range = 0, )
36 Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = tanh 1 x Domain = 1, 1 Range =,
37 Derivatives of Inverse Hyperbolic Functions Ex 13: Derive the formula for d dx cosh 1 x (in 2 ways)
38 Derivatives of Inverse Hyperbolic Functions Formulas d dx sinh 1 x = 1 d 1 + x 2 dx cosh 1 x = 1 x 2 1 d dx tanh 1 x = 1 d 1 x 2 dx coth 1 x = 1 1 x 2 d dx sech 1 x = 1 x 1 x 2 d dx csch 1 x = x 1 x 2 + 1
39 Mixed Examples (Sec , 3.11)
40 Examples Ex 14: Find y if a) y = ln cosh x + 4 x tan 1 (x) b) y = arctan 1 x 1+x c) y = log 3 (cos x) sinh 1 x 5 xlnx
41 Derivatives of Inverse Trig Functions Ex 15: Find dy dx if tan 1 x 2 y = x + xy 2
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
90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5 Logarithmic, Exponential, an Other Transcenental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential
Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
CHAPTER 1. DIFFERENTIATION 18. As x 1, f(x). At last! We are now in a position to sketch the curve; see Figure 1.4.
CHAPTER. DIFFERENTIATION 8 and similarly for x, As x +, fx), As x, fx). At last! We are now in a position to sketch the curve; see Figure.4. Figure.4: A sketch of the function y = fx) =/x ). Observe the
Differential and Integral Calculus
School of science an engineering El Akhawayn University Monay, March 31 st, 2008 Outline 1 Definition of hyperbolic functions: The hyperbolic cosine an the hyperbolic sine of the real number x are enote
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.
ENGI 3425 Review of Calculus Page then
ENGI 345 Review of Calculus Page 1.01 1. Review of Calculus We begin this course with a refresher on ifferentiation an integration from MATH 1000 an MATH 1001. 1.1 Reminer of some Derivatives (review from
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
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. y For example,, or y = x sin x,
Lecture Notes for Math 1000
Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: xswang@mun.ca Course
Lecture 3. Lecturer: Prof. Sergei Fedotov Calculus and Vectors. Exponential and logarithmic functions
Lecture 3 Lecturer: Prof. Sergei Fedotov 10131 - Calculus and Vectors Exponential and logarithmic functions Sergei Fedotov (University of Manchester) MATH10131 2011 1 / 7 Lecture 3 1 Inverse functions
FINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed.
Math 150 Name: FINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed. 135 points: 45 problems, 3 pts. each. You
MA 242 Review Exponential and Log Functions Notes for today s class can be found at
MA 242 Review Exponential and Log Functions Notes for today s class can be found at www.xecu.net/jacobs/index242.htm Example: If y = x n If y = x 2 then then dy dx = nxn 1 dy dx = 2x1 = 2x Power Function
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
( ) = 1 t + t. ( ) = 1 cos x + x ( sin x). Evaluate y. MTH 111 Test 1 Spring Name Calculus I
MTH Test Spring 209 Name Calculus I Justify all answers by showing your work or by proviing a coherent eplanation. Please circle your answers.. 4 z z + 6 z 3 ez 2 = 4 z + 2 2 z2 2ez Rewrite as 4 z + 6
Section 3.5: Implicit Differentiation
Section 3.5: Implicit Differentiation In the previous sections, we considered the problem of finding the slopes of the tangent line to a given function y = f(x). The idea of a tangent line however is not
Math 180 Prof. Beydler Homework for Packet #5 Page 1 of 11
Math 180 Prof. Beydler Homework for Packet #5 Page 1 of 11 Due date: Name: Note: Write your answers using positive exponents. Radicals are nice, but not required. ex: Write 1 x 2 not x 2. ex: x is nicer
Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review)
Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space
Trigonometric substitutions (8.3).
Review for Eam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Eam covers: 7.4, 7.6, 7.7, 8-IT, 8., 8.2. Solving differential equations
Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook
Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures
Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine
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
7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
Differential Equations DIRECT INTEGRATION. Graham S McDonald
Differential Equations DIRECT INTEGRATION Graham S McDonald A Tutorial Module introducing ordinary differential equations and the method of direct integration Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
Math F15 Rahman
Math - 9 F5 Rahman Week3 7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following definitions: sinh x = (ex e x ) cosh x = (ex + e x ) tanh x = sinh
m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule)
Chapter 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) = c f (x) = 0 g(x) = x g (x) = 1 h(x) = x n h (x) = n x n-1 (The Power Rule) k(x)
6.2. The Hyperbolic Functions. Introduction. Prerequisites. Learning Outcomes
The Hyperbolic Functions 6. Introduction The hyperbolic functions cosh x, sinh x, tanh x etc are certain combinations of the exponential functions e x and e x. The notation implies a close relationship
Monday, 6 th October 2008
MA211 Lecture 9: 2nd order differential eqns Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns 1/19 Class test next week... MA211 Lecture 9: 2nd order differential eqns 2/19 This morning
Chapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then
3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).
7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following
Math 2-08 Rahman Week3 7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following definitions: sinh x = 2 (ex e x ) cosh x = 2 (ex + e x ) tanh x = sinh
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
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:
Some functions and their derivatives
Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n
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
cos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =
MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)
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)
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
Hyperbolic functions
Roberto s Notes on Differential Calculus Chapter 5: Derivatives of transcendental functions Section Derivatives of Hyperbolic functions What you need to know already: Basic rules of differentiation, including
L Hôpital s Rule was discovered by Bernoulli but written for the first time in a text by L Hôpital.
7.5. Ineterminate Forms an L Hôpital s Rule L Hôpital s Rule was iscovere by Bernoulli but written for the first time in a text by L Hôpital. Ineterminate Forms 0/0 an / f(x) If f(x 0 ) = g(x 0 ) = 0,
cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:
6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan
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
Hyperbolic Functions: Exercises - Sol'ns (9 pages; 13/5/17)
Hyperbolic Functions: Exercises - Sol'ns (9 pages; 3/5/7) () (i) Prove, using exponential functions, that (a) cosh x sinh x = (b) sinhx = sinhxcoshx (ii) By differentiating the result from (i)(b), obtain
MIDTERM 1. Name-Surname: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts Total. Overall 115 points.
Name-Surname: Student No: Grade: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts 1 2 3 4 5 6 7 8 Total Overall 115 points. Do as much as you can. Write your answers to all of the questions.
1 + x 2 d dx (sec 1 x) =
Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating
CALCULUS: 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
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
B 2k. E 2k x 2k-p : collateral. 2k ( 2k-n -1)!
13 Termwise Super Derivative In this chapter, for the function whose super derivatives are difficult to be expressed with easy formulas, we differentiate the series expansion of these functions non integer
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
Throughout this module we use x to denote the positive square root of x; for example, 4 = 2.
Throughout this module we use x to denote the positive square root of x; for example, 4 = 2. You may often see (although not in FLAP) the notation sin 1 used in place of arcsin. sinh and cosh are pronounced
6.7 Hyperbolic Functions
6.7 6.7 Hyperbolic Functions Even and Odd Parts of an Exponential Function We recall that a function f is called even if f( x) = f(x). f is called odd if f( x) = f(x). The sine function is odd while the
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
JUST THE MATHS UNIT NUMBER DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) A.J.Hobson
JUST THE MATHS UNIT NUMBER 104 DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) by AJHobson 1041 Products 1042 Quotients 1043 Logarithmic differentiation 1044 Exercises 1045 Answers
Math 1431 Final Exam Review
Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back
CALCULUS Exercise Set 2 Integration
CALCULUS Exercise Set Integration 1 Basic Indefinite Integrals 1. R = C. R x n = xn+1 n+1 + C n 6= 1 3. R 1 =ln x + C x 4. R sin x= cos x + C 5. R cos x=sinx + C 6. cos x =tanx + C 7. sin x = cot x + C
Welcome to Math 104. D. DeTurck. January 16, University of Pennsylvania. D. DeTurck Math A: Welcome 1 / 44
Welcome to Math 104 D. DeTurck University of Pennsylvania January 16, 2018 D. DeTurck Math 104 002 2018A: Welcome 1 / 44 Welcome to the course Math 104 Calculus I Topics: Quick review of Math 103 topics,
Chapter 3: Transcendental Functions
Chapter 3: Transcendental Functions Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 32 Except for the power functions, the other basic elementary functions are also called the transcendental
Solution: APPM 1350 Final Exam Spring 2014
APPM 135 Final Exam Spring 214 1. (a) (5 pts. each) Find the following derivatives, f (x), for the f given: (a) f(x) = x 2 sin 1 (x 2 ) (b) f(x) = 1 1 + x 2 (c) f(x) = x ln x (d) f(x) = x x d (b) (15 pts)
FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS
Page of 6 FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS 6. HYPERBOLIC FUNCTIONS These functions which are defined in terms of e will be seen later to be related to the trigonometic functions via comple
Summary: Primer on Integral Calculus:
Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of
2 Recollection of elementary functions. II
Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic
Mathematics Notes for Class 12 chapter 7. Integrals
1 P a g e Mathematics Notes for Class 12 chapter 7. Integrals Let f(x) be a function. Then, the collection of all its primitives is called the indefinite integral of f(x) and is denoted by f(x)dx. Integration
D sin x. (By Product Rule of Diff n.) ( ) D 2x ( ) 2. 10x4, or 24x 2 4x 7 ( ) ln x. ln x. , or. ( by Gen.
SOLUTIONS TO THE FINAL - PART MATH 50 SPRING 07 KUNIYUKI PART : 35 POINTS, PART : 5 POINTS, TOTAL: 50 POINTS No notes, books, or calculators allowed. 35 points: 45 problems, 3 pts. each. You do not have
Chapter 2 Derivatives
Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,
Math 1 Lecture 22. Dartmouth College. Monday
Math 1 Lecture 22 Dartmouth College Monday 10-31-16 Contents Reminders/Announcements Last Time Implicit Differentiation Derivatives of Inverse Functions Derivatives of Inverse Trigonometric Functions Examish
Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit
Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A optional unit FP3.1 Unit description Further matrix algebra; vectors, hyperbolic
( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f
Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,
Calculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud
Calculus: Early Transcendental Functions Lecture Notes for Calculus 101 Feras Awad Mahmoud Last Updated: August 2, 2012 1 2 Feras Awad Mahmoud Department of Basic Sciences Philadelphia University JORDAN
Week 1: need to know. November 14, / 20
Week 1: need to know How to find domains and ranges, operations on functions (addition, subtraction, multiplication, division, composition), behaviors of functions (even/odd/ increasing/decreasing), library
Derivatives of Trig and Inverse Trig Functions
Derivatives of Trig and Inverse Trig Functions Math 102 Section 102 Mingfeng Qiu Nov. 28, 2018 Office hours I m planning to have additional office hours next week. Next Monday (Dec 3), which time works
JUST THE MATHS UNIT NUMBER INTEGRATION 1 (Elementary indefinite integrals) A.J.Hobson
JUST THE MATHS UNIT NUMBER 2. INTEGRATION (Elementary indefinite integrals) by A.J.Hobson 2.. The definition of an integral 2..2 Elementary techniques of integration 2..3 Exercises 2..4 Answers to exercises
m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule) thing CFAIHIHD fkthf.
. Chapter 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) c f (x) 0 g(x) x g (x) 1 h(x) x n h (x) n x n-1 (The Power Rule) k(x) c f(x) k (x)
x 2 + y 2 = 1. dx 2y and we conclude that, whichever function we chose, dx y2 = 2x + dy dx x2 + d dy dx = 2x = x y sinh(x) = ex e x 2
Implicit differentiation Suppose we know some relation between x and y, e.g. x + y =. Here, y isn t a function of x. But if we restrict attention to y, then y is a function of x; similarly for y. These
SOLUTIONS TO THE FINAL - PART 1 MATH 150 FALL 2016 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS
SOLUTIONS TO THE FINAL - PART MATH 5 FALL 6 KUNIYUKI PART : 5 POINTS, PART : 5 POINTS, TOTAL: 5 POINTS No notes, books, or calculators allowed. 5 points: 45 problems, pts. each. You do not have to algebraically
f(x)
Calculus m y_0e1^y jkdudtdaw ZS[oifntCwxaCrJej ilhl[cq.k i qatlplm mrpiyg^hztbsz YrmePsqeWrNvxeEdG. Calculus Ch. 3 Review Given the graph of f '(x), sketch a possible graph of f (x). 1) f '(x) f(x) 8 8
CHAPTER 1: FURTHER TRANSCENDENTAL FUNCTIONS
SSCE1693 ENGINEERING MATHEMATICS CHAPTER 1: FURTHER TRANSCENDENTAL FUNCTIONS WAN RUKAIDA BT WAN ABDULLAH YUDARIAH BT MOHAMMAD YUSOF SHAZIRAWATI BT MOHD PUZI NUR ARINA BAZILAH BT AZIZ ZUHAILA BT ISMAIL
10.7. DIFFERENTIATION 7 (Inverse hyperbolic functions) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 0.7 DIFFERENTIATION 7 (Inverse hyperbolic functions) by A.J.Hobson 0.7. Summary of results 0.7.2 The erivative of an inverse hyperbolic sine 0.7.3 The erivative of an inverse
Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response
3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
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
February 21 Math 1190 sec. 63 Spring 2017
February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative
1 Functions and Inverses
October, 08 MAT86 Week Justin Ko Functions and Inverses Definition. A function f : D R is a rule that assigns each element in a set D to eactly one element f() in R. The set D is called the domain of f.
TRIGONOMETRY OUTCOMES
TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.
5.2 Proving Trigonometric Identities
SECTION 5. Proving Trigonometric Identities 43 What you ll learn about A Proof Strategy Proving Identities Disproving Non-Identities Identities in Calculus... and why Proving identities gives you excellent
Implicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
MTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Solutions to Exam 2 April 9, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
Math Test #2 Info and Review Exercises
Math 180 - Test #2 Info an Review Exercises Spring 2019, Prof. Beyler Test Info Date: Will cover packets #7 through #16. You ll have the entire class to finish the test. This will be a 2-part test. Part
SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
CHAIN RULE: DAY 2 WITH TRIG FUNCTIONS. Section 2.4A Calculus AP/Dual, Revised /30/2018 1:44 AM 2.4A: Chain Rule Day 2 1
CHAIN RULE: DAY WITH TRIG FUNCTIONS Section.4A Calculus AP/Dual, Revised 018 viet.dang@humbleisd.net 7/30/018 1:44 AM.4A: Chain Rule Day 1 THE CHAIN RULE A. d dx f g x = f g x g x B. If f(x) is a differentiable
MTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam 2 November 5, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
CALCULUS II MATH Dr. Hyunju Ban
CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK August, 07 Introduction The Mathematical Tripos is designed to be accessible to students who are familiar with the the core A-level syllabus
Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?
. What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when
Hyperbolic Functions and the Twin Paradox
Hyperbolic Functions and the Twin Paradox Basics of hyperbolic functions The basic hyperbolic functions are defined as cosh x 1 2 (ex + e x ), sinh 1 2 (ex e x ). The notation ch, sh is also used (especially
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
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
IMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION Section.5 Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:47 AM.5: Implicit Differentiation 1 REVIEW Solve or d of 4 + 3 = 6 4 3 6 4 3 6 4 3 4 ' 3 3 7/30/018 1:47