Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Size: px
Start display at page:

Download "Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations"

Transcription

1 Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of functions. By te en of Section 3.5 we will be able to ifferentiate any algebraic or trigonometric function as a matter of routine witout reference to te limits use in Section 3.2. Differentiation of polynomials We first note tat if f is a first egree polynomial, say, f(x) = ax + b for some constants a an b, ten f is an affine function an ence its own best affine approximation. Tus f (x) = a for all x. In particular, if f is a constant function, say, f(x) = b for all x, ten f (x) = 0 for all x. Next we consier te case of a monomial f(x) = x n, were n is a positive integer greater tan 1. Ten Now f (x) 0 f(x + ) f(x) (x + ) n x n. (3.3.1) 0 (x + ) n = x n + nx n 1 + R(), (3.3.2) were R() represents te remaining terms in te expansion. Since every term in R() as a factor of raise to a power greater tan or equal to 2, it follows tat R() is o(). Hence we ave f x n + nx n 1 + R() x n (x) 0 nx n 1 + R() 0 ( nx n 1 + R() ) 0 = nx n 1 R() + lim 0 = nx n 1. Since from our previous result f (x) = 1 wen f(x) = x, tis formula also works in te case n = 1. Hence we ave te following proposition. Proposition For any positive integer n, x xn = nx n 1. (3.3.3) 1 Copyrigt c by Dan Slougter 2000

2 2 Differentiation of Polynomials an Rational Functions Section 3.3 If f(x) = x 3, ten f (x) = 3x 2, as we saw in an example in Section 3.2. Similarly, t t5 = 5t 4. Hence, for example, te equation of te line tangent to te curve x = t 5 at ( 1, 1) is or x = 5(t + 1) 1, x = 5t + 4. Once we establis results for te erivative of a constant times a function an for te erivative of te sum of two functions, similar to te results we ave for limits, we will be able to easily ifferentiate any polynomial. So suppose f is a ifferentiable function an let k(x) = cf(x), were c is any constant. Ten k (x) 0 k(x + ) k(x) 0 cf(x + ) cf(x) 0 c(f(x + ) f(x)) = c lim 0 f(x + ) f(x) = cf (x). Tat is, te erivative of a constant times a function is te constant times te erivative of te function. Proposition If f is ifferentiable an c is any constant, ten x (cf(x)) = c f(x). (3.3.4) x If f(x) = 14x 3, ten f (x) = (14)(3x 2 ) = 42x 2. Now suppose f an g are bot ifferentiable functions an let k(x) = f(x) +. Ten k k(x + ) k(x) (x) 0 (f(x + ) + g(x + )) (f(x) + ) 0 0 ( f(x + ) f(x) + f(x + ) f(x) 0 = f (x) + g (x). g(x + ) ) + lim 0 g(x + )

3 Section 3.3 Differentiation of Polynomials an Rational Functions 3 Hence te erivative of te sum of two functions is te sum of teir erivatives. A similar argument woul sow tat te erivative of te ifference of two functions is te ifference of teir erivatives. Proposition If f an g are bot ifferentiable, ten (f(x) + ) = x x f(x) + (3.3.5) x an (f(x) ) = x x f(x). (3.3.6) x Putting te preceing results togeter, we are now in a position to easily ifferentiate any polynomial, as te next examples will illustrate. Suppose f(x) = 3x 5 6x 2 + 2x 16. Ten f (x) = x (3x5 6x 2 + 2x 16) = x (3x5 ) x (6x2 ) + x (2x) x (16) = 3 x x5 6 x x2 + 2 x x 0 = (3)(5x 4 ) (6)(2x) + (2)(1) = 15x 4 12x + 2. Of course, it is not necessary to write out in etail all te steps in ifferentiating a polynomial as we i in te preceing example. For example, if g(t) = 3t 12 6t 2 + t, ten g (t) = (3)(12t 11 ) (6)(2t) + 1 = 36t 11 12t + 1. In particular, since g(1) = 2 an g (1) = 25, te best affine approximation to g at t = 1 is T (t) = 25(t 1) 2 = 25t 27. Differentiation of rational functions We next consier te problem of ifferentiating te quotient of two functions wose erivatives are alreay known. In particular, combining tis result wit our result for polynomials will enable us to easily ifferentiate any rational function. We migt ope tat, analogous to te last two results an te relate results for limits, te erivative of te quotient of two functions woul be equal to te quotient of teir erivatives. Tis turns out not to be true; neverteless, tere is a nice rule for ifferentiating quotients.

4 4 Differentiation of Polynomials an Rational Functions Section 3.3 Suppose f an g are bot ifferentiable functions an let k(x) = f(x). Ten, at all points were 0, k k(x + ) k(x) (x) f(x + ) g(x + ) f(x) f(x + ) g(x + )f(x) g(x + ) f(x + ) g(x + )f(x) 0 g(x + ) It turns out tat by aing an subtracting te term f(x) (a stanar matematical trick of aing 0) in te numerator, we can simplify tis limit into a form tat we can evaluate. Tat is, Now an k (x) 0 f(x + ) g(x + )f(x) g(x + ) f(x + ) f(x) + f(x) g(x + )f(x) 0 g(x + ) (f(x + ) f(x)) f(x)(g(x + ) ) 0 g(x + ) f(x + ) f(x) g(x + ) f(x). 0 g(x + ) f(x + ) f(x) lim 0 g(x + ) lim f(x) 0 = lim 0 f(x + ) f(x) = f(x) lim 0 g(x + ). = f (x), (3.3.7) = f(x)g (x), (3.3.8) lim g(x + ) = lim g(x + ) = = 0 0 ()2, (3.3.9) were te limits in (3.3.7) an (3.3.8) follow from te ifferentiability of f an g, wile te limit in (3.3.9) follows from te continuity of g (wic is a consequence of te ifferentiability of g). Putting everyting togeter, we ave a result known as te quotient rule. k (x) = f (x) f(x)g (x) () 2, (3.3.10)

5 Section 3.3 Differentiation of Polynomials an Rational Functions 5 Quotient Rule If f an g are bot ifferentiable, ten x at all points were 0. f(x) = Suppose f(x) = 2x + 1 x 2. Ten f(x) f(x) x x () 2 (3.3.11) (x 2) (2x + 1) (2x + 1) (x 2) f (x) = x x (x 2) 2 (x 2)(2) (2x + 1)(1) = (x 2) 2 2x 4 2x 1 = (x 2) 2 5 = (x 2) 2. Hence, for example, f(3) = 7 an f (3) = 5, so te equation of te line tangent to te grap of f at (3, 7) is y = 5(x 3) + 7, or y = 5x Suppose g(z) = 1 z 2. Ten g (z) = z 2 z (1) (1) z (z2 ) z 4 = (z2 )(0) 2z z 4 = 2 z 3. Note tat we may write tis result in te form wic is consistent wit our previous result z z 2 = 2z 3, z zn = nz n 1. However, we erive te latter uner te assumption tat n was a positive integer. We will now sow tat we can exten tis result to te case of negative integer exponents.

6 6 Differentiation of Polynomials an Rational Functions Section 3.3 Suppose f(x) = x n, were n is a negative integer. Ten, using te quotient rule an te fact tat n > 0, f (x) = x xn = 1 x x n x n (1) (1) = x x (x n ) x 2n = (x n )(0) ( nx n 1 ) x 2n = nx n 1 x 2n = nx n 1+2n = nx n 1. We can now state te more general result. Proposition For any integer n 0, x xn = nx n 1. (3.3.12) If f(x) = 1 x, ten f (x) = x x 1 = x 2 = 1 x 2. Similarly, 5 x x 3 = x (5x 3 ) = 15x 4 = 15 x 4. We will eventually see tat (3.3.12) ols for rational an irrational exponents as well. We will consier te rational case in Section 3.4, but we will not ave te tools for anling te irrational case until we iscuss exponential an logaritm functions in Capter 6. Differentiation of proucts We will close tis section wit a iscussion of a rule for ifferentiating te prouct of two functions. Since te prouct of two rational functions is again a rational function, tis will not exten te class of functions tat we know ow to ifferentiate routinely. However, tis rule will be very useful in te future an, even at te present point, can elp simplify some problems.

7 Section 3.3 Differentiation of Polynomials an Rational Functions 7 Suppose f an g are bot ifferentiable an k(x) = f(x). Ten k (x) 0 k(x + ) k(x) f(x + )g(x + ) f(x). (3.3.13) 0 Aing an subtracting f(x + ) in te numerator (again, te matematical trick of aing 0 in a useful manner) will elp simplify tis limit. Namely, k f(x + )g(x + ) f(x + ) + f(x + ) f(x) (x) 0 f(x + )(g(x + ) ) + (f(x + ) f(x)) 0 0 ( f(x + ) Now g(x + ) lim f(x+) 0 ( g(x + ) f(x + ) f(x) + 0 f(x+) lim 0 g(x + ) )). = f(x)g (x) (3.3.14) an f(x + ) f(x) lim 0 = lim 0 f(x + ) f(x) = f (x), (3.3.15) were, as wit te erivation of te quotient rule, we ave use te ifferentiability of f an g as well as te continuity of f in evaluating te limits. Putting everyting togeter, we ave k (x) = f(x)g (x) + f (x). (3.3.16) a result known as te prouct rule. Prouct Rule ten If If f an g are bot ifferentiable, ten f(x) = f(x) + f(x). (3.3.17) x x x f(x) = (x 4 3x 2 + 6x 3)(6x 3 + 2x + 5), f (x) = (x 4 3x 2 + 6x 3) x (6x3 + 2x + 5) + (6x 3 + 2x + 5) x (x4 3x 2 + 6x 3) = (x 4 3x 2 + 6x 3)(18x 2 + 2) + (6x 3 + 2x + 5)(4x 3 6x + 6). Of course, in tis example, f is just a polynomial so we coul also fin f by multiplying out te two factors of f an ifferentiating te polynomial term by term as usual. However,

8 8 Differentiation of Polynomials an Rational Functions Section 3.3 te prouct rule gives us a quicker route to te erivative. Altoug te result is not simplifie into te stanar form of a polynomial, for most applications tis form is just as useful as any oter. It is wort noting tat altoug we can now ifferentiate any rational function in teory, in practice our metos may not be te most useful. For example, te function f(x) = (x 2 + 1) 567 is a polynomial, an so we know ow to ifferentiate it. However, at tis point te only way we coul perform te ifferentiation woul be to expan f(x) into stanar polynomial form an ten ifferentiate term by term. In Section 3.4 we will learn ow to anle tis problem more irectly. At te same time we will exten te class of functions tat we can ifferentiate routinely to inclue all algebraic functions. Problems 1. Fin te erivative of eac of te following functions. (a) f(x) = x 3 + 6x (b) = 13x 5 6x (c) g(t) = 3t 6t 2 () y(t) = 4t 3 18t + 3 (e) f(t) = (3t 6) 2 (f) f(x) = (4x + 5)(6x 2 1) 2. Fin te erivative of eac of te following functions. (a) f(x) = (2x + 1) 2 (b) g(t) = (t 2 3) 3 (c) = x 3 2x + 5 (e) f(t) = 3t4 8t + 1 2t (g) (t) = 3 t (i) (z) = 8z 3 1 2z () (s) = 2s s2 s (f) x(t) = 3 t 3 16t2 () f(x) = 41 3x 7 (j) f(s) = 31 s s 2 16s 3. For eac of te following, make use of te prouct rule in fining te erivative of te epenent variable wit respect to te inepenent variable. (a) s = (t 2 6t + 3)(8t 4 + 6t 2 7) (b) q = (13t 4 + 5t)(3t 5 + 4t t 31) (c) y = (x 2 2x + 3)(2x x 6)(3x 2 4x + 1) () z = (x2 3x + 6)(8x 2 + 3x 2) x 2 6

9 Section 3.3 Differentiation of Polynomials an Rational Functions 9 4. Suppose f(2) = 2, f (2) = 6, g(2) = 3, an g (2) = 4. Fin k (2) for eac of te following. (a) k(x) = f(x) (b) k(x) = f(x) (c) k(x) = f(x)() 2 f(x) f(x) () k(x) = 5. Suppose an object moves along te x-axis so tat its position at time t is x = t + t3 6. (a) Fin te velocity, v(t) = ẋ(t), of te object. (b) Wat is v(0)? Wat oes tis say about te irection of motion of te object at time t = 0? (c) Wen is te object at te origin? Wat is te velocity of te object wen it is at te origin? () For wat values of t is te object moving towar te rigt? (e) For wat values of t is te object moving towar te left? (f) Wat is appening at te points were v(t) = 0? (g) Fin te acceleration of te object, a(t) = v(t). () Wen is te acceleration positive? Wen is it negative? (i) Notice tat v(1) < 0 an a(1) > 0. Wat oes tis say about te motion at time t = 1? 6. (a) Using only te prouct rule an te fact tat x = 1, sow tat x x x2 = 2x. (b) Now use te prouct rule to sow tat x x3 = 3x 2. (c) Let n > 1 an suppose we know tat x xm = mx m 1 for all m < n. Use te prouct rule to sow tat x xn = nx n 1.

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

More information

CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

More information

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,

More information

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line. Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

More information

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

More information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

More information

Math 113 HW #5 Solutions

Math 113 HW #5 Solutions Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

More information

ACT Math Facts & Formulas

ACT Math Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

More information

4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a

4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a Capter 4 Real Analysis 281 51. Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L

More information

Instantaneous Rate of Change:

Instantaneous Rate of Change: Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

More information

Rules for Finding Derivatives

Rules for Finding Derivatives 3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to

More information

20. Product rule, Quotient rule

20. Product rule, Quotient rule 20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that

More information

SAT Subject Math Level 1 Facts & Formulas

SAT Subject Math Level 1 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

More information

6. Differentiating the exponential and logarithm functions

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

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)

More information

Tangent Lines and Rates of Change

Tangent Lines and Rates of Change Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

More information

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

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1 Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

More information

The EOQ Inventory Formula

The EOQ Inventory Formula Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

More information

Chapter 7 Numerical Differentiation and Integration

Chapter 7 Numerical Differentiation and Integration 45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

More information

2 Limits and Derivatives

2 Limits and Derivatives 2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

More information

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

More information

SAT Math Facts & Formulas

SAT Math Facts & Formulas Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

More information

Average and Instantaneous Rates of Change: The Derivative

Average and Instantaneous Rates of Change: The Derivative 9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

CHAPTER 7. Di erentiation

CHAPTER 7. Di erentiation CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

2.1: The Derivative and the Tangent Line Problem

2.1: The Derivative and the Tangent Line Problem .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

More information

Compute the derivative by definition: The four step procedure

Compute the derivative by definition: The four step procedure Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function

More information

SAT Math Must-Know Facts & Formulas

SAT Math Must-Know Facts & Formulas SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas

More information

Differentiability of Exponential Functions

Differentiability of Exponential Functions Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an

More information

The Quick Calculus Tutorial

The Quick Calculus Tutorial The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,

More information

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution 1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

Using a table of derivatives

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

More information

Calculus Refresher, version 2008.4. c 1997-2008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/

Calculus Refresher, version 2008.4. c 1997-2008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Calculus Refresher, version 2008.4 c 997-2008, Paul Garrett, garrett@math.umn.eu http://www.math.umn.eu/ garrett/ Contents () Introuction (2) Inequalities (3) Domain of functions (4) Lines (an other items

More information

The modelling of business rules for dashboard reporting using mutual information

The modelling of business rules for dashboard reporting using mutual information 8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

More information

Geometric Stratification of Accounting Data

Geometric Stratification of Accounting Data Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

More information

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

More information

Chapter 10: Refrigeration Cycles

Chapter 10: Refrigeration Cycles Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area: Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force

More information

New Vocabulary volume

New Vocabulary volume -. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

College Planning Using Cash Value Life Insurance

College Planning Using Cash Value Life Insurance College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded

More information

An inquiry into the multiplier process in IS-LM model

An inquiry into the multiplier process in IS-LM model An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

More information

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat

More information

In other words the graph of the polynomial should pass through the points

In other words the graph of the polynomial should pass through the points Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

More information

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q = Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))

More information

Partial Fractions. p(x) q(x)

Partial Fractions. p(x) q(x) Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

Writing Mathematics Papers

Writing Mathematics Papers Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

More information

Definition of derivative

Definition of derivative Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example

More information

1 Lecture: Integration of rational functions by decomposition

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

More information

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

More information

The Derivative. Philippe B. Laval Kennesaw State University

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

More information

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Differentiation of vectors

Differentiation of vectors Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

More information

To differentiate logarithmic functions with bases other than e, use

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

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); 1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1 Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

More information

APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS

APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö

More information

Introduction to Integration Part 1: Anti-Differentiation

Introduction to Integration Part 1: Anti-Differentiation Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

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

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

More information

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

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

More information

Math Test Sections. The College Board: Expanding College Opportunity

Math Test Sections. The College Board: Expanding College Opportunity Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1) Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

More information

Equilibria in sequential bargaining games as solutions to systems of equations

Equilibria in sequential bargaining games as solutions to systems of equations Economics Letters 84 (2004) 407 411 www.elsevier.com/locate/econbase Equilibria in sequential bargaining games as solutions to systems of equations Tasos Kalandrakis* Department of Political Science, Yale

More information

How To Ensure That An Eac Edge Program Is Successful

How To Ensure That An Eac Edge Program Is Successful Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

More information

Limits and Continuity

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

More information

Practice with Proofs

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

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

More information

TOPIC 4: DERIVATIVES

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

More information

MATH 221 FIRST SEMESTER CALCULUS. fall 2007

MATH 221 FIRST SEMESTER CALCULUS. fall 2007 MATH 22 FIRST SEMESTER CALCULUS fall 2007 Typeset:December, 2007 2 Math 22 st Semester Calculus Lecture notes version.0 (Fall 2007) This is a self contained set of lecture notes for Math 22. The notes

More information

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade? Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

More information

13 PERIMETER AND AREA OF 2D SHAPES

13 PERIMETER AND AREA OF 2D SHAPES 13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a two-dimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter

More information

Distances in random graphs with infinite mean degrees

Distances in random graphs with infinite mean degrees Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

Theoretical calculation of the heat capacity

Theoretical calculation of the heat capacity eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

More information

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2 MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we

More information

26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order 26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information