Lecture 6: Calculus. In Song Kim. September 7, 2011

Size: px
Start display at page:

Download "Lecture 6: Calculus. In Song Kim. September 7, 2011"

Transcription

1 Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear function, y = a + b at a point is: m = y = y 2 y 2 = f( 2) f( ) 2 Note that the slope how the function changes as we move across values of oes not change. For a general curve, the slope varies from point to point. Definition A secant line of a curve is a line that intersects two or more points on the curve. Eample f() = 2. The slope of the secant between (0, 0) an (, ) is f( 2) f( ) 2 = =. The slope of the secant between (, ) an (2, 4) is f( 2) f( ) 2 = 3 = 3. How to calculate the slope of a curve at a certain point when this curve is not a straight line? Notice that we can raw many secants...ones that are closer to our point of interest give us a better approimation. Wouln t it be a reasonable iea to take a secant line an evaluate it at two points very close to the point we want to obtain the slope at?. Limit Review We have previously talke a little bit about limits. But lets o a little review an be a bit more technical as they will be important for us in talking about erivatives. Does a function f approach some number L as its input variable goes to some number c (often 0 or ± )? If so f() approaches L as approaches c. Formally we say lim f() = L. Eamples:. lim k = k Please o not istribute without permission. Ph.D. caniate, Department of Politics, Princeton University, Princeton NJ insong@princeton.eu

2 2. lim = c 3. lim = lim 2 = 6 3 Uniqueness: lim f() = L an lim f() = M L = M.. Properties of Limits Let f an g be functions with lim f() = A an lim g() = B. lim [f() + g()] = lim f() + lim g() = A + B What property oes this look like that we saw before? lim αf() = α lim f() = αa lim f()g() = [lim f()][lim g()] = AB lim f() Eample 2 g() = lim f() lim g() = A B, provie B 0 lim(2 3) = 2 lim 3 lim = = lim n = [lim ] [lim ] = c c = c n Eample 3 Limit eamples. Calculate the following limits at your esk. lim 2 ( ) = 5 5 lim 2 ( ) = 5 39 lim ( 3)( +3) lim 9 ( 9)(( +3)) ( lim + (factor an reuce, 0 ) 6 ) 3 + ln( + ) (get 0* at first...what to o? We ll have to use something calle L Hopital s Rule that we ll learn in a bit).2 Sequence Review Another helpful concept that we have seen a little of previously is the sequence. A sequence {y n } = {y, y 2, y 3,..., y n } is an orere set of real numbers, where y is the first term in the sequence an y n is the nth term. Generally, a sequence etens to n =. We can also write the sequence as {y n } n=. Sequences are similar to functions. Before, we ha y = f() with specifie over some omain. Now we have {y n } = {f()} with each value of having its own ine, n =, 2, 3,.... Thus the first number we put into our function gives us y. Three kins of sequences: 2

3 . Sequences that converge to a limit. 2. Sequences that increase or ecrease without boun. 3. Sequences like that neither converge nor increase without boun alternating over the number line. *Boar eamples of all 3*.2. The Limit of a Sequence We re often intereste in whether a sequence converges to a limit. Limits of sequences are conceptually similar to the limits of functions aresse in the previous lecture. Definition 2 (Limit of a sequence). The sequence {y n } has the limit L, that is lim n y n = L, if for any ɛ > 0 there is an integer N (which epens on ɛ) with the property that y n L < ɛ for each n > N. {y n } is sai to converge to L. If the above oes not hol, then {y n } iverges. Eample 4 { }. lim 2 n n = 2 2 Uniqueness: If {y n } converges, then the limit L is unique. Properties: Let lim n y n = A an lim n z n = B. Then. lim [αy n + βz n ] = αa + βb n 2. lim y nz n = AB n y 3. lim n n zn = A B, provie B 0 Fining the limit of a sequence in R n is similar to that in R. 2 The Derivative Definition 3 The erivative of a function f() is simply the slope of the secant of f() at a pair of points very close to (, f()): f f() f(a) f(a + h) f(a) (a) = lim = lim a a h 0 h This looks very similar to how we calculate the slope of a linear function. The erivative is evaluate at a point which converges to (i.e., h 0). Definition 4 A straight line is tangent to a curve, at some point, if both line an curve pass through the point with the same irection; such a line is the best straight-line approimation to the curve at that point. Eample 5 Graphical boar eamples Notation The slope of the tangent line of f() at any point is calle the erivative an we enote it f f() (), or in Leibniz notation :. Think of this as saying how is f() changing f() with an infinitesimally small change in. 3

4 3 Calculating Derivatives 3. Calculating Derivatives Lets start with the simple case where we have a function f() = k. Then the erivative is f () = k k. For eample, f() = 2, f () = 2. You ll use this over an over, but lets prove that this is actually the case. Before proving this, though, we nee a lemma. A lemma is an aitional statement (often proven elsewhere) that we nee in orer to prove the thing we are intereste in. Lemma 3. (Binomial epansion): For any positive integer k, ( + h) k = k + a k h +...a k h k + a k h k, where a j = k! j!(k j)!, for j =,...k Note a = k, a 2 = k(k ) 2, a k = For eample, ( + 3) 2 = With this in min lets simply apply this lemma to our epression of the erivative. Instea of using the notation f(), lets just use the k part: Proposition 3.2 f() = k f () = k k 3.2 Rules for calculating erivatives Theorem 3.3 (Algebraic Operations of Derivatives) Let f, g : X R be ifferentiable at c X an X R. Then,. (kf) (c) = kf (c) for all k R, 2. (f + g) (c) = f (c) + g (c), 3. (fg) (c) = f (c)g(c) + f(c)g (c) (Prouct Rule), ( ) 4. f g (c) = f (c)g(c) f(c)g (c) for g(c) 0 (Quotient Rule). g(c) 2 Eercise Prove (fg) (c) = f (c)g(c) + f(c)g (c) (Prouct Rule) ( ) Eercise 2 Prove f g (c) = f (c)g(c) f(c)g (c) for g(c) 0 (Quotient Rule). g(c) 2 Eample 6 4 Calculate the erivative of the following functions ( )2 (3 + 2)( 2 + ) 4

5 3.3 Chain Rule Reminer: A composite function is a function whose value epens on the output of another function. f() = 2, g() = (2 + ), hence f(g()) = (2 + ) 2 Definition 5 The Chain Rule is a formula for the erivative of the composite of two functions. In intuitive terms, if a variable, y, epens on a secon variable, u, which in turn epens on a thir variable,, then the rate of change of y with respect to can be compute as the prouct of the rate of change of y with respect to u multiplie by the rate of change of u with respect to. The chain rule may be state in any of several equivalent forms: Theorem 3.4 Suppose h() = f g. Then h () = (f(g())) = f (g())g (). or in the Leibniz notation: f = f u u. (Note: There shoul be some conitions for this theorem to work such as continuity of f an the eistence of f. For now, let s assume all the necessary conitions are met.) 4 Using Derivatives to Analyze Functions Lemma 4. A function is strictly increasing at if f is ifferentiable an its erivative is positive at. That is if f () > 0 Lemma 4.2 A function is strictly ecreasing at if f is ifferentiable an its erivative is negative at. That is if f () < 0 4. Steps for characterizing whether function is increasing or ecreasing. Calculate erivative 2. Fin where erivative is equal to 0 by solving f () = Sub in values of to the left an right of these points (or point) into the erivative f () an check sign 4. If positive then increasing in that region, if negative then ecreasing in that region Eample 7 f() = f () = 2 2, since 2 2 > 0 whenever >, f is increasing if >, an ecreasing if. f() = f() = f() =

6 5 The Secon Derivative In several of the eamples we i, we saw that the value of the erivative epene on the value of. Hence we also want to ask questions about how the erivative itself changes as a function of. I.e., we want to take the erivative of the erivative. Definition 6 The erivative of the first erivative is the secon erivative. We often use the notation f () or 2 y. The secon erivative is the slope of the line tangent to the first erivative 2 at the point. We can think of this as the change in change of the function. In physics, the first erivative is the spee of an object while the secon erivative is the acceleration of an object. 5. Calculating the 2n Derivative Simply take the erivative of the first erivative. All rule for ifferentiation continue to apply. Write neatly, things can get messy an complicate. 5.2 Using the 2n Derivative The secon erivative allows us to more completely characterize the behavior of a function. While the first erivative tells us whether a function is increasing or ecreasing at some point, it oesn t tell us whether the pace of increase/ecrease is changing. Two important properties of a function can be checke with the secon erivative. Definition 7 A function is conve (or concave up) in a region if a secant line in any two points of the region is above f. Formally, the function f : A R, efine on the conve set A R n is conve if f(α + ( α)) αf( ) + ( α)f() an A an all α [0, ]. Definition 8 A function is concave in a region if a secant line in any two points of the region is below f. Formally, the function f : A R, efine on the conve set A R n is concave if f(α + ( α)) αf( ) + ( α)f() an A an all αin[0, ]. Eample 8 boar eample To know if a function is conve, we o not nee to graph it or figure out the slope of all secant lines through any two of its points. We just check if the secon erivative is positive. Definition 9 A function is conve in a region if f () > 0 in that region. Definition 0 A function is concave in a region if f () < 0 in that region. Eample 9 f() = 2 f () = 2 f () = 2 > 0. So this function is conve everywhere. f() = 2 f () = 2 f () = 2 < 0. So this function is concave everywhere. 6

7 6 Graphing Using the first an secon erivatives we can sketch the graph of a function an ientify several important properties of the function. Steps to graph a function:. First fin the points at which f () = 0 or f is not efine. Such points are calle critical points of f. 2. Evaluate the function at each of these critical points an plot them in the graph. 3. Then, check the sign of f for each of the intervals efine by these critical points. 4. If f > 0 then raw the graph increasing over I, if f < 0 then raw the graph ecreasing over I. 5. Fin the points at which f () = 0 or f is not efine. Such points are calle secon orer critical points of f, or if the secon erivative actually changes sign there, inflection points of f. 6. Then, check the sign of f for each of the intervals efine by these critical points. 7. If f > 0 then raw the graph concave up (or conve) over I, if f < 0 then raw the graph concave own (or concave) over I. Eample 0 f() = f > 0 So the function is always increasing. f () = 6 So the function in concave for < 0 an conve for > 0. f(0) = 0, f (0) = 3 so slope 3 at the origin. Note: Draw in class. f() = (/3) f () 9 = ( + 3)( 3) So increasing on (, 3) an (3, ) an ecreasing on ( 3, 3). Note: Draw in class. Many times we will be intereste in fining the first an secon orer critical points in orer to fin maima an minima of a function. However, before we move on to this we will now iscuss several aitional rule of ifferentiation that will frequently arise. 6. Asymptotes Vertical asymptotes occur at points where the function is not efine. For eample f() = is not efine at = 0. 7

8 7 Optimization We can use calculus to easily fin the minimums an maimums of a function. Definition Looking at the graph, we note that the maimum an minimums occur where the function changes from being increasing to being ecreasing an vice versa. Since the erivative is positive when f is increasing, an negative when f is ecreasing, these points of minimums or maimums occur when the erivative is equal to 0. The points where f = 0 are calle Critical Points. Eample Fin the critical points of f() = f = 2 3 ( 2 ) = 2 3 ( )(+). So the critical points are at = 0, =, =. All the (interior) maimums an minimums are foun at critical points. The secon erivative helps etermine if a critical point is a maimum, a minimum, or neither. If f ( 0 ) = 0 an f ( 0 ) < 0 then 0 is a maimum of f. If f ( 0 ) = 0 an f ( 0 ) > 0 then 0 is a minimum of f. If f ( 0 ) = 0 an f ( 0 ) = 0 then we o not know, it might be a ma, a min or neither (These are calle sale points). See eamples. Eample 2 f() = f critical points at = 0, =, =. f f (0) = 0 f () = 24 f ( ) = 24 So local mins at = an =. Note: Make a table an to stuy the sign of f to make the graph of this function. Eample 3 f() = 3 f 2 Critical point at = 0 everywhere else positive so f increasing f () = 6 so f (0) = 0. Note: Draw graph. 7. Looking for Minimums an Maimums Steps:. Take erivative 2. Fin the such that the erivative function= 0 8

9 3. Evaluate the secon erivative at those critical points to etermine if at that there is a minimum or a maimum. Eample 4 ma 3 3 f 2 3 f () = 0 when = ± f () = 6, so f () = 6 > 0 an f ( ) = 6 < 0 So there is a local ma at = an a local min at =. Eample 5 f () = 2(+) 2 (+) 2 f () = 0 when = 0 an = 2 f () = (2+2)(+)2 2( 2 +2)(+) (+) 4 f (0) = 2 > 0 So there is a local min at = 0 f ( 2) = 2 < 0 So there is a local ma at = 2. ma 2 + Eample 6 Eample from Economics: Profit Maimizing [skippe] Let p enote the price the firm obtains for each of the units it prouces. Let c(q) enote the cost function of the firm. The cost function refers to the total cost of proucing q units. The erivative of the cost function is calle the Marginal Cost, it refers to the cost of proucing aitional unit when q nunits have been prouce. Let q enote the quantity prouce. This is the ecision variable of the firm. The problem of the firm can be enote as: ma pq c(q) q We solve it using calculus.we calculate the First Orer Conition for optimization, namely the q such that the first erivative equals 0. Then we obtain the secon erivative, evaluate it at q an check that it is negative, that way we know we are at a maimum. FOC: p = c (q) This is interprete as Marginal revenue (p) equals Marginal Cost. SOC: c (q) > 0 This is saying that we nee to have increasing marginal costs at q for this to be a maimum. This is the typical assumption of the perfect competition moel you will see in the fall. Note: Do the following numerical eample: Obtain the profit maimizing prouction when c(q) = q an p=5. 9

10 8 Aitional Differentiation Rules: Chain Rule an Eponential 8. Eamples f() = f () = 2 2 f() = f() = 2 2 ln() f() = (2 +8) f() = (e +) 3 2 f() = ( 2 + 8) 3 f() = 3ln(6 3 7) f() = g(3 2 7) f() = g()h() g(6) 8.2 Eponential an Log Derivatives We will frequently encounter logs (ln ) an eponentials (e ) in our stuies. Derivatives of these functions have special properties Derivatives of the natural log ln. ln = 2. ln k = k ln = k 3. ln u() = u () u() (by the chain rule) Lets prove property, as it is very important (S/B 94). Start with the stanar efinition, ln( + h) ln() h = h ln( + h ) = ln( + h ) h = ln( + Now we will use a common tactic to help simplify things, which is to efine a new variable as a more complicate function of something we alreay have. Let m = h. As h 0, m. Hence we have lim (ln( + m m )m ) = ln( lim ( + m m )m ) h ) h 0

11 We can interchange the limit an ln only because ln is a continuous function. I.e., if m o then ln m ln o. Furthermore, the efinition of e was e = lim m ( + m )m. Further, we ha the ientity e r = lim m ( + r m )m. Thus we let r =. We have Proposition 8. ln( lim ( + m m )m ) = ln e = f() = ln f () = Eample 7 ln( + ) Show lim = Derivatives of the eponential function: e. αe = αe 2. eu() = e u() u () *graph eamples on the boar* *how coul we prove?* Eercise 3 Suppose f() = log a. Show f () = lna Eamples: Fin y/ for. y = ln(4) 2. y = ln(e 2 ) 3. y = ln(ln ) 4. y = ln 2 ln 4 e Proposition 8.2 For any positive base b, a = (ln a) (a ). Eample 8 f() = 0 Take natural log on both sie. ln(f()) = ln 0 Taking the erivative of both sies we have f () f() = ln 0 f () = f() ln 0 Proposition 8.3 f() = e f () = e

12 8.3 L Hospital s Rule ( ) In stuying limits, we saw that lim f()/g() = lim f() g() 0, which will cause the limit to be unboune. lim ( ) / lim g(), provie that If both lim f() = 0 an lim g() = 0, then we get an ineterminate form of the type 0/0 as c. However, we can still analyze such limits using L Hospital s rule. Theorem 8.4 L Hospital s Rule: Suppose f an g are ifferentiable on a < < b an that either. lim f() = 0 an lim g() = 0, or a + a + 2. lim f() = ± an lim g() = ± a + a + Suppose further that g () is never zero on a < < b an that f () lim a + g () = L then f() lim a + g() = L Eample 9 Use L Hospital s rule to fin the following limits: Suppose > 0. lim 0 r take erivative of top an bottom wrt to : We have from the power rule above that lim 0 r r = lim 0 r ln r = ln r lim 0 r = ln r 9 Derivatives of Trigonometric Functions (sin) = cos (cos) = sin (tan) = sec2 (csc) = csc cot (sec) = sec tan (cot) = csc2 2

Math 1271 Solutions for Fall 2005 Final Exam

Math 1271 Solutions for Fall 2005 Final Exam Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly

More information

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin

More information

Calculus in the AP Physics C Course The Derivative

Calculus in the AP Physics C Course The Derivative Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.

More information

Section 2.1 The Derivative and the Tangent Line Problem

Section 2.1 The Derivative and the Tangent Line Problem Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

Chapter 2 Derivatives

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,

More information

MATH 205 Practice Final Exam Name:

MATH 205 Practice Final Exam Name: MATH 205 Practice Final Eam Name:. (2 points) Consier the function g() = e. (a) (5 points) Ientify the zeroes, vertical asymptotes, an long-term behavior on both sies of this function. Clearly label which

More information

3.2 Differentiability

3.2 Differentiability Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability

More information

Solutions to Practice Problems Tuesday, October 28, 2008

Solutions to Practice Problems Tuesday, October 28, 2008 Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what

More information

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0. Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?

More information

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)

x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x) Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane

More information

Differentiation ( , 9.5)

Differentiation ( , 9.5) Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the

More information

Differentiability, Computing Derivatives, Trig Review

Differentiability, Computing Derivatives, Trig Review Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute

More information

Chapter 1 Overview: Review of Derivatives

Chapter 1 Overview: Review of Derivatives Chapter Overview: Review of Derivatives The purpose of this chapter is to review the how of ifferentiation. We will review all the erivative rules learne last year in PreCalculus. In the net several chapters,

More information

Differentiability, Computing Derivatives, Trig Review. Goals:

Differentiability, Computing Derivatives, Trig Review. Goals: Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an

More information

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain

More information

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0. Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

Define each term or concept.

Define each term or concept. Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

MATH2231-Differentiation (2)

MATH2231-Differentiation (2) -Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha

More information

Chapter Primer on Differentiation

Chapter Primer on Differentiation Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.

More information

Final Exam Study Guide and Practice Problems Solutions

Final Exam Study Guide and Practice Problems Solutions Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making

More information

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we

More information

Tutorial 1 Differentiation

Tutorial 1 Differentiation Tutorial 1 Differentiation What is Calculus? Calculus 微積分 Differential calculus Differentiation 微分 y lim 0 f f The relation of very small changes of ifferent quantities f f y y Integral calculus Integration

More information

Integration Review. May 11, 2013

Integration Review. May 11, 2013 Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In

More information

by using the derivative rules. o Building blocks: d

by using the derivative rules. o Building blocks: d Calculus for Business an Social Sciences - Prof D Yuen Eam Review version /9/01 Check website for any poste typos an upates Eam is on Sections, 5, 6,, 1,, Derivatives Rules Know how to fin the formula

More information

Summary: Differentiation

Summary: Differentiation Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin

More information

(x,y) 4. Calculus I: Differentiation

(x,y) 4. Calculus I: Differentiation 4. Calculus I: Differentiation 4. The eriatie of a function Suppose we are gien a cure with a point lying on it. If the cure is smooth at then we can fin a unique tangent to the cure at : If the tangent

More information

February 21 Math 1190 sec. 63 Spring 2017

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

More information

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph

More information

Math Camp II. Calculus. Yiqing Xu. August 27, 2014 MIT

Math Camp II. Calculus. Yiqing Xu. August 27, 2014 MIT Math Camp II Calculus Yiqing Xu MIT August 27, 2014 1 Sequence and Limit 2 Derivatives 3 OLS Asymptotics 4 Integrals Sequence Definition A sequence {y n } = {y 1, y 2, y 3,..., y n } is an ordered set

More information

Section 7.1: Integration by Parts

Section 7.1: Integration by Parts Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the

More information

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects

More information

1 Definition of the derivative

1 Definition of the derivative Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms

More information

A. Incorrect! The letter t does not appear in the expression of the given integral

A. Incorrect! The letter t does not appear in the expression of the given integral AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question

More information

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

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

More information

(a 1 m. a n m = < a 1/N n

(a 1 m. a n m = < a 1/N n Notes on a an log a Mat 9 Fall 2004 Here is an approac to te eponential an logaritmic functions wic avois any use of integral calculus We use witout proof te eistence of certain limits an assume tat certain

More information

Module FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information

Module FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information 5548993 - Further Pure an 3 Moule FP Further Pure 5548993 - Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions

More information

Math 115 Section 018 Course Note

Math 115 Section 018 Course Note Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of

More information

Integration by Parts

Integration by Parts Integration by Parts 6-3-207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an

More information

THEOREM: THE CONSTANT RULE

THEOREM: THE CONSTANT RULE MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.

More information

102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2

102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2 Problems Calculus AB Stuents Shoul Know: Solutions. + ) = + =. chain rule ) e = e = e. ) =. ) = ln.. + + ) = + = = +. ln ) =. ) log ) =. sin ) = cos. cos ) = sin. tan ) = sec. cot ) = csc. sec ) = sec

More information

The Natural Logarithm

The Natural Logarithm The Natural Logarithm -28-208 In earlier courses, you may have seen logarithms efine in terms of raising bases to powers. For eample, log 2 8 = 3 because 2 3 = 8. In those terms, the natural logarithm

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

5.4 Fundamental Theorem of Calculus Calculus. Do you remember the Fundamental Theorem of Algebra? Just thought I'd ask

5.4 Fundamental Theorem of Calculus Calculus. Do you remember the Fundamental Theorem of Algebra? Just thought I'd ask 5.4 FUNDAMENTAL THEOREM OF CALCULUS Do you remember the Funamental Theorem of Algebra? Just thought I' ask The Funamental Theorem of Calculus has two parts. These two parts tie together the concept of

More information

Differentiation Rules Derivatives of Polynomials and Exponential Functions

Differentiation Rules Derivatives of Polynomials and Exponential Functions Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)

More information

23 Implicit differentiation

23 Implicit differentiation 23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For

More information

By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)

By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2) 3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power

More information

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) =

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) = Theorem 13 (i) If p(x) is a polynomial, then p(x) = p(c) 1 Limits 11 12 Fining its graphically Examples 1 f(x) = x3 1, x 1 x 1 The behavior of f(x) as x approximates 1 x 1 f(x) = 3 x 2 f(x) = x+1 1 f(x)

More information

The derivative of a constant function is 0. That is,

The derivative of a constant function is 0. That is, NOTES 3: DIFFERENTIATION RULES Name: Date: Perio: LESSON 3. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Eample : Prove f ( ) 6 is not ifferentiable at 4. Practice Problems: Fin f '( ) using the

More information

Derivatives and the Product Rule

Derivatives and the Product Rule Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives

More information

Lecture 16: The chain rule

Lecture 16: The chain rule Lecture 6: The chain rule Nathan Pflueger 6 October 03 Introuction Toay we will a one more rule to our toolbo. This rule concerns functions that are epresse as compositions of functions. The iea of a composition

More information

Math 210 Midterm #1 Review

Math 210 Midterm #1 Review Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function

More information

Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions 9-8-28 In this section, I ll iscuss its an erivatives of trig functions. I ll look at an important it rule first, because I ll use it in computing the erivative of

More information

Flash Card Construction Instructions

Flash Card Construction Instructions Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column

More information

SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3

SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3 SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 3 L P f Q L segments L an L 2 to be tangent to the parabola at the transition points P an Q. (See the figure.) To simplify the equations you ecie to place the

More information

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x . Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain

More information

The derivative of a constant function is 0. That is,

The derivative of a constant function is 0. That is, NOTES : DIFFERENTIATION RULES Name: LESSON. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Date: Perio: Mrs. Nguyen s Initial: Eample : Prove f ( ) 4 is not ifferentiable at. Practice Problems: Fin

More information

1 Lecture 20: Implicit differentiation

1 Lecture 20: Implicit differentiation Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation

More information

Solutions to Math 41 Second Exam November 4, 2010

Solutions to Math 41 Second Exam November 4, 2010 Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of

More information

Inverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that

Inverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that Inverse Functions Review from Last Time: The Derivative of y = ln Last time we saw that THEOREM 22.0.. The natural log function is ifferentiable an More generally, the chain rule version is ln ) =. ln

More information

lim Prime notation can either be directly applied to a function as previously seen with f x 4.1 Basic Techniques for Finding Derivatives

lim Prime notation can either be directly applied to a function as previously seen with f x 4.1 Basic Techniques for Finding Derivatives MATH 040 Notes: Unit Page 4. Basic Techniques for Fining Derivatives In the previous unit we introuce the mathematical concept of the erivative: f f ( h) f ( ) lim h0 h (assuming the limit eists) In this

More information

Additional Exercises for Chapter 10

Additional Exercises for Chapter 10 Aitional Eercises for Chapter 0 About the Eponential an Logarithm Functions 6. Compute the area uner the graphs of i. f() =e over the interval [ 3, ]. ii. f() =e over the interval [, 4]. iii. f() = over

More information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.

More information

Implicit Differentiation. Lecture 16.

Implicit Differentiation. Lecture 16. Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe

More information

Calculus Class Notes for the Combined Calculus and Physics Course Semester I

Calculus Class Notes for the Combined Calculus and Physics Course Semester I Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation - NSF-DUE-9752485 1 Section 0 2 Contents 1 Average

More information

UNDERSTANDING INTEGRATION

UNDERSTANDING INTEGRATION UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,

More information

Derivative of a Constant Multiple of a Function Theorem: If f is a differentiable function and if c is a constant, then

Derivative of a Constant Multiple of a Function Theorem: If f is a differentiable function and if c is a constant, then Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 1 Review of the Limit Definition of the Derivative Write the it efinition of the erivative function: f () Derivative of a Constant Multiple of a

More information

3.6. Implicit Differentiation. Implicitly Defined Functions

3.6. Implicit Differentiation. Implicitly Defined Functions 3.6 Implicit Differentiation 205 3.6 Implicit Differentiation 5 2 25 2 25 2 0 5 (3, ) Slope 3 FIGURE 3.36 The circle combines the graphs of two functions. The graph of 2 is the lower semicircle an passes

More information

18 EVEN MORE CALCULUS

18 EVEN MORE CALCULUS 8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;

More information

AP Calculus AB Ch. 2 Derivatives (Part I) Intro to Derivatives: Definition of the Derivative and the Tangent Line 9/15/14

AP Calculus AB Ch. 2 Derivatives (Part I) Intro to Derivatives: Definition of the Derivative and the Tangent Line 9/15/14 AP Calculus AB Ch. Derivatives (Part I) Name Intro to Derivatives: Deinition o the Derivative an the Tangent Line 9/15/1 A linear unction has the same slope at all o its points, but non-linear equations

More information

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam.

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam. MATH 2250 Calculus I Date: October 5, 2017 Eric Perkerson Optimization Notes 1 Chapter 4 Note: Any material in re you will nee to have memorize verbatim (more or less) for tests, quizzes, an the final

More information

Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10

Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10 Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10 This is a set of practice test problems for Chapter 4. This is in no way an inclusive set of problems there can be other types of problems

More information

Fall 2016: Calculus I Final

Fall 2016: Calculus I Final Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe

More information

Further Differentiation and Applications

Further Differentiation and Applications Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle

More information

0.1 The Chain Rule. db dt = db

0.1 The Chain Rule. db dt = db 0. The Chain Rule A basic illustration of the chain rules comes in thinking about runners in a race. Suppose two brothers, Mark an Brian, hol an annual race to see who is the fastest. Last year Mark won

More information

1. Find the equation of a line passing through point (5, -2) with slope ¾. (State your answer in slope-int. form)

1. Find the equation of a line passing through point (5, -2) with slope ¾. (State your answer in slope-int. form) INTRO TO CALCULUS REVIEW FINAL EXAM NAME: DATE: A. Equations of Lines (Review Chapter) y = m + b (Slope-Intercept Form) A + By = C (Stanar Form) y y = m( ) (Point-Slope Form). Fin the equation of a line

More information

Computing Derivatives Solutions

Computing Derivatives Solutions Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use

More information

More from Lesson 6 The Limit Definition of the Derivative and Rules for Finding Derivatives.

More from Lesson 6 The Limit Definition of the Derivative and Rules for Finding Derivatives. Math 1314 ONLINE More from Lesson 6 The Limit Definition of the Derivative an Rules for Fining Derivatives Eample 4: Use the Four-Step Process for fining the erivative of the function Then fin f (1) f(

More information

Exam 2 Review Solutions

Exam 2 Review Solutions Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon

More information

FINAL EXAM 1 SOLUTIONS Below is the graph of a function f(x). From the graph, read off the value (if any) of the following limits: x 1 +

FINAL EXAM 1 SOLUTIONS Below is the graph of a function f(x). From the graph, read off the value (if any) of the following limits: x 1 + FINAL EXAM 1 SOLUTIONS 2011 1. Below is the graph of a function f(x). From the graph, rea off the value (if any) of the following its: x 1 = 0 f(x) x 1 + = 1 f(x) x 0 = x 0 + = 0 x 1 = 1 1 2 FINAL EXAM

More information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x = 6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we

More information

Limits and Their Properties

Limits and Their Properties Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus

More information

Linear and quadratic approximation

Linear and quadratic approximation Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function

More information

2.1 Derivatives and Rates of Change

2.1 Derivatives and Rates of Change 1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x 2 0 2 x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an

More information

Chapter 1. Functions, Graphs, and Limits

Chapter 1. Functions, Graphs, and Limits Review for Final Exam Lecturer: Sangwook Kim Office : Science & Tech I, 226D math.gmu.eu/ skim22 Chapter 1. Functions, Graphs, an Limits A function is a rule that assigns to each objects in a set A exactly

More information

MA Midterm Exam 1 Spring 2012

MA Midterm Exam 1 Spring 2012 MA Miterm Eam Spring Hoffman. (7 points) Differentiate g() = sin( ) ln(). Solution: We use the quotient rule: g () = ln() (sin( )) sin( ) (ln()) (ln()) = ln()(cos( ) ( )) sin( )( ()) (ln()) = ln() cos(

More information

CHAPTER 3 DERIVATIVES (continued)

CHAPTER 3 DERIVATIVES (continued) CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The

More information

The Explicit Form of a Function

The Explicit Form of a Function Section 3 5 Implicit Differentiation The Eplicit Form of a Function The normal way we see function notation has f () on one sie of an equation an an epression in terms of on the other sie. We know the

More information

Linear First-Order Equations

Linear First-Order Equations 5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)

More information

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,

More information

y. ( sincos ) (sin ) (cos ) + (cos ) (sin ) sin + cos cos. 5. 6.. y + ( )( ) ( + )( ) ( ) ( ) s [( t )( t + )] t t [ t ] t t s t + t t t ( t )( t) ( t + )( t) ( t ) t ( t ) y + + / / ( + + ) / / /....

More information

Economics 205 Exercises

Economics 205 Exercises Economics 05 Eercises Prof. Watson, Fall 006 (Includes eaminations through Fall 003) Part 1: Basic Analysis 1. Using ε and δ, write in formal terms the meaning of lim a f() = c, where f : R R.. Write the

More information

Formulas From Calculus

Formulas From Calculus Formulas You Shoul Memorize (an I o mean Memorize!) S 997 Pat Rossi Formulas From Calculus. [sin ()] = cos () 2. [cos ()] = sin () 3. [tan ()] = sec2 () 4. [cot ()] = csc2 () 5. [sec ()] = sec () tan ()

More information

Antiderivatives and Indefinite Integration

Antiderivatives and Indefinite Integration 60_00.q //0 : PM Page 8 8 CHAPTER Integration Section. EXPLORATION Fining Antierivatives For each erivative, escribe the original function F. a. F b. F c. F. F e. F f. F cos What strateg i ou use to fin

More information

Trigonometric Functions

Trigonometric Functions 72 Chapter 4 Trigonometric Functions 4 Trigonometric Functions To efine the raian measurement system, we consier the unit circle in the y-plane: (cos,) A y (,0) B So far we have use only algebraic functions

More information

Derivatives and Its Application

Derivatives and Its Application Chapter 4 Derivatives an Its Application Contents 4.1 Definition an Properties of erivatives; basic rules; chain rules 3 4. Derivatives of Inverse Functions; Inverse Trigonometric Functions; Hyperbolic

More information