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# Calculus for Physical Scientists I (GT MATH 160

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This 11 page Study Guide was uploaded by Melvina Keeling on Monday September 21, 2015. The Study Guide belongs to MATH 160 at Colorado State University taught by Cassandra Williams in Fall. Since its upload, it has received 15 views. For similar materials see /class/210088/math-160-colorado-state-university in Mathematics (M) at Colorado State University.

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M 160 Study Guide for Chapter 3 Differentiation Summer 2008 This Study Guide describes everything you are expected to know understand and be able to do from Chapter 3 Questions on Exam 2 covering this material will ask you to do one or more of the tasks described in this Study Guide i State the definition of derivative of a function y fx at a specific point x c as a limit Illustrate and explain each part of the definition graphically using secant lines tangent lines etc Explain why the definition requires taking a limit and what the limit means in this setting Comment Definition of the derivative of a function is on page 147 Formula for the slope of a curve on page 137 also defines the derivative at a specific point Best graphical interpretation of the definition is on pages 135136 This idea is the subject of a Concept Quiz Explain how to tell from its graph whether a function is differentiable or not Given the graph of a function indicate the points xvalues where the function is differentiable and where it is not differentiable Estimate the derivative at points where the function is differentiable Pages 156 715739 35 7 44 The derivative of a function y fx is another function y f x Explain in terms of the graph of y x what the derivative y f x tells you Explain in terms of physical quantities eg time and position altitude and air pressure or something else what the function y f x tells you Given the graph of a function y x sketch the graph of the derivative y f x Representative homeworkproblems Page15639 27 7 30 33 34 Representative homework problems Page 14039 1 7 439 Use the definition of a function being differentiable at a point x c and the definition of continuity at a point to show that a function that has a derivative at a point x 0 must also be continuous at that point see Theorem 1 pg 154 Give examples by graphs andor equations of functions that are continuous but not differentiable at a point Representative homeworkproblems Page 157 7 15839 39 7 44 Use the definition of the derivative of a function at a specific point x c to determine whether a function given by a specific formula perhaps defined piecewise is or is not differentiable at a given point x 0 Explain how you see from the definition that the function is or is not differentiable at x c The language of onesided derivatives can be useful Confirm your conclusion by examining the graph of the function Representative homeworkproblems Page 15839 54 and 58 Page 236 65 7 67 i Write the differentiation formulas for the sum difference and product of two or more differentiable functions and for the quotient of two differentiable functions Use these differentiation formulas to find the value of the derivative of a combination of functions at a given point xvalue from information about the values of the functions and their first derivatives at that point Representative homeworkproblems Page 16939 39 40 Page 23639 55 56 i Use the differentiation formulas to calculate first and second derivatives of functions defined by expressions that involve constant multiples sums differences products andor quotients of power and root functions Representative homeworkproblems Page 16939 1 7 38 Page 23539 1 7 4 6 9 10 31 Find an equation in pointslope form for the line tangent to the graph of a differentiable function at a given point Find the points on the graph of a differentiable function where the tangent line has specified slope Representative homeworkproblems Page 14039 23 7 26 Page 16939 41 7 44 Given an expression for a function that describes the motion of a body along a straightline path a find the displacement and b find the average velocity of the body over a time interval Find functions that describe the velocity and acceleration of the moving body Then find the position the velocity and speed not the same as velocity and the acceleration of the body at a given time Representative homeworkproblems Page 17939 1 7 16 Page 18339 31 7 34 i From the graph of a function that describes the motion of a body along a straight line path make reasonable estimates of the position and velocity of the body at a given time and determine whether the velocity of the body is increasing or decreasing at that time Explain the basis for your estimates Representative homeworkproblems Page 181 7 18239 17 18 20 7 22 Given a function y fx that models the a physical situation eg volume as a function of time volume as a function of radius find the instantaneous rate of change of the dependent variable with respect to the Page 1 independent variable Use this result to analyze and answer questions about how the dependent variable changes as the independent variable changes Representative homeworkproblems Page 182 7 18339 25 7 30 Prerequisite facts from trigonometry 1 Sketch the graphs of the six basic trig functions with out a calculator 2 Express the sine cosine and tangent of angles in terms of the lengths of the sides of a right triangle 3 Know how to represent all six trig functions in terms of sine and cosine and the Pythagorean identityies 7 Use the graphs of the sine and cosine functions to explain how one might conjecture the formulas for the derivatives of the sine and cosine functions Derive the formulas for the derivatives of the tangent cotangent secant and cosecant from the derivatives of the sine and cosine 7 Calculate derivatives of functions formed from trigonometric functions and power functions with positive or negative rational exponents by adding subtracting multiplying dividing and forming compositions Find equations for tangent lines to graphs of these functions in pointslope form of course Find the points on the graph of a differentiable function where the tangent line has specified slope Illustrateverify graphically that your tangent lines are correct using a calculator Representative homeworkproblems Page 18839 1 7 26 age 20139 select problems involving trig functions from 9 7 48 Pages 235 7 23639 select problems involving trig functions from 11 7 40 Suggested problems for practicing nding tangent lines Page 18839 27 7 38 Page 23739 69 7 72 No exam questions will involve normal lines 7 Given a function that may be defined piecewise and may involve trigonometric functions use the definitions to determine whether the function is continuous and whether it is differentiable at specified points Given a function that may be defined piecewise by expressions involving parameters determine whether the parameters may be assigned values so as to make the function be continuous andor differentiable at specified points If the function has a removable discontinuity at a point x 0 determine what value should be assigned to fc so as to make the function be continuous at the point x c If the function is differentiable use the definition to find its derivative Representative homeworkproblems Page 18939 48 Page 23739 68 Page 24239 15 7 18 7 Given two functions f and g that may be the same function form the composite functions fog and g0 f Given a function h write it as a nontrivial composition fog perhaps in more than one way Nontrivial means neither f nor g is the identity function y x Representative homework problems Page 201 1 7 18 7 Write a complete statement of the Chain Rule as a theorem with hypothesis and conclusion 7 Use the Chain Rule perhaps in combination with other differentiation formulas to calculate derivatives of functions formed by adding subtracting multiplying dividing and composing constants power functions with positive or negative rational exponents and trigonometric functions Representative homeworkproblems Page 201 7 20239 select from 1 7 58 63 64 Page 21139 1 7 18 Pages 235 7 236 Practice Exercises select from 7 7 40 7 Given information about the values of two functions f and g and their derivatives at certain points find the value of the derivative of functions formed from f g andor additional specific functions by using algebraic operations andor composition Functions may not be given explicitly but information about function values and values of the derivative at specific points are given or can be found from graphs of the functions Representative homeworkproblems Page 20239 59 60 Page 23639 55 56 Study recommendation A Concept Quiz on the Chain Rule asked you to do this 7 Solve related rate problems in which time t is the independent variable there are two or three dependent variables and an equation connecting the dependent variables can be found by using at most two of the following relations a Pythagorean Theorem b similar triangles c area formulas for simple plane figures eg circle triangle rectangle parallelogram trapezoid d volume formulas for familiar solids eg sphere cylinder cone rectangular parallelepiped Page 2 e surface area formulas for a sphere or cube and f definition of the sine cosine or tangent function in terms of a right triangle Representative hamewarkprablems Pages 218 7 219 10 11 7 18 20 21 7 23 30 31 33 34 35 NOTE You are required to know the Pythagorean Theorem properties of similar triangles formulas for areas of simple plane figures formulas for volumes for spheres cylinders cones and rectangular parallelepipeds boxes and the definitions of the trigonometric functions in terms of right triangles Section 36 Implicit Differentiation This section will probably not be tested but may be assessed in a quiz 7 Given an equation F x y c c a constant the graph of this equation and a point x0 yo on the graph a identify e g by encircling or shading a portion of the graph that defines a function y fx Whose derivative gives the slope of the line tangent to the graph of F x y c at the point x0 yo See discussion and examples on pp 205 7 206 b find the derivative of the function y fx defined in a by implicit differentiation and c find the equation for the tangent line at x0 yo in pointslope form of course d Explain Why the formula for the derivative of the function y f x obtained by implicit differentiation the slope of the tangent line at each point on the graph F x y c not just at the point x0 yo used to get the function y f x Representative hamewarkprablems Page 211 47 7 56 and 59 7 62 perhaps with extended instructions 0 exam questions will involve the normal line 7 Given an equation F x y c c a constant that can be solved explicitly for y and given a point x0 yo on the graph of Fx y c a find an explicit expression for a differentiable function y fx that is defined implicitly by F x y c and Whose graph includes the point x0 yo b compute the derivative of the function y fx explicitly using the expression found in a c compute the derivative of the function y fx by implicit differentiation and d demonstrate that the results of b and c are the same Study Suggestion Study Examples 1 amp 2 on page 206 7 Calculate first and second derivatives by implicit differentiation Representative hamewarkprablems Page 211 19 7 36 Page 211 37 7 44 Page 3 MATH 160 Study Guide for Chapter 2 Limits amp Continuity Summer 2008 The rst midterm exam is Thursday June 26 Midterm I will include all ofChapter 2 This Study Guide describes everything you are expected to know understand and be able to do from Chapter 2 Exam questions on this material will ask you to do one or more of the tasks described in this Study Guide 7 Explain what it means to say that a function y x has limit L as x approaches a Relate your explanation to the formal 87 6 de nition 7 Illustrate e and 5 graphically as in the formal 2376 de nition of limit 7 Given that limx7c fx L where x is a speci c not necessarily linear function and L and c are speci c numbers and given a speci c error tolerance 8 nd an associated 5 graphically using a graphing calculator Representative problems Page 9839 7 7 14 Page 9939 in 15 7 26 nd 5 graphically 7 Given that limx7c fx L wherex mx b is a speci c linear function c is a speci c number andL may or may not be given and given a speci c error tolerance 8 nd the limitL if not given by inspection and nd an associated 5 Then show algebraically that the number 5 you found meets the requirements of the formal de nition 139 e that when 1x 7 cl lt 5 we have lx 7L1 lt e Representativeproblems Page 9939 l5 16 31 32 You can also make up problems like thisforyourself 7 Give examples that show why statements commonly used to describe limits are wrong Explain how your example shows these statements are wrong Representativeproblems Page 10039 53 54 7 Give graphical examples of functions that do not have a limit at a speci ed point x c Explain how to see from your graph that no matter what number L is chosen your example function is not related to the number L in the way the de nition of limx7c fx L requires Representativeproblems Page 10039 57 58 59 60 7 Explain the difference between evaluating the limit of a function y x as x approaches a and evaluating the function at x a Illustrate with examples Representative problems Page 8239 7 7 10 7 Interpret the limit concept and the formal de nition of limit in realworld situations 7 Given a function y x nd the average rate of change of the dependent variable with respect to the independent variable over a given interval Interpret the average rate of change physically in terms of motion and veloci d graphically as the slope of a secant line Explain the difference between average rate of change and instantaneous rate of change Representativeproblems Page 8339 29 7 34 and 35 7 40 Page 141 Questions to Guide Review 1 2 7 Explain why the idea of limits is needed Give examples of problems that can t be solved using only arithmetic and algebra but can be solved using the idea of a limit Explain how these problems are solved using limits Representativeproblems Page 10039 55 Page 14439 4 5 6 7 Evaluate limits as x approaches c graphically and numerically including the possibility that the limit might not exist Explain why a table showing exact values of a function y fx for x values close to c cannot in itself conclusively establish the value of the limit of the function as x approaches c Representativeproblems Page 8139 1 7 4 Page 8239 11 7 20 Page 9139 59a 60a 7 Use the Limit Theorems called Limit Laws in sec 22 to evaluate limits algebraically including the possibility that the limit might not exist or the limit may be in nite Explain speci cally how you used the Limit Theorems to evaluate a speci c limit One or more of the following algebraic techniques may be needed factoring and dividing out common factors addingsubtracting rational functions dividing rational functions andor numerical fractions rationalizng numerator or denominator Representativeproblems Page 8939 1 7 34 Page 9039 35 36 37 38 Page 9039 49 50 51 52 7 Given information about the limits of several functions as x approaches some number c use the Limit Theorems to nd the limits of combinations of these functions Representativeproblems Page 9039 39 7 42 Page 9139 55 7 58 Page 14539 14 20 7 Evaluate onesided limits graphically and algebraically Explain the connection between onesided limits and limits aka twosided limits Use information about the onesided limits to infer the existence or nonexistence of a limit Representativeproblems Pages 111 7 11339 1 718 Page 11439 63 64 Page 12239 1 26 Page 1 of 2 MATH 160 Study Guide for Chapter 2 Summer 2008 7 Explain what it means for a function to have a limit L as x approaches plus or minus in nity Use a graph to illustrate your explanation Explain what is meant by a horizontal asymptote of a graph Explain how limits as x approaches i so and horizontal asymptotes are related Evaluate limits as x approaches i 00 numerically including the possibility that the limit might not exist Explain why a table showing exact values of a function y fx for very large x values cannot in itself conclusively establish the value of the limit of the function as x approaches i 00 Representative problems Page 11339 37 7 46 Change instructions to t this statement Page 11439 70 Evaluate the limits of rational functions as x approaches in nity or minus in nity algebraically Show the details of the algebraic work as in Examples 8 and 9 page 109 Explain how to see that the limit of the ratio of two polynomials of the same degree is the ratio oftheir leading coef cients Representativeproblems Page 113 7 11439 47 7 56 Explain what it means when we say a function has in nity or minus in nity as its limit as x approaches a number a and as x approaches a number a from the le or from the right Explain the difference between having a limit and having an in nite limit Explain what is meant by a vertical asymptote in terms of limits Explain how in nite limits and vertical asymptotes are related Use graphs to illustrate your explanations 7 Sketch the graph of a function that has speci ed vertical and horizontal asymptotes and perhaps speci ed values at certain points Asymptotes may be given explicitly or have to be inferred from statements about limits Representative problems Page 12339 39 7 46 State the mathematical de nition of the phrase a function y x is continuous at x e Identify points where a function is continuous and where it is not continuous from the graph of the function Explain in terms of the de nition why the function is or is not continuous at the points in you identi ed Representativeproblems Page 13239 1 7 10 7 Give graphical examples of functions that i are not continuous at a point e but do have a limit as x approaches e39 ii are not continuous at a point e but are de ned at the point x e39 iii are not continuous at a point e but have both a le hand and a righthand limit as x approaches e If it is not possible to give such an example explain why 7 Use the facts theorems that algebraic combinations and compositions of continuous functions are continuous to explain us how to see that a given function is continuo Then use the fact that the function is continuous to justify evaluating its limit by substitution Representative problems Page 13339 11 7 28 Given a function that may be de ned piecewise use the de nition to determine whether the function is continuous at speci ed points Explain how your conclusion follows from the de nition If the function has a removable discontinuity at a point x e determine what value should be assigned to fe so as to make the function be continuous at the point x e Representative problems Page 13339 35 7 40 State the de nition of derivative of a function y x at a speci c point x x0 as a limit Illustrate and explain each part of the de nition graphically using secant lines tangent lines etc ng Explain why the de nition requires taking a limit and what the limit means in this setti 7 Estimate the derivative of a function at a given point from a graph of the function Representative problems Page 14039 1 7 4 Find the slope of the tangent to the graph of a given function y fx at a speci c point x x0 by using the de nition of the slope as the limit of slopes of secant lines If the relevant limit does not exist explain what this tells you about the existence or nonexistence of a tangent line If there is a tangent line write an equation for the tangent line in p oint sl up e f a rm Representativeproblems Page 14039 5 7 22 Page 140 7 14139 31 7 44 Given function y fx that relates two physical quantities eg time and position elevation and atmospheric pressure radius and volume use the de nition of the instantaneous rate of change as the limit of average rates of change to nd the instantaneous rate of change at a speci ed value for the independent variable Representative problems Page 14039 27 7 30 Page 2 of 2 M 160 Study Guide for Chapter 4 Applications of Derivatives Summer 2008 This Study Guide describes everything you are expected to know understand and be able to do from Chapter 4 except Section 48 Exam questions on this material will askyou to do one or more ofthe tasks described in this Study Guide Sec 42 The Mean Value Theorem 7 a Write a complete statement of the Mean Value Theorem pg 257 b Explain what the Mean Value Theorem says graphically by sketching the graph of a function and the relevant secant and tangent lines and then explaining how these lines are related and b Explain what the Mean Value Theorem says physically by interpreting the hypotheses and the conclusion of the theorem in terms of an object moving along a straight line path Study Suggestion Graphical interpretation is discussed on page 2577 258 Physical interpretation is discussed on pages 258 Representative homework problems Page 260 1 7 4 think graphically Page 261 45 7 49 think physically Page 262 56 think about hx fx 7gx and 59 7 Explain how Rolle s Theorem and the Mean Value Theorem are related 7 Given a function defined on a given closed interval a b by an expression y fx determine whether the Mean Value Theorem is applicable or not In both cases use calculus and algebra to EITHER locate the point or points that satisfy the conclusion of the Mean Value Theorem OR to explain how to see that there are no such points Representative homework problems Page 260 5 7 8 In these exercises if there is a point c that satisfies the conclusion of the Mean Value Theorem nd it Mathophiles tryPage 262 51 52 amp 58 7 Apply the fact that two functions that have the same derivative must differ by a constant and your knowledge of the basic differentiation formulas to a find all functions that have a given polynomial power function or simple trigonometric function as first derivative b show that two seemingly different functions differ only by a constant Representative homeworkproblems Page 261 277 40 Page 262 53 Page 319 17 18 Sec 41 Extreme Values ofFunctions 7 Write complete definitions of the following words and phrases a absolute maximum aka global maximum of a function pg 244 b absolute minimum aka global minimum of a function pg 244 c local maximum aka relative maximum pg 247 e local minimum aka relative minimum pg 247 f critical point of a function pg 248 Illustrate each of the above by sketching a graph Given a sketch of the graph of a function identify and label all of the above Notice that whether a function has absolute and local extrema depends on the domain of the function as well as the equationformula for the function Study Suggestion Study Sec 41 Representative homework problems Page 252 1 7 10 7 Write a complete statement of the Extreme Value Theorem pg 246 Illustrate the Extreme Value Theorem graphically Give graphical examples to show that when any of the hypotheses of the Extreme Value Theorem are not satisfied the function may or may not have an absolute maximum or minimum Study Suggestion Study Theorem 1 in Sec 41 Find examples among exercises 1 7 14 pg 252 Give examples to show that it is possible for a function not to have a local extremum at a point where its derivative is 0 Study Suggestion Study Theorem 2 and the discussion of Finding Extrema in Sec 41 7 Explain the connection between critical points and local extrema of a function Representative homework problems Page 252 11 7 14 Page 1 7 Describe a procedure for finding the absolute extrema of a continuous function on a closed interval of finite length by finding the critical points and endpoints and evaluating the function at these points summarized in the box top of page 249 Use this procedure to find the absolute maximum and minimum values of a given function and the points where these extreme values are attained Representative homeworkproblems Page 253 15 7 34 and 35 7 44 7 Give examples to show that a function can fail to have a local extremum at a critical point and at an end point of its domain Representative homework problems Pages 253 7 254 53 65 68 69 70 Sec 43 Monotonic Functions and the First Derivative Test 7 Write a complete definition of what it means for a function to be increasing and for a function to be decreasing on an interval pg 263 Explain how to use the derivative to determine the intervals on which a function is increasing and on which it is decreasing Explain how to use the first derivative to determine whether a function has a local maximum a local minimum or neither at its critical points and the end points of its domain Illustrate your explanations with graphs pg 263 Study Suggestion Study Sec 43 Related homeworkproblems Page 267 43 7 46 7 Given a function defined by an expression y fx calculate and analyze its first derivatives using algebra to find the critical points of the function Knowing the critical points of a function use the First Derivative Test for Monotonic Functions page 263 to determine the intervals where the function is increasing and the intervals where the function is decreasing Use the First Derivative Test for Local Extrema page 265 to determine whether the function has a local maximum a local minimum or neither at each critical point and if applicable at the end point of its domain Representative homeworkproblems Page 266 1 736 Page 267 37 38 41 42 47 48 Sec 44 Concavity and Curve Sketching 7 Given the graph of a function identify and indicate a the intervals where the function is increasing b the intervals where the function is decreasing c the critical points d the intervals where the function is concave up e the intervals where the function is concave down and f the in ection points Study Suggestion Read the examples Section 4 4 Representative homework problems Page 274 7 275 1 7 8 and page 276 71 72 74 7 Given a function defined by an expression y fx which may involve constants designated by letters analyze its first and second derivatives to determine a the intervals where the function is increasing b the intervals where the function is decreasing c the critical points d the intervals where the function is concave up e the intervals where the function is concave down and f the in ection points If possible graph the function on your calculator and verify that your conclusion in a 7 f are correct Representative homework problems Page 275 97 40 Page 277 79 80 81 83 84 7 State the Second Derivative Test for Local Extrema Explain how to see that the sign of the second derivative reliably indicates whether a function has a local maximum or local minimum at a point where the first derivative is Give examples showing that if the first and second derivative are both zero at a point x c then the function may or may not have a local extremum at the point x 0 Representative homework problems Page 277 82 7 Sketch the graph of a function whose first andor second derivatives have certain properties Properties may be given symbolically algebraically or graphically or you may have to figure them out from an expression y x for the function or for its derivative Representative homework problems Pages 275 7 276 63 7 70 75 76 77 78 Page 2 Sec 45 Applied Optimization Problems 7 Solve optimization problems similar to the exercises in this section of the textbook Representative homeworkproblems Page 253 55 7 64 Page 285 17 37 Also look atPage 320 63 7 71 NOTE You are required to know the Pythagorean Theorem properties of similar triangles area formulas for simple plane figures volume formulas for spheres cylinders cones and rectangular parallelepipeds boxes and the definitions of the trigonometric functions Sec 47 Newton s Method This topic is explored in Calculator Lab 2 It will not be on the exam Given the graph of a function y fx and an initial approximation x0 for a zero of the function find the next three Newton approximates graphically and an illustrate the process of finding the these approximates on the graph Study Suggestion Study sec 4 7 and the Calculator Lab on Solving Equations by Newton s Method 7 Given a specific function y fx implement Newton s method on a calculator to calculate an estimated value for a specified zero or zeros of the function to a specified number of decimal places initial estimate xO may or may not be given Sketch a graph of the function and illustrateexplain the calculations graphically Study Suggestion Page 305 1 7 10 12 Page 3 MATH 160 Study Guide for Integration Summer 2008 This Study Guide describes everything from Section 48 Chapter 5 and Chapter 6 that you will be expected to know understand and be able to do on the nal exam Exam questions on integration will ask you to do one or more of the tasks described in this Study Guide Of course to do the things listed here you must be able to do almost everything listed in previous Study Guides Don t be surprised if some exam questions require you to use ideas or skills from earlier in the course Sec 48 Antiderivatives i 1 Explain what is meant by an antiderivative and by the indefinite integral for a function What is the difference if any between an antiderivative and the indefinite integral of a function Explain why an arbitrary constant is added in the indefinite integral i 2 Determine whether a given function y gx or family of functions y gx C is an antiderivative or the indefinite integral for a given function y x Representative Homework Problems Page 31539 55 7 60 and 61 7 64 i 3 Find the indefinite integral of functions that can be written as sumsdifferences of constant multiples of functions whose antiderivatives are known ie functions listed in Table 42 pg 308 by algebraic manipulations perhaps using trigonometric identities You are expected to know the basic trigonometric identities how to represent all six trig functions in terms of the sine and cosine the Pythagorean identity and the double angle formula for the sine Representative Homework Problems Page 314 31539 1 7 16 and 177 54 dy i 4 Interpret the general solution to a first order initial value problem E 7 gx graphically Explain graphically how an initial value selects one solution from the infinite number of possibilities Representative Homework Problems Page 315 7 316 39 65 amp 66 89 7 92 VI i 5 Solve initial value problems d y n gx 1 Sn 3 3 where gx can easily be antidifferentiated dx Representative Homework Problems Page 31639 677 86 93 7 97 amp 99 Sec 51 Estimating with Finite Sums i 1 Use finite sums to calculate reasonable estimates for the area of a region enclosed by the graph of a positivevalued function y fx the xaxis and vertical lines y a and y b and the distance a body travels during a given time interval from a graph of its velocity Representative homeworkproblems Selectfrom pages 333 7 33539 1 7 8 12 Page 388 1 2 Sec 52 Sigma Notation and Limits of Finite Sums VI i l A sum ofthe form SP ZfckAxk is called a Riemann sum for f on the interval a b kl Use words and pictures to explain how the interval a b enters into the Riemann sum and what each of the VI symbols n 2 P f xk Axk and ck in this expression means kl i 2 Given a function y fx defined on an interval a b and given a partition a x0 ltx1 lt ltxn1 ltx1 b of the interval choose appropriate evaluation points c1 c2 c3 c1 write an expression for the associated Riemann sum in expanded form and calculate the numerical value of the Riemann sum Proficiency with summation notation is not required but expect to see summation notation in later courses Representative homework problems Select from pages 342 7 34339 29 7 32 Study suggestion CalculatorLab on Riemann Sums i 3 Write an expressionequation that defines the Riemann integral in terms of Riemann sums Explain using words and pictures what this definition means and why it is necessary to take the limit as n approaches infinity in this definition b Explain why the variable of integration in I f x dx is called a dummy variable a i 4 Use a Riemann sum to calculate a reasonably accurate approximate value for a given definite integral Study suggestion CalculatarLab an Riemann Sums Sec 53 The Definite Integral i 1 Use properties of the definite integral summarized in Table 53 entries 3 5 to evaluate definite integrals by rewriting the integral as a combination of integrals that either have given values or can be evaluated by interpreting the integral as positive or negative area Representative homework problems Pg 353 select from add numbered problems from 9 7 26 i 2 Estimate integrals by using inequalities 6 and 7 in Table 35 Suggestedprablems Selectfram page 355 63 7 72 In 63 amp 64 assume a lt b In 68 the integral shauldbefram 0 t0 1 not to 0 Sec 54 The Fundamental Theorem of Calculus i 1 Write a differentiation formula for functions F x Ix f t dt defined by an integral with continuous a integrand Explain why it is appropriate to call this theorem The Fundamental Theorem of Calculus This theorem is on page 358 of the textbook where it is called The Fundamental Theorem of Calculus Part 1 Use this formula to compute derivatives of functions defined by integrals with the independent variable as the upper limit of integration Representative homework problems Page 365 31 amp 32 Page 390 77 i 2 Use the differentiation formula from 3 with the Chain Rule and properties of the integral to calculate the b derivatives of functions F x lg x f t dt and F x I f t dt where the upper or lower limit of 1 gr integration is a differentiable function y gx Representative hamewarkprablems Page 365 277 36 Page 367 63 amp 64 Page 390 78 79 80 b i 3 Explain the difference between an indefinite integral I f x dx and a definite integral I f x dx a State a Theorem that gives a connection between indefinite integrals antiderivatives and definite integrals Show how this result follows from The Fundamental Theorem of Calculus Part 1 from 3 above In the textbook this result is called The Fundamental Theorem of Calculus Part2 i 4 Infer information about the motion of an object moving along a coordinate axis from a graph of its velocity Representative homework problems Page 366 59 amp 60 Page 388 1 2 i 5 Use The Fundamental Theorem of Calculus Part 2 to evaluate definite integrals Suggested problems Page 365 1 7 26 Sec 55 Indefinite Integrals and Substitution i 1 Write an integration formula corresponding to the Chain Rule Explain how integration by substitution is used to apply or implement this integration formula Study suggestion Pages 368 7 371 afthe text Representative homework problems Pages 3 74 7 375 1 7 12 Page 376 61 62 i 2 Evaluate indefinite integrals by usubstitution After you have evaluated an indefinite integral by usubstitution identify the functions f and g that connect your work with the Chain Rule Express the u you used in the substitution process in terms of one or more of the functions f and g and their derivatives Representative homework problems Select from pages 374 7 3 75 13 7 52 i 3 Solve initial value problems that require usubstitution method to find the required antiderivatives Representative homework problems Pages 375 7 3 76 53 7 60 NOTE You will not be asked to derive or to remember the integration formulas for sin20 and cos20 pg 373 on the Final Exam These two integration formulas will be given if needed to answer an exam question Sec 56 Substitution and Area between Curves i 1 Use change of variables usubstitution including changes in the limits of integration to evaluate definite integrals Representative homework problems Pages 383 1 7 24 i 2 Use definite integrals to calculate the area of a region enclosed by the graph of a given function y x the xaxis and vertical lines y a an y Representative hamewarkprablems Page 365 7 366 377 46 and 55 Page 384 7 385 25 7 29 i 3 Given a description of a region bounded by the graphs of continuous functions and perhaps vertical andor horizontal lines find the area of the region by sketching the region and deciding whether it is better to integrate with respect to x or with respect to y sketching a representative area element used to nd the integral for the area writing the definite integral or integrals that represents the area and evaluating the integrals Representative homework problems Pages 384 7 386 30 7 80 Sec 61 Volumes by Slicing and Rotation About an Axis i 1 Given a description of a solid from which a formula Ax for a typical cross section can be found find the volume of the solid by sketching the solid and a typical cross section39 finding a formulaAx for a typical cross sectional volume element39 writing an definite integral that represents the volume including correct limits of integration and evaluating the integral using the Fundamental Theorem Suggested problems Selectfram pages 4067 407 1 7 12 i 2 Given a description of a solid of revolution formed by rotating a region enclosed by the graphs of two or more equations including the possibilities of horizontal or vertical lines around a vertical line x a or a horizontal line y 1 find the volume of the solid by sketching the solid and a typical crosssectional volume element used to find the volume by the method of crosssectional slices washers or discs 39 writing an definite integral that represents the volume including correct limits of integration and evaluating the integral using the Fundamental Theorem Suggestedprablems Selectfram pages 4067 407 13 7 48 49 50 51 55 58 Sec 63 Lengths of Plane Curves i 1 Given a curve specified as the graph of a given continuously differentiable function y x over a specified interval a Ex 3 sketch the curve sketch a typical segment of length used to find the integral that gives the length of the curve39 express the length of this typical segment of length in terms of f x and dx39 write a definite integral that represents the length of the curve including correct limits of integration and find the length of the curve by evaluating the definite integral by the Fundamental Theorem or numerically using a calculator as directed Suggested problems Select from pages 423 7 424 77 24 Sec 65 Areas of Surfaces of Revolution and the Theorem of Pappus i 1 Find the lateral surface area of a solid formed by revolving the graph of a nonnegative valued differentiable function y x a f x f 1 around the xaxis Sketch the surface and a representative area element used to find the integral to evaluate

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All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.