M160 Study Guide for Chapter 6 Applications of Integration



MATH 160 Final Exam Study Guide Fall, 2008

The final exam covers the entire course, though the material since the third midterm (Sec 5.5, Sec 5.6, and topics from Chapter 6) will get more emphasis. Expect two or three exam questions about Applications of Integration (sec 5.6, 6.1, 6.2, 6.3, and 6.6). This Study Guide describes everything you are expected to know, understand, and be able to do for the Fall, 2008, MATH 160 Final Exam.

Final exam questions will ask you to do one or more of the tasks described in this Study Guide. You may need to use information described in several different parts of the Study Guide to answer some questions or solve some problems on the final exam. When you have mastered an item on the Study Guide, check it off. When every item is checked, you are ready for any question that might appear on the final exam.

Topics from Chapter 5 since Exam 3:

Sec 5.5 Indefinite Integrals and Substitution

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 – 371 of the text.

Representative homework problems: Pages 374 – 375; #1 – 12. Page 376; #61, 62.

2. Evaluate indefinite integrals by u-substitution.

After you have evaluated an indefinite integral by u-substitution, 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/or their derivatives.

Representative homework problems: Select from pages 374 – 375; #13 – 52.

3. Solve initial value problems that require u-substitution method to find the required antiderivatives.

Representative homework problems: Pages 375 – 376; #53 – 60.

NOTE: You will not be asked to derive or to remember the integration formulas for sin2θ and cos2θ (pg 373). These two integration formulas will be given if needed to answer an exam question.

Sec 5.6 Substitution and Area between Curves

1. Use change of variables (u-substitution), including changing the limits of integration, to evaluate definite integrals.

Representative homework problems: Pages 383; #1 – 24.

2. Rewrite definite integrals involving arbitrary functions (indicated only by function notation), including changing the limits of integration, by making a change of variables (that is, by making a u-substitution). (You might or might not be able to evaluate the resulting integral.)

Representative homework problems: Pages 386 – 387; #83, 84, 87, 88, 89.

3. Use a definite integrals to calculate the area of a region enclosed by the graph of a given function y =f(x), the x-axis, and vertical lines y = a and y = b.

Representative homework problems: Page 365 – 366; #37 – 46 and #55. Page 384 – 385; #25 – 29.

4. Given a description of a region bounded by the graphs of continuous functions and (perhaps) vertical and/or 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 find the integral for the area,

• writing the definite integral (or integrals) that represents the area, and

• evaluating the integral(s).

Representative homework problems: Pages 384 – 386; #30 – 80.

Topics from Chapter 6 since Exam 3:

Sec 6.1 Volumes by Slicing and Rotation About an Axis

1. Given a description of a solid from which a formula A(x) for the area of a typical cross section can be found, find the volume of the solid by

• sketching the solid and a typical cross section;

• finding a formula A(x)dx for a typical cross sectional volume element;

• writing an definite integral that represents the volume (including correct limits of integration); and

• evaluating the integral using the Fundamental Theorem.

Suggested problems: Select from pages 406 – 407, #1 – 12.

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 = b), find the volume of the solid by

• sketching the solid and a typical cross-sectional volume element used to find the volume by the method of

cross-sectional slices (washers or discs);

• writing an definite integral that represents the volume (including correct limits of integration); and

• evaluating the integral using the Fundamental Theorem.

Suggested problems: Select from pages 406 – 407, #13 – 48, #49, 50, 51, 55, 58

Sec 6.2 Volumes by Cylindrical Shells

1. 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 = b), find the volume of the solid by

• sketching the solid and a typical cylindrical volume element used to find the volume by the method of

cylindrical shells,

• writing an definite integral that represents the volume (including correct limits of integration); and

• evaluating the integral using the Fundamental Theorem.

Suggested problems: Select from pages 414 – 415; 1 – 24.

2. Determine which method (slices or shells) is preferable for calculating the volume of a given solid of revolution (as described above). Then calculate the volume of the solid.

Suggested problems: Select from pages 415 – 416; 25 – 36.

Sec. 6.3 Lengths of Plane Curves (Length of parameterized curves will not be on the Final Exam.)

1. Given a curve specified as the graph of a given continuously differentiable function y = f(x) over a specified interval a < x < b,

• sketch the curve (calculator allowed),

• sketch a typical segment of length used to set up the integral that gives the length of the curve;

• express the length of this typical segment of length in terms of f′(x) and dx;

• 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 – 424; #7 – 24.

Sec. 6.6 Work

1. Solve work problems involving compressing or stretching a spring; lifting a rope, chain, or cable (perhaps with an object of constant mass attached); or pumping liquid into or out of a container, tank or reservoir.

Suggested problems: Select from pages 452 – 453; #1 – 24.

Topics from earlier in the course. Refer to earlier Study Guides for suggested problems.

Limits and Continuity

• Understand the idea of limit. Explain accurately in non-technical language what it means to say that a function y = f(x) has limit L as x approaches a. Illustrate with examples.

• Explain why the idea of limits is needed. Explain how the idea of limit is used to define continuity of a function y = f(x) at a point x = a, derivatives, and integrals. Illustrate with examples.

• Is evaluating the limit of a function y = f(x) as x approaches a the same as evaluating the function at x = a ? If so, why have both ideas? If not, explain the difference and illustrate with examples.

• Give graphical examples of functions that do not have a limit as x approaches a specified point x = a.

Explain how to see from the graph that the function does not have limit as x approaches the specified point.

• Evaluate limits as x approaches a number a and as x approaches plus or minus infinity numerically and graphically (including the possibility the limit might not exist).

• Use algebraic manipulation and the Limit Theorems (sec 2.2, Theorems 1 – 4) to evaluate limits as x approaches a number a and as x approaches plus or minus infinity (including the possibilities that the limit might not exist or might be ±∞). Show details of algebra.

• Evaluate one-sided limits. Explain the connection between one-sided limits and limits (aka two-sided limits). Use one-sided limits to investigate the existence (or non-existence) of a limit. Show the details of the analysis.

• Explain how infinite limits and vertical asymptotes are related. Use algebra to locate points where the graph of a given rational function might have vertical asymptotes. Then evaluate the relevant limits (without a calculator if so directed) to determine whether the graph actually has vertical asymptotes at these points.

• State the mathematical definition of the phrase “a function y = f(x) is continuous at x = a”.

Identify points where a function is continuous and where it is not continuous from the graph of the function.

Explain in terms of the definition why the function is or is not continuous at specified points (and at the point(s) you identified).

• Give graphical examples of functions that

(i) are not continuous at a point a but do have a limit as x approaches a ;

(ii) are not continuous at a point a but are defined at the point x = a ;

(iii) are not continuous at a point a but have both a left-hand and a right-hand limit as x approaches a .

If it is not possible to give such an example, explain why.

• Given an expression for a function (perhaps defined piecewise), use the definition to determine whether the function is continuous or can be defined so as to be continuous at specified points. If the function has a removable discontinuity at a point x = a, determine what value should be assigned to f(a) so the function is continuous at x = a. Show details of the analysis.

Derivatives

• State the definition of derivative of a function y = f(x) at a point x = c (as the limit of a difference quotient).

Interpret 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.

• Explain how to tell from its graph whether a function is differentiable or not. Given the graph of a function, indicate the points (x-values) where the function is differentiable and where it is not differentiable.

• Use the definition of the derivative of a function at a point (as a limit) to determine whether a given function (perhaps defined piecewise) is or is not differentiable at a given point x = c. If the function is differentiable at x = c, find f′(c). Show details.

• Estimate the derivative of a function at a specified point x = a from the graph of the function.

Sketch the graph of the derived function y = f′(x) from the graph of y = f(x).

• Use the differentiation formulas (especially the Product, Quotient, and Chain Rules) to calculate first, second, and higher derivatives of functions defined by expressions that involve combinations (sums, differences, products, quotients, compositions) of polynomials, power and root functions, and/or trigonometric functions.

• Given the values of two functions f and g and their derivatives at several x-values, find the values of the derivatives of functions that are combinations (sums, differences, products, quotients, compositions) of f

and g at these x-values.

• Find an exact equation in point-slope form for the tangent line at a given point on the graph of a differentiable function. Find the equation without a calculator; check your answer with a calculator.

• Use implicit differentiation to calculate the derivative of a function defined implicitly by an equation relating

x and y. Find an equation for the line tangent to the graph of an implicitly defined function at a given point.

Applications of Derivatives

• Given a position function y = s(t), calculate the velocity and acceleration functions. Use these functions to answer questions about the motion of the object (including displacement and distance traveled).

• Solve Related Rate Problems.

(Know the distance formula and Pythagorean Theorem; perimeter and area formulas for simple plane figures; volume formulas for spheres, cylinders, cones, rectangular boxes; surface area formulas for spheres, cylinders, and rectangular boxes; relationships among sides and angles of similar triangles; the definitions of the trigonometric functions; and relationships among trigonometric functions.)

• State the Mean Value Theorem. Interpret the Mean Value Theorem graphically and physically.

Given either the graph of or an expression for a function y = f(x) defined on a closed interval, determine whether the hypotheses of the Mean Value Theorem are satisfied. Either locate the point or points that satisfy the conclusion of the Mean Value Theorem or explain how to see that there are no such points.

• From the graph of a function y = f(x) infer the values (intervals of x-values) where f′(x) is > 0, = 0, and < 0 and where f′′(x) is > 0, = 0, and < 0. Identify critical points and inflection points. Explain what you see in the graph that tells you the signs of f′(x) and f′′(x) or that you have a critical point or an inflection point.

• Given an expression for a function y = f(x) find the intervals of x-values where f′(x) is > 0, = 0, and < 0 and where f′′(x) is > 0, = 0, and < 0. Identify critical points and inflection points. Explain what the signs of f′(x) and f′′(x) , the critical points, and the inflection points tell you about the graph of the function.

• Sketch an accurate graph of a function from information about the value of the function and/or its derivatives at selected points and about the sign of f′(x) and/or f′′(x) on various intervals.

• Use calculus (not a calculator) to find and analyze the points where a given function could have an extremum. Use the first-derivative or second-derivative test as directed to determine whether the function has a local maximum, local minimum, or neither at these points. Determine whether the function has absolute extrema. If the function has an absolute maximum and/or absolute minimum, determine where these extrema occur and explain how you know these are the extreme values.

• Solve Optimization Problems.

(Know the distance formula and Pythagorean Theorem; perimeter and area formulas for simple plane figures; volume formulas for spheres, cylinders, cones, rectangular boxes; surface area formulas for spheres, cylinders, and rectangular boxes; relationships among sides and angles of similar triangles; the definitions of the trigonometric functions; and relationships among trigonometric functions.)

Integration

• Explain what is meant by the indefinite integral of a function. Explain the connection between antiderivatives and indefinite integrals. Check whether an indefinite integral is correctly evaluated by differentiating.

• Solve initial value problems that require finding a function from its first or second derivative and values at specified points. Problem may be presented in narrative form.

• Use the indefinite integration theorems for constant multiples, sums, and differences of functions and integration formulas for polynomials, power and root functions, and trigonometric functions, perhaps in combination with algebraic manipulation and/or u-substitution, to evaluate indefinite integrals.

• Explain how the substitution method (aka u-substitution) for evaluating integrals is related to the Chain Rule.

• State the definition of the Riemann (definite) integral of a function over an interval [a, b] in terms of Riemann sums. Interpret each part of the definition (including the final value of the integral) graphically.

Explain why the definition requires taking a limit and what the limit means in this setting.

• Use Riemann sums to calculate an approximate value for a given definite integral. (Calculator required.)

• Evaluate definite integrals by identifying the integral with the areas of one or more regions having known area. (Keep in mind that areas of regions below the x-axis are counted negatively.)

• Evaluate definite integrals numerically by using a calculator.

• State the Fundamental Theorem of Calculus, Part 1. Use this theorem to write an antiderivative of a given continuous function (when none of the techniques for evaluating an indefinite integral listed above work).

• Use the Fundamental Theorem of Calculus, Part 1 (perhaps in combination with the Chain Rule) to differentiate functions defined by integrals with variable upper limits.

• State the Fundamental Theorem of Calculus, Part 2. Explain why the Fundamental Theorem, Part 2 allows us to use any antiderivative of the integrand to evaluate a definite integral.

• Explain the difference between indefinite and definite integrals. Explain the connection between the two.

• Use u-substitution to rewrite a definite integral, including changing the limits of integration.

• Use the Fundamental Theorem of Calculus, Part 2 to evaluate definite integrals.

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