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HARVARD MATH 1A - Final Exam Review II

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Math 1aFinal Exam Review IIT. JudsonJanuary 9, 2006Resources for ReviewFinal Exam Review Guide http://www.courses.fas.harvard.edu/~math1a/exams/finalf05review.pdf.pdfExams and solutions from previous years http://www.courses.fas.harvard.edu/~math1a/prevexams/Solutions to the Chapter Review Exercises http://www.courses.fas.harvard.edu/~math1a/exams/Exam ParticularsSaturday, January 14 at 9:15 AM to 12:15 PMStudents whose last name begins with A-L will take the exam in Science Center CStudents whose name begins will M-Z will take the exam in Science Center B.No calculators allowedAll out-of-sequence exams must be approved by the Final Exams OfficeWhat to ExpectApproximately fifteen questions (some with several parts)The Exam is comprehensiveRefer to the Final Exam Review Guide for detailsA Strategy for OptimizationDraw the picture.Name the variables. Write down the function that needs to be maximized or minimized.Write down the relationships between the variables.Reduce the function that is to be optimized to a function in one variable using the relationships in the previous step.Find the critical values of the function.Apply the First or Second Derivative Test to the Function. Don't forget to test the endpoints.Write down the final answer.An Optimization Problem11. A trough that is 20 feet long is to be constructed from a sheet of me tal ofwidth 6 feet by bending up one-third of the sheet on each side throughan angle θ. How should θ be chosen so that the trough will hold themaximum amount of water? (10 points)qq2 ft2 ft 2 ftgraphics.nb 120Read the problem and draw a picture.Name the variables. These are the quantities that are changing.Write down the rates of change that you know and the rates of change that you want to know. Write down an equation that relates the variables.Implicitly differentiate the equation with respect to the appropriate variable (usually time).Substitute the quantities you know into the equation from the previous step.Solve for any unknown variables.Write down the final answer. Does your answer make sense?A Strategy for Related RatesRelated Rates16. A super-absorbant square sponge is place in water, and immediatelystarts growing. It’s perimeter changes at a constant rate of 2 inchesper minute.How quickly is the area of the square sponge changing when the perime-ter is 12 inches?17. Two ships sail from the same island port, one going north at 16 knotsand the other east at 20 knots. The northbound ship departed at 9:00A.M. and the eastbound ship left at 11:00 A.M. How fast is the distancebetween them increasing at 2:00 P.M.?18. Let A(x) be the area functionA(x) =!x2f(t) dt,defined for 0 ≤ x ≤ 6. Here f(t) is given by the following graph!"!"!!" ! #$ %& '()'*(a) Find A(0)(b) For which x is A(x) increasing? Explain.(c) Find local minima and local maxima of A(x).(d) Find global minima and maxima of A(x). Explain.(e) For which x is A(x) concave down? Explain.7Curve Sketching1. Evaluate the following limits. (10 points)• limx→01 − cos 5x4x + 3x2• limx→0+x2ln x• limx→1(ex+ x)1/x• f(x) = x + 2 sin x• Suppose that limx→5f(x) = 1. Find limx→5f(x)3f(x) − x.• Suppose that g(x) is a decreasing function, and limx→−10g(x) = 5. Find limx→−10+g(x)g(x) − 5.• limx→3x ln x − 3 ln 3x − 3• limx→(π/2)−(sin x)tan x2. Let f (x) = e−xx2. (10 points)(a) What are the critical points of f?(b) Classify the critical points as local maxima, local minima, or neither.(c) Where is the graph of f concave up?(d) What is limx→∞f(x)? Show your work.1-4 p -3 p -2 p -p p 2 p 3 p 4 px-10-5510x + 2sin xUntitled-1 1l’Hospital’s Rule2 L’Hˆospit al’s RuleIflimx→af(x)g(x)is of the form 0/0 or ∞ /∞ andlimx→af"(x)g"(x)exists, thenlimx→af(x)g(x)= limx→af"(x)g"(x).L’Hˆospital’s rule holds if a is replaced by ∞, −∞, a+, or a−.Examples1. limx→∞2x3− x + 1x3+ 3x,2. limx→0+ln xx,3. limθ→π/21 − sin θcsc θ,4. limx→∞x3e−x2,3 Why l’Hˆospital’s Rule WorksConsider the special case oflimx→af(x)g(x),3Some Examples1. Evaluate the following limits. (10 points)• limx→01 − cos 5x4x + 3x2• limx→0+x2ln x• limx→1(ex+ x)1/x• limx→0cos xx• Suppose that limx→5f(x) = 1. Find limx→5f(x)3f(x) − x.• Suppose that g(x) is a decreasing function, and limx→−10g(x) = 5. Find limx→−10+g(x)g(x) − 5.• limx→3x ln x − 3 ln 3x − 3• limx→(π/2)−(sin x)tan x2. Let f(x) = e−xx2. (10 points)(a) What are the critical points of f?(b) Classify the critical points as local maxima, local minima, or neither.(c) Where is the graph of f concave up?(d) What is limx→∞f(x)? Show your


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