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HARVARD MATH 1A - Review Guide for Midterm II

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Math 1a. Introduction to CalculusReview Guide for Midterm IIThomas W. JudsonHarvard UniversityFall 2005Midterm DetailsThe first midterm will primarily cover Chapter 3 and 4 (through 4.8), but you will also be responsible forany previous material covered in the course. The midterm exam will be on Tuesday, December 13 at 7-9PM in Science B. Ther e will also be a course-wide review session on Thursday, December 8 at 7-9 PM inScience D. We plan to videota pe the review session. You should be able to access the video by clicking onLecture Videos at the course website.Studying and Reviewing• You can find copies of old midterms as well as solutions by c licking on Previous Exams at the coursewebsite.• You should also try working some of the problems in the review sections of Chapter 3 (pp. 255–257)and Chapter 4 (pp. 335–338). We will post solutions to these problems on the course website on theExams page.• Be sure to take advanta ge of the TF office hours, CA sectio ns , and the MQC in Loker Commons.Topics for Midterm II• To understand and be able use the power rule and the derivative of ex. These rules should be developedfrom the definition of the derivative (Section 3.1).• To understand and be able to apply the definition of e (Section 3.1).• To understand and be able the product and quotient rules for differentiating functions (Section 3.2).• To understand and be able to applying the concept of a derivative to applications in the natural andsocial sciences (Section 3.3).• To understand and be able to evaluate the derivatives of trigonometr ic functions (Section 3.4).1• To understand and be able to apply the chain rule when differentiating composite functions (Section3.5).• To understand and be a ble to apply the chain rule when differentiating implicitly defined functions(Section 3.6).• To understand and be able to a pply the derivatives of the inverse trigonometric functions (Section 3.6).• To understand and be able to determine when two curves are orthogonal (Section 3.6).• To understand and be able to apply the basic logarithmic differentiation formula (Section 3.7).• To understand and be able to apply the technique of logarithmic differentiation (Section 3.7 ).• To understand and be able to apply the concept o f e as a limit (Section 3.7).• To understand and be able to apply the process o f linearizing a function at x = a (Section 3.8).• To understand and be able to the differential as the difference between the linearization of a functionand the function itself (Section 3.8).• To understand and be able to the concept of related rates (Section 4.1 ).• To understand the definition of local and absolute extrema both intuitively and precisely (Section 4.2).• To understand and b e able to apply the Extreme Value Theorem and Fermat’s Theorem (Section 4.2).• To understand and be able to find critical values and use the closed interval method (Section 4.2).• To be able to use the first derivative to determine whether a function is increasing or decreasing (Section4.3).• To understand and be able to apply the First and Seco nd Derivative Tests for lo cal maxima and minima(Section 4.3).• To understand and be able to apply the Mean Value Theorem (Section 4.3).• To understand the relationship between concavity and the behavior of the first derivative (Section 4.3).• To be able to use the se c ond derivative to determine concavity and points of inflection (Section 4.3).• To be able to use calculus to sketch the graphs of functions (Section 4.3 ).• To recognize the indeterminate for ms 0/0, ∞/∞, and ∞ − ∞ and be able to apply L’Hˆospital’s Rulewhen evaluating limits of these indeterminate forms (Section 4.5).• To understand l’Hospital’s Rule in terms o f relative rates of change (Section 4.5).• To recognize the indeterminate forms 0 · ∞, 1∞, ∞0, 0∞, and 00and be able to apply l’Hospital’s Rulewhen evaluating limits of these indeterminate forms (Section 4.5).• To be able to set up a nd solve optimization problems using calculus (Section 4.6).• To be able to apply calculus to problems in business and e conomics (Section 4.7).• To be able to use the Newton-Raphson algorithm to approximate roots of functions (Section


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HARVARD MATH 1A - Review Guide for Midterm II

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