| Date |
Topics covered in class |
Readings |
Other links |
| Jun 30 |
Intro, class website, pretest |
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| Jul 1 |
Definition of set, finite and infinite set, definition of function: rule, domain, range, examples of functions, rules that are not functions;
even and odd functions; types of functions: polynomial, power, exponential, trigonometric, graphs of functions, exponential laws.
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Sections 1.1 and 1.2 |
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| Jul 2 |
Transformations of functions: translation in X and Y, scaling in X and Y, reflection across X and Y axes, graphing of transformations
of functions; composition of functions.
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Section (1.3) |
Hw1 out. |
| Jul 3 |
Composition of functions: domain, range, examples; a problem involving composition and transformation of functions; concept of one-one functions, examples; inverse functions with examples, graphing of the inverse function and its relation to reflection across the line y = x; logarithms and some rules dealing with logarithms. |
Section (1.3), (1.5), (1.6) |
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| Jul 7 |
Inverse trigonometric functions: plotting and simplication; CHAPTER 2: Introduction to limits: example of estimating the tangent to a curve at a point, and the velocity of ball at a given time instant; a more formal definition of a limit |
Section 1.6, 2.1 (example 1 and 3); section 2.2 (def 1, example 1, 2, 4, 5,6), see figure 7 in section (2.2). |
Practice problems on inverse trigonometric functions |
| Jul 8 |
Review of chapter 1: some questions regarding functions and one-one functions, properties of logarithms and inverse functions; Analysis of cases where the method of tables for computing limits fails: (1) finite precision of calculators, and (2) Non-existence of limits (eg: f(x) = sin(pi/x)); Basic laws governing limits. |
Examples in section (2.1) and (2.2). Laws of limits from section (2.3). |
Hw1 due.
Homework 1 solutions.
Hw 2 out.
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| Jul 9 |
Quiz 1; quick review of laws of limits; concept of left-hand and right-hand side limits, and a new definition of limit in terms of left-hand and right-hand limits; conditions for non-existence of limits (either the limit is infinity, or the left-hand and right-hand limits are unequal); problems from section (2.3). |
Examples in section (2.3) |
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| Jul 10 |
Problems from section (2.3); problems on left-hand and right-hand limits; concept of floor, ceiling, round functions; problems involving non-trivial polynomial factorization; change of variables in problems dealing with limits. |
Examples in section (2.3) |
|
| Jul 14 |
Section 2.5: definition of continuous functions, examples; examples of three types of discontinuity: removable, infinite and jump; concept of left and right continuity, and continuity over an interval; properties of continuous functions: sum, difference, product, quotient, composition; intermediate value theorem: use of the intermediate value theorem to determine whether a polynomial has roots in a given interval, the monk puzzle. |
Section 2.5.
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Hw3 is out. Due on Mon 21st July.
(1)Application of the intermediate value theorem in geography: http://en.wikipedia.org/wiki/Intermediate_value_theorem (read the section on applications only).
(2)Statement of the monk problem.
(3)Applications in root finding (Read just the first paragraph on this link).
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| Jul 15 |
Section 2.6: Vertical and horizontal asymptotes to a function f(x), limits with x tending to infinity, careful use of sum/difference/product/quotient laws for limits.
Section 2.7 and 2.8: definition of the derivative of f(x) at x = a, computation of slope of a tangent to the curve y = f(x) at a point, and its relation to the derivative; derivative as a function; obtaining derivatives from first principles using limits.
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Section 2.6, 2.7, 2.8.
Review of chapter 2 (page 165/166) and true false quiz (page 166).
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Hw3 is updated.
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| Jul 16 |
QUIZ 2; Section 2.8: Notion of differentiability of a function, example of a continuous function that is not differentiable, proof of the fact that a differentiable function is always continuous.
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Section 2.8.
Review of chapter 2 (page 165/166) and true false quiz (page 166).
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| Jul 17 |
Discussion of QUIZ 2 questions; Section 2.8: When is a function not differentiable? Concept and examples of higher order derivatives, obtaining the derivatives of a function using first principles (example: sin(x) and tan(x)).
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Section 2.8. Reference page (very first page) of your textbook for trigonometric formulae.
Review of chapter 2 (page 165/166) and true false quiz (page 166).
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| Jul 21 |
Midterm review.
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Hw3 solutions: part 1 and part 2.
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| Jul 22 |
MIDTERM.
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| Jul 23 |
Section 3.1, 3.2 and 3.3: Definition of a derivative (Recap from section 2.8). Derivatives of functions such as f(x) = x^n, f(x) = a^x, f(x) = e^x, and trigonometric functions. Applications in obtaining the equation of the tangent to the curve y = f(x) at a given point, and then the normal to the curve at that point. Example dealing with displacement, velocity, acceleration and jerk (i.e. higher order derivatives).Sum and difference rule, Product rule and quotient rule, proof of the product rule.Examples using the product rule and quotient rule. |
Sections 3.1, 3.2 and 3.3. (Skip the proofs of limits of trigonometric functions).
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| Jul 24 |
Section 3.4: Chain rule for derivative of F(x) = f(g(x)). Two alternative (equivalent) notations. Proof of the chain rule. Examples of the chain rule. Example for the derivative of F(x) = f(g(k(x))). Section 3.6: Derivative of f(x) = ln (x) and f(x) = log_a (x). Comparison of derivatives of f(x) = x^n, f(x) = a^x and f(x) = x^x, and review of commonly made mistakes while obtaining these derivatives. Use of logarithms in simplifying the calculations in derivatives of complicated functions. |
Sections 3.4 and 3.6.
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HW4 out and due on Wed 30th July.
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| Jul 28 |
Section 3.5: Implicit functions and implicit differentiation, derivatives of inverse trigonometric functions, problems dealing with a combination of implicit differentiation, chain rule and inverse trigonometric functions. |
Section 3.5.
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| Jul 29 |
Section 3.3: Geometric applications of limits; Section 3.9: Definition of a related rate, examples of geometric problems involving related rates. |
Section 3.3 (read it again) and Section 3.9.
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| Jul 30 |
Section 4.1: Definition of absolute (global) maximum and minimum, examples of global maxima and minima of a function, examples of functions without a global maximum and/or a global minimum, extreme value theorem with examples, pitfalls in using the extreme value theorem, Fermat's theorem that characterises the derivative of a function at a local maximum or local minimum, proof of Fermat's theorem, pitfalls in using Fermat's theorem. |
Section 4.1.
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Homework 5 is out. Solutions to HW4.
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| Thursday Jul 31 |
QUIZ 3; Section 4.1: Definition of a critical point, Examples of closed interval method to find absolute minima and maxima of a function within a closed interval. |
Section 4.1
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Solutions to quiz3 are here and here.
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| Monday Aug 4 |
Review for Final Exam |
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| Tuesday Aug 5 |
Final Exam (COMPREHENSIVE) |
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| Aug 6 |
Distribution of Graded Final Exam Answer Sheets and Discussion of Solutions. |
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Aug 5: To calculate your final grade: G1 = Add up your grade on homeworks 1 through 4 (30 points each) and divide by 12. G2 = Add up your grade on quizzes 1 through 4 (36 points each) and divide by 14.4. G3 = grade for class participation (out of 5) and note-taking (out of 5). G4 = grade on midterm (out of 75) divided by 3. G5 = grade on final exam (out of 100) multiplied by 0.35. Final grade = G1 + G2 + G3 + G4 + G5 (out of 100). See grading policy above. Homeworks are worth 10%, quizzes 20%, class participation and note-taking is for 10%, midterm is 25% and final is 35%. The curve is as follows: 85 and above is an A. 75 and above (and less than 85) is a B+. 70 and above (and less than 75) is a B. 65 and above and less than 70 is a C+. 55 and above (and less than 65) is a C+. Less than 55 is a C.