Homework: Appendix A: 1, 2, 3, 5, 6, 7, 8, 11, 13-33(odd), 34, 37, 38, 44, 45, 49, 51, 56. Appendix B: 3, 6, 7, 9, 11, 14, 16-21, 24, 29, 33, 36, 37, 42. Appendix D: 1, 2, 4, 9, 11-20, 23, 26, 28, 29, 30, 32, 35, 38, 43, 44, 46. Chapter 1: o 1.1: 1, 5, 6, 7, 13, 21, 23, 27, 31. o 1.2: 1, 3, 4, 7, 9, 16. o 1.3: 1, 3, 9, 11, 13, 14, 23, 35, 39, 43. o 1.5: 3, 13, 15, 25. o 1.6: 1, 4-7, 10, 11, 17, 20, 23, 24, 27, 35, 39-41, 47, 50, 51, 53, 63, 65. Chapter 2: o 2.1: 3, 5. o 2.2: 4, 7, 13, 14, 15, 23, 25, 27, 32. o 2.3: 1, 3, 10, 11-30(odds), 35. o 2.5: 3, 5, 11, 17, 21, 23, 27, 31, 37, 41. o 2.6: 5, 7, 11, 13-34(odds), 37, 41. o 2.7: 1, 3, 7, 11, 13, 17. o 2.8: 7, 13, 17, 19, 23, 25. o 2.9: 1, 5, 9, 14, 21, 23, 29, 31, 37. Chapter 3: o 3.1: 3-17(odd), 21, 23, 31, 39, 45, 47. o 3.2: 1, 3, 5, 13-23(odd), 31, 35. o 3.4: 3, 5, 9, 13, 17, 21, 29, 33. o 3.5: 9, 11, 15, 19, 21, 23, 27, 31, 33, 39, 41, 43, 53. o 3.6: 7, 13, 17, 23, 41, 59. o 3.7: 3, 9, 15, 23, 29, 33, 35, 43. 49. o 3.8: 3, 5, 13, 17, 23, 35, 39, 43, 47. o 3.9: 1, 7, 9, 31, 33. o 3.10: 1, 9, 11, 17, 25, 31. o 3.11: 5, 9, 15, 17, 23, 43. Chapter 4: o 4.1: 3, 7, 11, 15, 19, 25, 31, 37, 45, 47, 53. o 4.2: 3, 6, 13, 15, 25. o 4.3: 1, 5, 7, 11, 23, 29, 41, 45, 49. o 4.4: 5, 9, 11, 15, 23, 41, 49, 51, 53, 57. o 4.5: 5, 9. o 4.7: 3, 9, 11, 17, 21, 27. o 4.9: 4, 5, 13, 41. o 4.10: 3-13 (odds), 21, 27, 32, 41, 45. Chapter 5: o 5.1: 1, 5, 11, 17, 19.
o 5.2: 5, 17, 21, 23, 29, 33, 36, 39. o 5.3: 5, 9, 19, 23, 25, 29, 39, 41, 47. o 5.4: 5, 11, 13, 19, 23, 25, 31, 35, 39, 45, 53. o 5.5: 3, 5, 7, 11, 15, 19, 21, 23, 31, 35, 41, 51, 55, 61, 67. Chapter 7: o 7.1: 3, 5, 7, 9, 15, 19, 21, 23, 27, 33, 51. o Review for final. Exams: Exam 1 -- Fri. September 30th, 2005 (Appendix A, B, D, and Ch 1) Exam 2 -- Fri. October 7th, 2005 (Ch 2) Exam 3 -- Fri. October 14th, 2005 (Ch 3) Exam 4 -- Fri. October 21st, 2005 (Ch 4) Final Exam -- Fri. Oct 28th, 2005 (cummulative including Ch 5 and section 7.1). The final exam is a 2- hour exam that can be taken from 11am-1pm or noon-2pm. If we need more time to review the chapters for the final exam, the Fri class will be used for review, and the exam will take place on Mo, Oct 31st from 1-3pm. Topics: Tu. Sept 27th: 1.1 Four ways to represent a function: o function o domain and range of a function o independent and dependent variable o graph o the 4 ways to represent a function (verbally, numerically, visually, algebraically) o vertical line test o piecewise functions o even function versus odd function o increasing versus decreasing functions 1.2 Mathematical Models: A Catalog of Essential Functions: o what is a mathematical model o polynomials o degree of a polynomial o coefficients of the polynomial o terms of a polynomial o know the types of functions: linear, quadratic and cubic functions (formulas and graphs) power functions (f(x) = x^a, for different types of powers a) rational functions (fractions of polynomials) algebraic functions (such as functions that contain roots, quotient or products of polynomials,...) trig functions (functions containing trig identities) exponential functions (variable is the exponent of a number) logarithmic functions (functions containing logarithms)
Wed. Sept 28th: 1.3 New Functions from Old Functions: o vertical and horizontal shifts o vertical and horizontal stretching and reflecting o composition of functions (not commutative --Ex 7 p44) 1.5 Exponential Functions: o negative and positive exponents o fractional exponent o laws of exponents o applications (double time --p58, and half-life -- p59) o the number e Th. Sept 29th: 1.6 Inverse Functions and Logarithms: o one-to-one and horizontal line test o what is and how to find the inverse function o relationship between the domain and codomain of the function and its inverse o composition of the function and its inverse o inverse functions of some common functions: log, exponential, trig functions o laws of logs (in particular rules for natural log) o change of base formula o inverse trig functions (restriction of the domain to have an inverse) Fri. Sept 30th: Exam 1 over Appendices and Chapter 1. Mo. Oct 3rd: 2.1 The Tangent Problem: o finding the equation of a line tangent to a curve o instantaneous velocity 2.2 The Limit of a Function: o defn. (and meaning) of the limit o guess the value of a limit considering table of values o one sided limits o infinite limits o vertical asymptotes 2.3 Calculating Limits Using the Limit Laws: o limit laws o evaluating limits o Thm.1 p109 o greatest integer function (aka "floor function") o Thm. 2 p110 o Squeeze Thm. Tu. Oct 4th: 2.5 Continuity: o function continuous at a number o function continuous on its domain o discontinuous functions o continuous from one side only o linear combinations of continuous functions are continuous (Thm 4 p127)
o continuous functions: polynomials, rational functions, root functions, trig functions, inverse trig functions, exponential functions, and their inverse functions (logarithmic functions), composition of continuous functions --p129. o Intermediate Value Theorem (IVT) 2.6 Limits at Infinity: o Horizontal Asymptotes difference between infinite limits (and vertical asymptotes--section 2.2) and limits at infinity (and horizontal asymptotes) o Thm 5 p138 o infinite limits at infinity: Example 9 p142: cannot subtract infinity from infinity. The alternatives are: (a) factor out to get a product of infinity and infinity, or (b) cancel: divide the numerator and the denominator by the highest power of x in the denominator (so that the denominator takes a finite value) o skip pages 143-146 Wed. Oct 5th: 2.7 Tangents, Velocities, and Other Rates of Change o slope as a limit o average velocity versus instantaneous velocity (as a limit) o average rate of change versus instantaneous rate of change 2.8 Derivatives o derivative of a function at a point o derivative of a function ("slope of the tangent line" or "instantaneous rate of change") Th. Oct 6th: 2.9 The Derivative as a Function: o formula for the derivative of a function (versus the derivative of a function at a point as given in section 2.8) o differentiable function on an interval (if the derivative exits everywhere on the interval. Examples and counterexamples?) o f is differentiable implies f is continuous o f is continuous does not imply f is differentiable (counterexample?) o not differentiable functions (find examples for each case): 1. there is no slope at the point, i.e the function has a corner 2. the function is not continuous at the point 3. there is a vertical tangent line at the point Fri. Oct 7th: Exam over Ch 2 Mo. Oct 10th: HOLIDAY - COLUMBUS DAY Tu. Oct 11th: 3.1 Derivatives of Polynomials and Exponential Functions: o derivatives of: constant functions, x^n o constant multiplication, sun and difference of derivatives o definition of number "e" o derivative of e^x o finding points where (1) a curve has horizontal tangent lines, or (2) the tangent line is parallel or perpendicular to a given line 3.2 The Product and Quotient Rules: o product and quotient rules o equation of the tangent line using the derivative to find the slope o table on page 197
3.4. Derivatives of the Trig Functions: o finding the derivative of sin x o equation 2 page 312 and applications (Example 4 and 5) o derivatives of the trig functions (page 314) 3.5 The Chain Rule: o the chain rule on p 218 and examples o the power rule combined with the chain rule o derivative of a^x, for any constant Wed. Oct 12th: 3.6 Implicit Differentiation: o implicit differentiation = finding derivatives of functions without solving for y since it is easier o orthogonal trajectories (families of curves that are orthogonal, i.e. they have perpendicular tangent lines at any point of intersection) o derivatives of inverse trig functions (p 233) 3.7 Higher Order Derivatives: o finding the second derivative by taking the derivative of the first derivative, and higher derivatives o finding the nth derivative for a function o n! 3.8 Derivatives of Logarithmic Functions: o derivatives of the log and ln o logarithmic differentiation o power rule o e as a limit Th. Oct 13th: 3.9 Hyperbolic Functions o hyperbolic function and hyperbolic identities o derivatives of hyperbolic functions o skip inverse hyperbolic functions 3.10 Related Rates Steps in solving problems using related rates.: find a formula that relates the variables differentiate both sides of the formula to find the related rates substitute the given values positive rate means that the variable increases in time vs. negative rate which means that the variable decreases in time. Make sure you check this part after solving problems. 3.11 Linear Approximations & Differentials o linear approximation and its applications (linear approximations are used to approximate a function near a point) o approximations of sin x and cos x o differentials (dy is the rise or fall of the tangent line when the change in x is dx) o use of differentials to find maximum computational errors Fri. Oct 14th: Exam 3. Mo. Oct 17th: 4.1 Maximum & Minimum Values o extreme values: absolute maximum and absolute minimum (the value f(c) as a y-value is the min/max) o local max and local min. o Extreme Value Thm. (function needs to be continuous)
o Fermat's Thm: f'(c) = 0 then f MAY HAVE a local max or min at x = c (i.e. f(c) is the local max or min) o to find min and max for f look for values in the domain that are critical numbers: 1. if f'(c) = 0, then it is possible for f to have a max or min at c 2. if f'(c) does not exist (DNE), then it is possible for f to have a max or min at c o to find the absolute max or min: Closed Interval Method (check the end points of the interval, plus the points where f' = 0 or f' DNE) 4.2 Mean Value Theorem o Rolle's Thm --used to find local min/max by finding the zero of the derivative o MVT --used to find a point "c" with a particular derivative ( the tangent line at c has the same slope as the secant line that connects the end points of the interval considered) o if f'(x) = 0 for all values of x in (a, b), then f is constant on (a, b) -- used to show that the sum or difference of two functions can be a constant (Corollary 7 and Example 6) Tu. Oct 18th: 4.3 How Derivatives Affect the Shape of Graphs o f'(x) : 1. Increasing and decreasing test (helps decide where the function is increasing (f'>0) or decreasing (f'<0)) 2. 1st derivative test (helps decide when f'(x) = 0 produces a local min/max) o definitions of concave up and concave down o inflection point (f changes concavity when f"=0 or when f"dne) o f''(x): concavity test--helps decide the concavity of the function (up (f">0) or down(f"<0)) 2nd derivative test :finding max and mins of a function (f'=0) depending on the value of f'': 1. f">0 gives a min 2. f"<0 gives a max 3. f"=0 gives a POSSIBLE max, min or inflection point use the tests above + vertical asymptotes (VA) and horizontal asymptotes (HA) to sketch the graph of a function. 4.4 Indeterminate Forms & L Hospitals Rule o When evaluating limits, one can use cancellation laws (section 2.3) or if the limit is at infinity divide by the highest power of x in the denominator (section 2.6). Not all functions will work that way, so these are possible alternative methods for each indeterminate form. Wed. Oct 19th: 4.5 Curve Sketching o Guidelines for sketching a curve: Domain, intercepts, symmetry, asymptotes, f' (increasing, decreasing, max and mins), f" (concavity and inflection points). 4.7 Optimization Problems o steps in modeling a problem: draw a diagram, introduce notation, relate by equations the variables introduced, reduce the equations to one equation with one variable only and use derivatives to find absolute max or min (optimization). 4.9 Newton's Method (how calculators find roots) o Steps in Newton's method. Th. Oct 20th: 4.10 Antiderivatives o defn. of antiderivative F(x)
o general antiderivative F(x) + C o table of antiderivatives o using antiderivatives to solve differential equations (answer is a general antiderivative like Example 2) o using antiderivatives to solve differential equations with an initial condition (answer is a particular antiderivative like Example 3) o finding the graph of the antiderivative of f (the function f gives the slope of F at each point which creates a direction field) o applications of solving differential equations by antidifferentiation Fri. Oct 21st: Exam 4 Mo. Oct 24th: 5.1 Areas & Distances o finding the area of the region under a curve by approximating the area with rectangles o if the rectangles have their heights to be the right endpoint of the rectangle: R_n o if the rectangles have their heights to be the left endpoint of the rectangle: L_n o more rectangles we consider, better approximation we obtain. If n goes to infinity, then we obtain the real value of the area under the curve versus an approximation (definition 2 page 374, which would work with L_n as well). o sigma notation o formula 1 page 372 (and its sigma notation formula page 375) o application: approximating the distance traveled by an object 5.2 Definite Integrals & Riemann Sums o definition of integral o terminology: Riemann integral, integrand, limits of integration (upper and lower limit) o finding the sample points for the integral formula (use either the right endpoints, the left endpoints, or midpoints -- often we use the right endpoints) o formulas 4-10 page 383 (they are used in finding the limit that will give the area under the curve) o standard problem: #2 page 383 o properties of the integrals (their meaning, too): formulas 1-8 pages 388 and 389. Tu. Oct 25th: 5.3 Fundamental Theorem of Calculus o Fundamental Theorem of Calculus (FTC) parts 1 and 2 (review antiderivatives) and their meaning 5.4 Indefinite Integrals & the Net Change Theorem o difference between definite integral (a number) and indefinite integral (a function) see page 405 in red. o table of integrals page 406 (see connection to the derivatives) o Net Change Theorem -- FTC2 5.5 Substitution Rule o substitution rule for definite versus indefinite integrals (for definite integrals you need to change the limits) o make sure that you only have one variable within an integral, and it matches the "dx" or "du" part of the integral o if it seems hard to impossible to solve integrals, check for the symmetry of the function and use info on page 419
Wed. Oct 26th: 7.1 Integration by Parts o integration by parts formula for indefinite integrals page 476 o integration by parts formula for definite integrals page 479 Th. Oct 27th: Review Fri. Oct 28th: Final Exam Review For Final: 1. 3-5 limits (left-hand and right-hand limits, limits at infinity (2.6.), limits laws (2.3.), and L Hospital(4.4.)) 2. 3-5 derivatives (including second derivative(3.7.), logarithmic (3.8.) and implicit differentiation(3.6)) 3. 3-5 integrals (includes integration by parts (7.1.) and substitution rule(5.5)) 4. find the equation of the tangent line, or line perpendicular or parallel to a given line 5. show that a function is continuous 6. find H.A. and V.A. (2.6.) 7. applications: word problems (3.10) and optimization problems (4.7) 8. find min/max/inflection points of a function (chapter 4) 9. Evaluate definite integral using Riemann sum (5.2)