﻿ Calculus I Mathematics Daily Syllabus homepage: Dr. Carol JVF Burns

# Calculus I (MAT 136) Daily Syllabus (with homework)

 WEEK #2 CLASS # DATE CLASS CONTENT HOMEWORK 4 M, Jan 23 Sample Prerequisite Problems (PreCalculus) (Solutions included) Questions over 8–15? Questions over web exercises? STUDY FOR PREREQUISITE REVIEW QUIZ! 5 W, Jan 25 PREREQUISITE REVIEW MAJOR QUIZ (80 pts) 6 Th, Jan 26 1ab average rate of change, the tangent problem in-class activity: Average ROC versus Instantaneous ROC Worksheet Worksheet SOLUTIONS 2ab position functions; derivative of position is velocity; derivative of velocity is acceleration in-class activity (finish as part of homework): Position and Velocity Functions Worksheet SOLUTIONS (1,2,3) Worksheet SOLUTIONS (4) finish worksheets, as needed web exercise: Average Rate of Change short in-class quiz on Monday WebWorks: Average Rate of Change (2 problems, 4 pts each, 8 pts) DUE: Friday, Jan 27, end of day WebWorks: Rate of Change (5 problems, 2 pts each, 10 pts) DUE: Friday, Jan 27, end of day 7 F, Jan 27 3ab intuitive derivative (notation); finding derivative values 4ab limit of a function; 3 cases where the limit of $f(x)$, as $x$ approaches $a$, equals $\ell$ 5ab one-sided limits; the Squeeze (Pinching) Theorem text, Exercises 2.4.15 and 2.4.16, finding limit values (finish for homework, as needed) WebWorks: Intuitive Derivative (12 pts) DUE: Friday, Feb 3, end of day WebWorks: Limits (13 pts) DUE: Friday, Feb 3, end of day web exercise: Introduction to Limits

 WEEK #3 CLASS # DATE CLASS CONTENT HOMEWORK 8 M, Jan 30 6ab continuity at a point; three ways a function can fail to be continuous at $c$ 7ab key idea: if $f$ is continuous at $c$, then evaluating a limit is as easy as direct substitution 8ab an important type of continuity problem; relationship between differentiability and continuity short in-class quiz; Average Rate of Change web exercise WebWorks: Continuity (7 problems, 2 pts each, 14 pts) DUE: Friday, Feb 3, end of day DUE: Monday, Feb 6, end of day 9 W, Feb 1 9ab limit rules; techniques for evaluating limits (examples) 10ab indeterminate forms Samples: recognizing indeterminate forms in-class: worksheet #1 on recognizing indeterminate forms in-class: worksheet #2 on recognizing indeterminate forms worksheet #1 solutions worksheet #2 solutions WebWorks: Limit Rules (20 problems, 1 pt each, 20 pts) DUE: Friday, Feb 3, end of day DUE: Monday, Feb 6, end of day 10 Th, Feb 2 11ab the derivative of $f$ at $a$ 12ab more examples: using the definition of derivative WebWorks: Basic Derivatives (8 problems, 1 pt each, 8 pts) DUE: Friday, Feb 3, end of day DUE: Monday, Feb 6, end of day 11 F, Feb 3 13ab the linearization of $f$ at $a$ 14ab local/global max/min (extreme values) WebWorks: Local Linearization (6 problems, 2 pts each, 12 pts) DUE: Friday, Feb 10, end of day WebWorks: Extreme Values (6 problems, 2 pts each, 12 pts) DUE: Friday, Feb 10, end of day

 WEEK #4 CLASS # MWThF DATE CLASS CONTENT HOMEWORK 12 M, Feb 6 15ab strictly increasing and decreasing functions; getting inc/dec behavior from the first derivative 16ab concavity; getting concavity info from the second derivative WebWorks: Direction and Sign of the Derivative (2 problems, 2 pts each, 4 pts) DUE: Friday, Feb 10, end of day Hint: you will need to know the vertex formula for a quadratic function WebWorks: Concavity (9 problems, 2 pts each, 18 pts) DUE: Friday, Feb 10, end of day Suggestion: Do these homework sets before the first exam! 13 W, Feb 8 Worksheet, page 1: Exploring Increasing/Decreasing Function/Slope Behavior Worksheet, page 2: Exploring Increasing/Decreasing Function/Slope Behavior Worksheet Solutions, page 1 Worksheet Solutions, page 2 Worksheet, page 1: More Practice with Function/First Derivative/Second Derivative Behavior Worksheet, page 2: More Practice with Function/First Derivative/Second Derivative Behavior Worksheet Solutions, page 1 Worksheet Solutions, page 2 Finish the in-class worksheets. Study for Exam #1. Bring good questions to class tomorrow! 14 Th, Feb 9 catch-up; review 15 F, Feb 10 EXAM #1

 WEEK #5 CLASS # DATE CLASS CONTENT HOMEWORK 16 M, Feb 13 17ab the Power Rule for differentiation; examples of using the power rule 18ab partial proof of the Power Rule (nonnegative integers) WebWorks: Power Rule (10 problems, 1 pt each, 10 pts) DUE: Friday, Feb 17, end of day 17 W, Feb 15 19ab derivative of a constant times a function; finding $\frac{d}{dx}(Kx^n)$ 20ab derivatives of sums and differences; proof WebWorks: Derivatives of Linear Combinations (11 problems, 1 pt each, 11 pts) DUE: Friday, Feb 17, end of day 18 Th, Feb 16 21ab differentiating composite functions—motivation; differentiating composite functions—the idea 22ab the Chain Rule (prime notation); the Chain Rule (Leibnitz notation) 23ab Why is it called the Chain Rule? Using the Chain Rule to differentiate $(f(x))^n$ 24ab pattern for generalizing all the basic differentiation formulas; examples WebWorks: Derivatives of Compositions (5 problems, 2 pts each, 10 pts) DUE: Friday, Feb 17, end of day 19 F, Feb 17 25ab the Product Rule for differentiation; Proof of the Product Rule 26ab When is $\displaystyle\lim_{h\rightarrow 0} f(x+h) = f(x)\,$? When is this true? As $x\rightarrow c\,$, $f(x) \rightarrow f(c)$ 27ab the Quotient Rule for Differentiation; Proof of the Quotient Rule WebWorks: Derivatives of Products and Quotients (19 problems, 1 pt each, 19 pts) DUE: Friday, Feb 24, end of day Practice this web exercise to firm up your understanding of limits: Introduction to Limits Also practice this web exercise: Basic Differentiation Shortcuts Next week (Wednesday and Thursday) there will be quizzes over these two web exercises (in the form of printed, randomly-generated worksheets)

 WEEK #6 CLASS # DATE CLASS CONTENT HOMEWORK 20 M, Feb 20 28ab the irrational number e; equivalent limit statements defining e 29ab evaluating a limit involving e; why is $\displaystyle\lim_{n\rightarrow\infty} [(1+\frac 1n)^n]^{2k} = {\text{e}}^{2k}\,$? 30ab derivatives of logarithms; proof that $\displaystyle\frac{d}{dx} \ln x = \frac 1x$ 31ab more notation for derivatives; notation: evaluating a derivative at a point WebWorks: Natural Base e (3 problems, 1 pt each, 3 pts) DUE: Friday, Feb 24, end of day WebWorks: Derivatives of Logarithmic Functions (6 problems, 1 pt each, 6 pts) DUE: Friday, Feb 24, end of day web exercise: Introduction to Limits (quiz on Wednesday) web exercise: Basic Differentiation Shortcuts (quiz on Thursday) 21 W, Feb 22 32ab differentiating $\,{\text{e}}^x\,$; differentiating $\,a^x$ web exercise QUIZ: Introduction to Limits WebWorks: Derivatives of Exponential Functions (4 problems, 1 pt each, 4 pts) DUE: Friday, Feb 24, end of day 22 Th, Feb 23 33ab what is $\displaystyle\,\lim_{x\rightarrow 0}\frac{\sin(x)}{x}\,$; an example 34ab derivatives of sine and cosine; derivatives of tangent and cotangent 35ab derivatives of secant and cosecant; a useful observation—derivatives of co-functions web exercise QUIZ: Basic Differentiation Shortcuts WebWorks: Derivatives of Trigonometric Functions (20 problems, 1 pt each, 20 pts) DUE: Friday, Feb 24, end of day 23 F, Feb 24 36ab finding the derivative of an inverse function; two methods of finding $\,(f^{-1})'(x)$ 37ab What does $\,(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}\,$ really mean? WebWorks: Derivatives of Inverse Functions (4 problems, 1 pt each, 4 pts) DUE: Friday, Mar 2, end of day Prepare for next week's Gateway Differentiation Quizzes

 WEEK #7 CLASS # DATE CLASS CONTENT HOMEWORK 24 M, Feb 27 short Gateway Differentiation Quiz at end of class (20 pts) 38ab derivatives of the inverse trigonometric functions; a typical derivation WebWorks: Derivatives of Inverse Trigonometric Functions (6 problems, 1 pt each, 6 pts) DUE: Friday, Mar 2, end of day 25 W, Feb 29 short Gateway Differentiation Quiz at end of class (20 pts) 39ab differentiating variable stuff to variable powers—the log trick; example (differentiating $x^x$) WebWorks: the Log Trick (5 problems, 1 pt each, 5 pts) DUE: Friday, Mar 2, end of day 26 Th, Mar 1 short Gateway Differentiation Quiz at end of class (20 pts) catch-up; review 27 F, Mar 2 EXAM #2

 WEEK #8 CLASS # DATE CLASS CONTENT HOMEWORK 28 M, Mar 5 40ab l'Hopital's Rule (for investigating indeterminate forms); motivation for the ‘$\frac 00$’ case 41ab examples: using l'Hopital's Rule (basic; more advanced) WebWorks: l'Hopital's Rule (6 problems, 1 pt each, 6 pts) DUE: Friday, Mar 9, end of day 29 W, Mar 7 42ab where can a function change its sign (from positive to negative, or negative to positive); negating ‘and’ and ‘or’ sentences 43ab information given by the sign of $f$, $f'$, and $f''$; a basic sign analysis of a function 43cd, math language material: implications; the contrapositive of an implication continue WebWorks: l'Hopital's Rule (6 problems, 1 pt each, 6 pts) DUE: Friday, Mar 9, end of day 30 Th, Mar 8 44ab Where can a function have a local max/min? At a horizontal tangent line; at a place where the function is not differentiable; at an endpoint of the domain 45ab definition: critical numbers (points) for a function; these give the candidates for places where there are max/min; careful—you may not have max/min at critical points 46ab inflection points; Where can a function have an inflection point? At a place where $f''(x) = 0$ or where $f''(x)$ does not exist begin WebWorks: Function Analysis (9 problems, 1 pt each, 9 pts) DUE: Friday, Mar 9, end of day 31 F, Mar 9 47ab an algorithm for thorough function analysis; example: implementing the algorithm for thorough function analysis (creating a derivative chart) 48ab deciding if candidates are local max/min: the first derivative test; the second derivative test Have a wonderful Spring Break!! (March 12–16) continue WebWorks: Function Analysis (9 problems, 1 pt each, 9 pts) DUE: Friday, Mar 9, end of day

 WEEK #9 CLASS # DATE CLASS CONTENT HOMEWORK 32 M, Mar 19 49ab open/closed intervals; bounded subsets of $\Bbb R$ 50ab review of absolute/global max/min; the Extreme Value Theorem 51ab optimization problems; finding the absolute max/min of a continuous function on a closed bounded interval 52ab a first optimization problem; you try it! begin WebWorks: Optimization (8 problems, 3 pts each, 24 pts) Keep working on it all week! DUE: Friday, Mar 23, end of day 33 W, Mar 21 53ab an optimization problem; you try it! 34 Th, Mar 22 53cd a useful observation (if $f \gt 0$ and $f$ has a max/min at $c$, then $f^2$ also has a max/min at $c$) 54ab an optimization problem; you try it! 35 F, Mar 23 55ab explicit versus implicit; implicit differentiation WebWorks: Implicit Differentiation (6 problems, 2 pts each, 12 pts) DUE: Friday, Mar 30, end of day

 WEEK #10 CLASS # DATE CLASS CONTENT HOMEWORK 36 M, Mar 26 56ab related rate problems; example (falling ladder) WebWorks: Related Rates (3 problems, 4 pts each, 12 pts) DUE: Friday, Mar 30, end of day 37 W, Mar 28 57ab Is the car speeding? (another related rate problem) 38 Th, Mar 29 58ab the Mean Value Theorem; interpretation and intuition 59ab the Mean Value Theorem: examples WebWorks: the Mean Value Theorem (4 problems, 2 pts each, 8 pts) DUE: Friday, Mar 30, end of day 39 F, Mar 30 60ab bounded functions; the definite integral of a function 61ab a definite integral may or may not exist; notation for the definite integral 62ab dummy variables in definite integrals; simple definite integral examples WebWorks: Definite Integrals as Signed Area (7 problems, 2 pts each, 14 pts) DUE: Friday, Apr 6, end of day

 WEEK #11 CLASS # DATE CLASS CONTENT HOMEWORK 40 M, Apr 2 Optimization problem quiz (10 minutes, 20 points) 63ab properties of the definite integral; linearity of the integral 64ab definite integral examples continue working on: WebWorks: Definite Integrals as Signed Area (7 problems, 2 pts each, 14 pts) DUE: Friday, Apr 6, end of day 41 W, Apr 4 more practice with complete function analysis study for Exam #3 42 Th, Apr 5 review study for Exam #3 43 F, Apr 6 EXAM #3

 WEEK #12 CLASS # DATE CLASS CONTENT HOMEWORK 44 M, Apr 9 65ab how might we estimate areas beneath a curve; general notation for the area problem 66ab summation notation; left and right estimates using summation notation 67ab precise definition of a definite integral; Riemann sums 68ab estimating a definite integral using a Riemann sum; finding a very simple definite integral using the definition of definite integral Read and study this online web exercise: Summation Notation Print out a worksheet, write in all the answers, and pass it in as tomorrow's Quick Quiz. There will be a QUIZ (a randomly-generated worksheet) on THURSDAY, APRIL 19. WebWorks: Riemann Sums (6 problems, 2 pts each, 12 pts) DUE: Friday, Apr 13, end of day 45 W, Apr 11 69ab the Fundamental Theorem, Part I; proof of the Fundamental Theorem 70ab antiderivatives; ‘undoing’ differentiation—antidifferentiation 71ab every differentiation formula gives an antidifferentiation formula; some antiderivatives you should know 72ab examples: using the Fundamental Theorem 73ab a very useful formula; integrating a rate of change gives total change WebWorks: Fundamental Theorem (9 problems, 2 pts each, 18 pts) DUE: Friday, Apr 13, end of day 46 Th, Apr 12 74ab connection between area and antiderivatives; making it precise 75ab continuing: making it precise practice: print out this worksheet and fill in the blanks 76ab summary: the Fundamental Theorem of Calculus 77ab examples: derivatives involving integrals WebWorks: Antiderivatives (7 problems, 2 pts each, 14 pts) DUE: Friday, Apr 13, end of day 47 F, Apr 13 78ab indefinite integrals (general antiderivatives); more antiderivative formulas 79ab finding a particular antiderivative; example 80ab generalizing the formula for the derivative of $\ln x$; the antiderivatives of $\frac 1x$ WebWorks: Indefinite Integrals (6 problems, 2 pts each, 12 pts) DUE: Friday, Apr 20, end of day

 WEEK #13 CLASS # DATE CLASS CONTENT HOMEWORK 48 M, Apr 16 81ab the substitution technique for integration; format for substitution problems 82ab a substitution example; same problem; more compact version 83ab substitution with definite integrals; two approaches WebWorks: Substitution (7 problems, 2 pts each, 14 pts) DUE: Friday, Apr 20, end of day 49 W, Apr 18 Lots of substitution problems! Study for tomorrow's quizzes! 50 Th, Apr 19 more substitution problems; homework questions; catch-up day QUIZ: Summation Notation randomly-generated worksheet (11 problems, 2 pts each; 22 pts) QUIZ: Fill-in-the-blank worksheet: connection between area and antiderivatives (16 blanks, 1 pt each; 16 pts) 51 F, Apr 20 84ab integration by parts; strategies for using parts 85ab format for integration by parts problems; using parts with a definite integral get started on: WebWorks: Integration by Parts (8 problems, 2 pts each, 16 pts) DUE: Friday, Apr 27, end of day

 WEEK #14 CLASS # DATE CLASS CONTENT HOMEWORK 52 M, Apr 23 86ab periodically differentiable functions; using parts with products of polynomials and periodically differentiable functions 87ab inverse functions with simpler derivatives; using parts for inverse functions 88ab using parts with products of periodically differentiable functions; it doesn't always work continue working on: WebWorks: Integration by Parts (8 problems, 2 pts each, 16 pts) DUE: Friday, Apr 27, end of day 53 W, Apr 25 89ab analyzing falling objects: $m x''(t) = mg$; integrate to find velocity and position (height) Integration problems—all mixed up! (from textbook, Section 5.8) WebWorks: Integration Mixed Practice (6 problems, 2 pts each, 12 pts) DUE: Friday, Apr 27, end of day 54 Th, Apr 26 review for Exam #4 (68b and 69b will be EXTRA CREDIT on this exam) Study for Exam #4!! 55 F, Apr 27 EXAM #4 This exam will include a worksheet from: Differentiation Formula Practice Study for the first mixed integration quiz! Study for the final exam!

 WEEK #15 (end-of-term week) CLASS # DATE CLASS CONTENT HOMEWORK 56 M, Apr 30 mixed integration quiz #1 (18 problems, 25 pts, recorded as 20 pts) review for final exam (first hour exam) Study for the second mixed integration quiz! Study for the final exam! 57 W, May 2 mixed integration quiz #2 (5 problems, 25 pts, recorded as 20 pts) review for final exam (second hour exam) Study for the third mixed integration quiz! Study for the final exam! 58 Th, May 3 mixed integration quiz #3 (5 problems, 25 pts, recorded as 20 pts) review for final exam (third hour exam) Study for the last mixed integration quiz! Study for the final exam! 59 F, May 4 mixed integration quiz #4 (5 problems, 25 pts, recorded as 20 pts) review for final exam (third hour exam) review for final (fourth hour exam) Bring INDEX CARDS to FINAL EXAM to be graded: -- they MUST have a rubber-band around them, or be in a plastic bag or zippered pouch -- your name and number must clearly appear on the top of the pile -- they must be in INCREASING ORDER Study for the final exam! final for 8:00 class: Monday, May 7, 7:30–9:30 AM, ROOM 221 final for 11:30 class: Wednesday, May 9, 10:00–noon, ROOM 147