MAC 2311 - CALCULUS 1 Section 003 - Fall 2018
This page is devoted to Calculus 1 - Section 003 of Fall 2018 at USF.
Note: All the material (notes, homework, reviews, quizzes, slides, etc.) you will find in this page has been realized by me, unless otherwise specified (the videos has been recorded by Robert Connelly). I am more than happy if you want to use part of it with your students, but in this case please give credits by sharing the URL of this webpage.
Time and location of class:
MW: 12:30 pm – 13:45 pm in CHE 217
F: 12:30 pm – 1:20 pm in CHE 217 (peer leading session)
Office hours (CMC 110):
Monday 11:00 am - 12:15 pm
Wednesday 11:00 am - 12:15 pm and 2:00 pm - 3:00 pm
Contents:
The course in organized in the following way:
- Functions and limits
- Derivatives
- Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions
- Applications of Differentiation
- Integrals
Textbook: Essential Calculus: Early Transcendentals, 2nd edition, by Stewart.
Here, you can find more information about the course of Calculus 1 at USF.
- The final exam will take place on Saturday, December 1 from 3 – 5 pm
- On Thursday, November 29, I am holding a review session for the final exam from 2pm to 6pm in CMC 110.
- The third test will take place on Saturday, November 3, in CHE 100, from 9 – 10:15 a.m.
- On Thursday, November 1, I am holding a review session for the third test from 2pm to 6pm in NES 104.
- The second test will take place on Saturday, October 13, in CHE 100, from 9 – 10:15 a.m.
- On Thursday, October 11, I am holding a review session for the second test from 2pm to 6pm in NES 104.
- The second homework is due on Wednesday October 10 at the beginning of the class.
- On Thursday, September 13 I am holding a review session for the first test from 2pm to 6pm in NES 104.
- The first test will take place on Saturday, September 15, in CHE 100, from 9 – 10:15 a.m.
- The first homework is due on Wednesday September 12 at the beginning of the class.
- Here or at the bottom of this page you can find the course documents (notes, quizzes, homework, etc.) of Calculus 1 - Spring 2018 & Fall 2017 .
Logbook:
- 08/20/18: Presentation of the course. First day attendance. Introduction to calculus. Notation. Function: definition, domain, range, different representations, graph of a function, example. Vertical line test. Exercise (find the domain of a function).
Introduction to calculusNotationReview - 1.1, 1.2Slides - Introduction to calculus - 08/22/18: Operations between functions (example about the composition of two functions). Limit of a function (idea, notation, intuitive way with table(s) of values, examples, definition). One-sided limits (left-hand, right-hand limit).
Limit of a function - 1.3Video - 1.3 - 08/27/18: Limit laws. Direct substitution property. Classical examples: rational function $``\frac{0}{0}"$, conjugate, special limit $\lim_{x\to 0} \frac{\sin(x)}{x}=1$, limit of a piecewise-defined function. Examples.
Calculating limits - 1.4Video - 1.4 - 08/29/18: Quiz 1. Comments on Quiz 1. Squeeze theorem. Continuity: definition, kinds of discontinuity (removable, jump, infinite) and examples.
Continuity - 1.5Video - 1.5 - 09/05/18: Quiz 2. Theorems about continuity (sum, difference, product, quotient, composition of continuous functions). Exercise about continuity (piecewise defined function with a parameter). Intermediate Value Theorem. Exercise about IVT (show that an equation has at least one solution in an interval).
- 09/10/18: Limits of the form $``\frac{1}{0}"$ (examples). Limits at $\infty$ and $-\infty$. "Operations" with $\infty$ and $-\infty$ (indeterminate forms). Limit at infinity of a rational function $\frac{P(x)}{Q(x)}$ (three cases with examples: $\deg(P)=\deg(Q)$, $\deg(P)<\deg(Q)$ and $\deg(P)>\deg(Q)$). Definition of vertical and horizontal asymptote.
Limits involving infinity - 1.6Video - 1.6 - 09/12/18: Introduction to derivatives: instantaneous velocity, slope of the tangent line to a graph at a point, instantaeous rate of change. Definition of derivative and derivative function (examples). Definition of differentiability. Examples: $f(x)=\sqrt{x}, |x|$. Theorem: differentiability implies continuity. How a function can fail to be differentiable.
Introduction to derivatives - 2.1Derivative as a function - 2.2Video - 2.1Video - 2.2 - 09/13/18: Review session Test 1.
- 09/17/18: Logic implication . Proof of the theorem: differentiability implies continuity. Lagrange and Leibniz' notation for the derivative function. Derivatives of higher order. Differentiation rules: derivative of a constant function, power rule (examples: $x,x^2,x^3,x^{-1}=\frac{1}{x},x^{\frac{1}{2}}=\sqrt{x}$), derivatives of sine and cosine, constant multiple rule. Examples.
Slides - Logic implicationDifferentiation rules - 2.3, 2.4Video - 2.3Video - 2.4 - 09/19/18: Differentiation rules: sum/difference rule (with proof), product rule (with proof), quotient rule and examples. Derivative of $\tan(x)$. Introduction to chain rule.
- 09/24/18: More examples about chain rule. Double chain rule. Functions explicitely defined. Functions implicitely defined: example $x^2+y^2=1$ (algebraic and geometric interpretation). Problem: find the tangent line to the curve $x^2+y^2=1$ at $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Resolution without implicit differentiation.
Chain rule - 2.5 - 09/26/18: Quiz 3. Comments on quiz 3. Method of implicit differentiation (description in 4 steps). Exercise 1: find an equation of the tangent line to the curve described by the equation $y^2+xy=\cos(y)-\sin(x)$ at the point $(\frac{\pi}{2},0)$. Exercise 2: find an equation of the tangent line to the heart curve described by the equation $(x^2+y^2-1)^3=x^2y^3$ at the point $(1,1)$.
Implicit differentiation - 2.6Video - 2.6 - 10/01/18: Review of the definition of the instantaneous rate of change: with respect to time (rate of change of the position function with respect to time -> velocity; rate of change of the velocity function with respect to time -> acceleration) or other quantities (rate of change of the pressure of an object submerged in a fluid with respect to the depth of the object). Presentation of a related rate problem through steps. Examples: square with the length of the sides increasing, air pumped into a spherical ballon, ladder sliding down.
Related rates - 2.7Video - 2.7 - 10/03/18: Quiz 4. Comments on quiz 4. How does an exponential function look like? How to define $2^{\sqrt{2}}$? Review: definition of exponentiation with natural numbers as exponent and properties; extension of the range of the exponent to integer, rational and real numbers.
Exponential functions - 3.1 - 10/08/18: The exponential function $f(x)=a^x$ and its graph depending on the base $a$. The "most convenient" base: the Euler's constant $e$ (definition as a limit). Natural exponential function and properties. Definition of one-to-one functions, horizontal line test, examples. Definition of the inverse function of a one-to-one function, cancellation laws. The logarithmic function with base $a$, properties, cancellations laws and laws of logarithm. Notation $\log$, $\ln$. The natural logarithmic function and properties. Derivatives of logarithmic and exponential function and examples. Proof of the derivative of $e^x$ with logarithmic differentiation.
Inverse functions and logarithms - 3.2Derivatives of $\log_a(x)$ and $a^x$ - 3.3Video - 3.3 - 10/10/18: Derivative of $f(x)=g(x)^{h(x)}$: easy cases (either $g$ or $h$ constant), method 1 (rewrite $f(x)=g(x)^{h(x)}=e^{h(x)\ln(g(x))}$), method 2 (logarithmic differentiation). Examples: $f(x)=x^x$, $f(x)=x^{\cos(\pi x)}$. Method of linear approximation for approximating the value of a function with examples.
Linear approximation - 2.8 - 10/11/18: Review session Test 2.
- 10/15/18: Definition of absolute/local maximum/minimum value for a function $f$. Geometrical interpretation and examples. Extreme Value Theorem and remarks. Fermat's theorem, proof and remarks. Definition of critical numbers. Generalized version of Fermat's theorem.
Maximum and minimum values - 4.1 - 10/17/18: Closed interval method and example. Introduction to Rolle's theorem (a ball thrown in the air). Rolle's theorem (with proof). The Mean Value Theorem and its geometrical interpretation.
Mean Value Theorem - 4.2Video - 4.2 - 10/22/18: Review statement of the Mean Value Theorem and two related exercises. Proposition: if $f'(x)=0$ on $(a,b)$ then $f$ constant on $(a,b)$ (with proof). Definition of strictly increasing/decreasing. Increasing/decreasing test (with proof).
- 10/24/18: Quiz 5. Comments on Quiz 5. Review statement of the Mean Value Theorem and example. Proposition: $f'(x)=0$ on $(a,b)$ then $f$ constant on $(a,b)$ (with proof). Definition of strictly increasing/decreasing. Increasing/decreasing test (with proof). First derivative test. Example.
Derivatives and shapes of the graphs - 4.3Video - 4.3 Part 1Video - 4.3 Part 2 - 10/29/18: Definition of concave up/down and inflection point. Concavity test. Example. Exercises 1,2, and 3 from In-class review session.
Video - 4.5 Part 1Video - 4.5 Part 2 - 10/31/18: Quiz 6. Comments on Quiz 6. Exercises 5 and 6 from In-class review session.
- 11/05/18: Review: definition of one-to-one function and inverse function. Inverse trigonometric functions: $\sin^{-1}$ ($\arcsin$), $\cos^{-1}$ ($\arccos$), $\tan^{-1}$ ($\arctan$). Domain and range, graph, cancellation equations. Examples of simplification: $\tan\left(\arcsin\left( \frac{1}{3}\right)\right)$, $\cos(\tan^{-1}(x))$, $\sin(\cos^{-1}(2x))$.
Inverse trig. functions - 3.5Video - 3.5 - 11/07/18: Derivatives of inverse trigonometric functions (proof of the derivative of $\sin^{-1}(x)$). Indeterminate forms and examples. Statement of l'Hospital's rule.
- 11/14/18: Quiz 7. Comments on quiz 8. L'Hospital's rule and idea behind the proof. Examples. The case $0\cdot \infty$ and $0^0, 1^{\infty}, \infty^0$ with examples.
L'Hospital's rule - 3.7Video - 3.7 - 11/19/18: Quiz 8. Comments on quiz 8. Definition of an antiderivative. The most general antiderivative. Table of antiderivatives. Remark on the antiderivative of $\frac{1}{x}$. Exercise: Find the antiderivative of a function that satisfies a given condition.
Antiderivatives - 4.7Video - 4.7 - 11/21/18: Find the position function given the acceleration function, the initial velocity and the initial displacement. The area problem: easy cases ($f(x)=c$, $f(x)=\frac{1}{2}x$); $f(x)=x^2$ on $[0,1]$ and approximation through rectangles; general case of a positive function $\rightarrow$ Riemann sums (left, right, midpoint). The sum notation $\Sigma$. Definition of the definite integral as the limit of Riemann sums.
Riemann sums and definite integral - 5.1, 5.2Video - 5.1Video - 5.2 - 11/26/18: Review of the definition of the definite integral. The definite integral of a negative function. Defintion and properties of the indefinite integral. Exercises: approximation of an integral with Riemann sums; computation of the integral of a function whose graph is made by lines and semicircles. The integral function $g(x)=\int_{a}^{x}f(t) dt$. The integral function when $f(t)=c$ and $f(t)=mt$. Statement of the Fundamental Theorem of Calculus (Part 1 and Part 2).
The Fundamental Theorem of Calculus - 5.3, 5.4Video - 5.3, 5.4 part aVideo - 5.3, 5.4 part b - 11/28/18: Fundamental Theorem of Calulus: proof of Part 2 using Part 1. Exercises: evaluation of definites integrals, derivative of an integral function (with chain rule). Defintion and properties of the indefinite integral.
- 11/29/18: Review session Final Exam.
Course documents:
- Notes:Introduction to calculusNotationReview - 1.1, 1.2Limit of a function - 1.3Calculating limits - 1.4Continuity - 1.5Limits involving infinity - 1.6Introduction to derivatives - 2.1Derivative as a function - 2.2Differentiation rules - 2.3, 2.4Chain rule - 2.5Implicit differentiation - 2.6Related rates - 2.7Exponential functions - 3.1Inverse functions and logarithms - 3.2Derivatives of $\log_a(x)$ and $a^x$ - 3.3Linear approximation - 2.8Maximum and minimum values - 4.1Mean Value Theorem - 4.2Derivatives and shapes of the graphs - 4.3Inverse trig. functions - 3.5L'Hospital's rule - 3.7Antiderivatives - 4.7Riemann sums and definite integral - 5.1, 5.2The Fundamental Theorem of Calculus - 5.3, 5.4
- Slides: Introduction to calculusLogic implication
- Videos: (by Robert Connelly): 1.31.41.51.62.12.22.32.42.62.73.34.24.3 - Part 14.3 - Part 24.5 - Part 14.5 - Part 23.53.7
- Sheets: Def. & Th. - Test 1DerivativesSteps - Impl. diff., related rates, log. diff.
- Quizzes: Quiz 1Q1 solutionQuiz 2Q2 solutionQuiz 3Q3 solutionQuiz 4Q4 solutionQuiz 5Q5 solutionQuiz 6Q6 solutionQuiz 7Q7 solutionQuiz 8Q8 solutionQuiz 9
- Homework: Homework 1HW 1 solutionHomework 2HW 2 solution
- Reviews: Review Test 1Review Test 2In class review Test 3Review Test 3Review Final Exam
Documents from Calculus 1 Section 001 - Spring 2018:
- Notes: ReviewLimit of a functionCalculating limitsContinuityLimits involving infinityIntroduction to derivativesDerivative as a functionDifferentiation rulesChain ruleImplicit differentiationRelated ratesExponential functionsInverse function and logarithmsDerivatives of $\log_a(x)$ and $a^x$Linear approximationInverse trig. functionsL'Hospital's ruleMaximum and minimum valuesMean Value TheoremDerivatives and shapes of the graphsCurve SketchingAntiderivativesRiemann sums and the definite integralFundamental Theorem of Calculus
- Slides: Introduction to calculusLogic implication
- Videos (by Robert Connelly): 1.31.41.51.62.12.22.32.42.52.62.73.33.53.74.24.3 - Part 14.3 - Part 24.5 - Part 14.5 - Part 2
- Sheets: DerivativesSteps - Impl. diff., related rates, log. diff.
- Quizzes: Quiz 1Q1 solutionQuiz 2Q2 solutionQuiz 3Q3 solutionQuiz 4Q4 solutionQuiz 5Q5 solutionQuiz 6Q6 solutionQuiz 7Q7 solutionQuiz 8Q8 solutionQuiz 9Quiz 10
- Homework: Homework 1HW 1 solutionHomework 2HW 2 solutionHomework 3HW 3 solution
- Reviews: Review Test 1In class review Test 2Review Test 2In class review Test 3Review Test 3Review integralsIn class review Final ExamReview Final Exam