MAC 2311 - CALCULUS 1 Section 003 - Fall 2018


This page is devoted to Calculus 1 - Section 003 of Fall 2018 at USF.

SYLLABUS - slidesSCHEDULE


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.


Announcements
  • 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.