 ### MAC 2311 - Calculus 1 Section 001 - Spring 2018

This page is devoted to Calculus 1 - Section 001 of Spring 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: 11:00 am – 12:15 pm in CMC 130
F: 11:00 am – 11:50 am in CHE 303 (peer leading session)

Office hours (CMC 110):

Monday 12:30-1:30 pm
Wednesday 9:45-10:45 am and 2:00-3:30 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.

Announcements
• Here or at the bottom of this page you can find the course documents (notes, quiz, homework, etc.) of Calculus 1 - Fall 2017.
• The third homework is due on Wednesday April 4 at the beginning of the class.
• On Friday April 6 I am holding a review session for the third test from 2pm to 6pm in NES 104.

#### Logbook:

• 01/08/18: 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).
• 01/10/18: Composition of two functions. Limit of a function (idea, notation, intuitive way with table of values, examples, definition). One-sided limits (left-hand, right-hand limit). Limit Laws.
• 01/17/18: Quiz 1. Comments on quiz 1. Direct substitution property. Classical examples: rational function $\frac{0}{0}"$, conjugate, special limit $\lim_{x\to 0} \frac{\sin(x)}{x}=1$. Examples.
• 01/22/18: Limit of a piecewise function. Squeeze theorem. Continuity: definition, kinds of discontinuity (removable, jump, infinite) and examples. Theorems about continuity (sum, difference, product, quotient, composition of continuous functions).
• 01/24/18: Quiz 2. Exercise about continuity (piecewise function with a parameter). Intermediate Value Theorem. Exercise about IVT (show that an equation has at least one solution in an interval). Limits of the form $\frac{1}{0}"$ (examples).
• 01/29/18: 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. Introduction to derivatives (position functon and istantaneous velocity).
• 01/31/18: Quiz 3. Comments on Quiz 3. Istantaneous 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.
• 02/01/18: Review session Test 1.
• 02/05/18: Logic implication . Proof of the theorem: differentiability implies continuity. Differentiation rules: derivative of a constant function, power rule (ex: $x,x^2,x^3,x^{-1}=\frac{1}{x},x^{\frac{1}{2}}=\sqrt{x}$), derivatives of sine and cosine, sum rule (with proof), product rule (with proof). Examples.
• 02/07/18: Quotient rule and examples (derivative of $\tan(x)$). Chain rule (Lagrange and Leibniz notation) and examples. Introduction to implicit differentiation.
• 02/12/18: When do we need chain rule?: $\cos(7x)$, $7\cos(x)$,$\sin(x)\tan(x)$,$\sin(\tan(x))$,$(\sqrt{x})^2$,$\frac{1}{\cos(x)}.$ Other examples of derivatives ($\sqrt{\tan(x^2)},\frac{x\cos(2x)}{\sin(x)}$). 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.
• 02/14/18: Quiz 4. 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 $(1,1)$ (it is Valentine's day afterall!) .
• 02/19/18: 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.
• 02/21/18: Quiz 5. 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. 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).
• 02/26/18: 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. Logarithmic differentiation ($f(x)=x^x$). Proof of the derivative of $e^x$ with logarithmic differentiation.
• 02/28/18: Method of linear approximation for approximating the value of a function with examples. Review session Test 2.
• 03/01/18: Review session Test 2.
• 03/05/18: Inverse trigonometric functions: $\sin^{-1}$ (or $\arcsin$), $\cos^{-1}$ (or $\arccos$), $\tan^{-1}$ (or $\arctan$). Domain and range, graph, cancellation equations. Examples of simplification. Derivatives (proof of the derivative of $\sin^{-1}(x)$ and $\tan^1(x)$).
• 03/07/18: Indeterminate forms. Statement of l'Hospital's rule and idea behind the proof. Examples. The case $0\cdot \infty$ and $0^0, 1^{\infty}, \infty^0$ with examples.
• 03/19/18: Definition of absolute/local maximum/minimum value. Geometrical interpretation. Examples. Extreme Value Theorem and remarks. Fermat's theorem, proof and remarks. Definition of critical numbers and example. Generalized version of Fermat's theorem.
• 03/21/18: Quiz 6. Quick review of the previous class. Closed interval method and example. Rolle's theorem and Mean Value Theorem. Interpretation of MVT with a position function.
• 03/26/18: 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. Definition of concave up/down and inflection point.
• 03/28/18: Quiz 7. Comments on the bonus question of the quiz. Review of the definition of concave up/down and inflection point. Concavity test (idea of the proof). Example. Beginning of the sketching of the graph of $f(x)=xe^{-x}$.
• 04/02/18: Sketching of the graph of $f(x)=xe^{-x}$. Exercise: getting information about a function $f$ from the graph of its derivative. Optimization problem.
• 04/04/18: Quiz 8. Comments on Quiz 8. In-class review session Test 3.
• 04/06/18: Review session Test 3.
• 04/09/18: Definition of antiderivative. The most general antiderivative. Table of antiderivatives. Exercises: Find the antiderivative of a function that satisfies a given condition - find the position function given the acceleration function, the initial velocity and the initial displacement.
• 04/11/18: 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). Definition of the definite integral as the limit of Riemann sums.
• 04/16/18: Review of the definition of the definite integral. Sum notation: $\Sigma$. Computation of $\int_{0}^{1}x^2\, dx$ with the definition (limit of a Riemann sum). The definite integral of a negative function. Exercises: approximation of an integral with Riemann sums; computation of the integral of a function whose graph is made by lines and semicircles.
• 04/18/18: Quiz 9. Comments on Quiz 9. Properties of the definite integral and exercise. 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).
• 04/23/18: 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. Example.
• 04/24/18: Review Integrals.
• 04/25/18: The last quiz. Comments on the quiz. In-class review session Final Exam.
• 04/26/18: Review session Final Exam.

#### Course documents:

• Notes:
• Slides:
• Videos (by Robert Connelly):
• Sheets:
• Quizzes:
• Homework:
• Reviews:

• Notes:
• Sheets:
• Quizzes :
• Homework :