Calculus I
Contents
Class Quiz
Class Quiz
Class Comment Card
Comment Card
Syllabus
You can find the link to the general syllabus and the course syllabus below:
General Syllabus
Course Syllabus
Student Help Resources
Mental Health Resources: Click here to find a list of mental health resources available to you at Syracuse University as well as national hotlines. Never hesitate to make use of these services if you need them. If you know someone in distress, say something and be sure to make them aware of these resources.
Mathematics Help Resources: Click here to find Mathematics help offered through the Mathematics Department and click here to find Mathematics help offered through CLASS (Center for Learning and Student Success).
Past Final Exams: Click here to find past MAT 295 final exams.
Handouts
Reference Tables: I created a reference table which is not just useful for Calculus, but most of the more applied mathematics courses you could take as an undergraduate.
Other Resources
WolframAlpha: WolframAlpha is a useful online computation device. WolframAlpha takes written commands, interprets the input, and performs the computations. WolframAlpha is an online ‘watered down’ version of Mathematica, which at this time is offered to Syracuse University students for free! However, WolframAlpha will take many Mathematica commands, as I outline in this document.
Symbolab: Symbolab is a useful online computational device, which will not only integrate but show steps. This can be a useful study tool, but be careful how you use it!
Geogebra Calculus Applets: These GeoGebra applets help you to explore various course concepts. They work in your browser, and they also come with motivating questions.
Other Lectures/Notes:
3Blue1Brown
Professor Leonard
OCW MIT
UMKC
Gil Strang
Steven Strogatz
Homework
Homework is completed on WebAssign.
Quizzes
Quizzes: Solutions
Exams
Wolfram Demonstrations
The Wolfram Demonstrations page contains over 12,000 different demonstrations of applications of Mathematica programs to dozens of fields, including Calculus. You can interact with these demonstrations by downloading the Wolfram CDF player for your browser. Feel free to use these to help you engage with course concepts, and explore other topics. You can find a list of particularly useful, relevant, or interesting demonstrations in the drop down tab below.
Demonstrations List-
Limits
Investigating Limits Numerically
Limits: A Graphical and Numerical Approach
Finite Limit at a Finite Point
Infinite Limit at a Finite Point
Finite Limit at Infinity
Infinite Limit at Infinity
Investigating Limits at Infinity Numerically
Limit of $\sin x/x$ as $x$ Tends to Zero
Squeeze Theorem
Exercises in Limits from Above and Below
Limit Laws
Inset Plot Magnifier
Epsilon-Delta Definition of Limit
A Case of the Epsilon-Delta Definition of a Limit
Sine Wave Example of the Epsilon-Delta Definition of Limit
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Derivative Rules
Derivatives of Sine
Derivatives of Trig and Hyperbolic Functions
Derivatives of Exponential Functions
The Product Rule
The Quotient Rule
Chain Rule
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Implicit Differentiation
The Tangent Line of an Implicit Relation
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Related Rates
A Snowball’s Rate of Change
Person Walking Away from Spotlight
A Boat Approaching a Dock
Related Rates: Triangle Angle and Area
Related Rates Clock
Calculating the Path of a Point on a Sliding Ladder
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Linear Approximation/Differentials
Differential of a Function
Geometric Difference between a Finite Difference and a Differential
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Optimization
Maximizing the Volume of a Box
Tin Box with Maximum Volume
Maximizing Volume Cone with Given Slant
The Maximum Rectangular Area under Different Curves
Shortest Time Problem
Minimizing the Time for a Dog to Fetch a Ball in Water
Minimum Area between a Semicircle and a Rectangle
Largest Isosceles Triangle Inscribed in a Circle
Largest Triangle Inscribed in a Circle
Maximum Rectangle Inscribed in a Circle
Maximum Area Field with a Corner Wall
Maximizing the Volume of a Cup Made from a Square Sheet of Paper
Maximizing the Area of a Rectangle with Fixed Perimeter
Maximizing the Area of Some Geometric Figures of Fixed Perimeter
Maximizing the Viewing Angle of a Painting
Swim, Swim and Walk, or Walk?
Moving a Couch around a Corner
The Ladder around the Corner Problem
Optimize the Length of the Crease of a Folded Piece of Paper
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Numerical Methods
Bisection Method
Learning Newton’s Method
Square Roots with Newton’s Method
Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values
Equally Spaced Multipoint Method for Differentiation
Numerical Integration using Rectangles, the Trapezoidal Rule, or Simpson’s Rule
Numerical Integration by Simpson’s 1/3 and 3/8 Rules
Numerical Integration Examples
Comparing Basic Numerical Integration Methods
Numerical Integration: Romberg’s Method
Integral Evaluation Using the Monte Carlo Method
Mean Value Theorem for Integrals and Monte Carlo Integration
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Derivatives
How the Area of a Disk Grows
Velocity of a Falling Object
Average Rate of Change: Exploring More Functions
Instantaneous Rate of Change: Exploring More Functions
The Tangent Line Problem
Slope between Two Points on a Curve
Secant and Tangent Lines
Secant Approximations
The Definition of the Derivative
Instantaneous Rate of Change
Derivative as a Function
Tangent to a Curve
Approximating the Tangent to a Curve with Secants
Tangent Line Using Many Different Limit Configurations
Family of Curves, Tangents, and Intuition
A Look at Graphs
Exploring More Functions with the First and Second Derivatives
Derivatives of Quintic Polynomials
Multiple Derivatives
Successive Derivatives of a Polynomial
Tangent Line to a Parabola
Graphing Derivatives
Visualizing Jerk: Change of Acceleration
Car Traveling at Night
Pathological Derivatives
Bolzano’s Continuous but Nowhere Differentiable Function
Bolzano’s Function
Riemann’s Example of a Continuous but Nowhere Differentiable Function
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l’Hôspital’s Rule
L’Hôspital’s Rule for 0/0 Forms
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Intermediate/Mean Value Theorem
Intermediate Value Theorem
Bolzano’s Theorem
Mean Value Theorem
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Curve Sketching
Simple Rational Functions
Rational Functions of Small Degree
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Integrals
Integration Is a Sum
Riemann Sums
Integration by Riemann Sums
Riemann Sums: A Simple Illustration
Common Methods of Estimating the Area under a Curve
Signed and Unsigned Area under a Curve
Area Under a Curve
A Basic Property of Integrals
Area Function for a Piecewise Curve
Area between Curves
Average Value of a Function
Average Value via Integrals
The Trapezoidal Rule for Increasing Functions
Visual Proof of Two Integrals
Visual Computation of an Integral
Visual Computation of an Integral (II)
Continuous Functions Are Integrable
The Integral Mean Value Theorem: An Illustration
Integral Mean Value Theorem
Two Integral Mean Value Theorems