Understanding Calculus: Problems, Solutions and Tips

Rated 2 out of 5 by from Needs a proofreader I bought this, along with Calc 2 and Multivariable to keep mentally active in retirement. Overall, the presentation is pretty good. The explanations clear, the samples good. But the Workbook is awful - at least my copy is. The typesetting of the mathematical statements is often mangled and incomprehensible. There is hardly any page with some typo and it's very, very frustrating. There are also some errors in the lectures. Lesson 21 in particular has a doozy of a goof. The work is all wrong, the factoring impossible, the arithmetic in error - yet the correct answer! (-5/6) Go figure. Someone seriously needs to proof read and correct these materials. A reprint of the workbook should be mailed out to purchasers. Looking ahead to Calc 2 I've already spotted many typos in the math. Unless these errors are corrected I would NOT recommend this for self-study; you'll doubt your own intelligence, but it's not you - it's the lack of quality control.
Date published: 2020-09-17
Rated 5 out of 5 by from Excellent Course! Professor Edwards rates at the top as a Great Courses professor. He was very understandable. He provided guidance on the importance of the homework and how to learn the material. He is an outstanding communicator on the subject of math. I felt I got a strong grasp of the material. I'm looking forward to taking his Multivariable Calculus.
Date published: 2020-09-02
Rated 5 out of 5 by from I learned a lot of foundational calculus skills. I found Dr. Edwards communicated the material very well. I had taken a first year calculus course 44 years ago and the instructor was so bad I had the feeling of "unlearning". I really learned a lot from this course and found it pretty addictive. I have purchased Calculus 2 and 3 and am starting to work on them. The course problems were very good for reinforcing the material. I found Dr. Edwards entertaining and effective.
Date published: 2020-06-13
Rated 5 out of 5 by from Great lectures, difficult workbook. I studied calculus 50 years ago and wanted a refresher course to keep my mind active. The professor's lectures are very well done. However, around lesson 11 some of the workbook problems were difficult to solve. Thank goodness for the answers in the back. I believe I would have to solve many more problems to feel procient.
Date published: 2020-04-28
Rated 5 out of 5 by from Excellent course I just wanted to say "Bravo" to Professor Edwards for this course. He was able to present the material at a very well paced rate with good explanations. I think his students at the University are lucky to have him as their professor. His visual graphs and drawings help in making the material clear. If Calculus is new to you, I suggest you follow the Professor's advice to "practice, practice, practice" solving the problems in the text and not just watch the video. After watching a video and working through the problems in the text, I was able to feel confident about the material presented to move on to the next chapter. I enjoyed finishing the 36 chapters and am now ready to take Professor Edwards Calculus II course.
Date published: 2020-04-19
Rated 4 out of 5 by from Good course with an appropriate title. This course has done much to further my understanding of calculus. The explations of various functions and their formulas and graphs are well done. The only thing that would make it even clearer (less abstract) would be examples of real world problems each formula could be used to solve.
Date published: 2020-04-17
Rated 5 out of 5 by from The title explains it all. I am 85 and this is to keep my mind active, and it is. I didn't realize how much fun it would be. It's definitely keeping me out of the bars!
Date published: 2020-02-26
Rated 5 out of 5 by from Best Introductory Calculus Course I've taken a number of "Calculus 101" courses (yeah, I know), but this one differs in that Professor Starbird eliminates most of the complex methods to calculate deriviatives and integrals, choosing to use the simpliest to explain what calculus is and how it works. Exceptional approach.
Date published: 2020-01-22
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Understanding Calculus: Problems, Solutions and Tips
Course Trailer
A Preview of Calculus
1: A Preview of Calculus

Calculus is the mathematics of change, a field with many important applications in science, engineering, medicine, business, and other disciplines. Begin by surveying the goals of the course. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits.

33 min
Review-Graphs, Models, and Functions
2: Review-Graphs, Models, and Functions

In the first of two review lectures on precalculus, examine graphs of equations and properties such as symmetry and intercepts. Also explore the use of equations to model real life and begin your study of functions, which Professor Edwards calls the most important concept in mathematics.

30 min
Review-Functions and Trigonometry
3: Review-Functions and Trigonometry

Continue your review of precalculus by looking at different types of functions and how they can be identified by their distinctive shapes when graphed. Then review trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians rather than degrees.

30 min
Finding Limits
4: Finding Limits

Jump into real calculus by going deeper into the concept of limits introduced in Lecture 1. Learn the informal, working definition of limits and how to determine a limit in three different ways: numerically, graphically, and analytically. Also discover how to recognize when a given function does not have a limit.

31 min
An Introduction to Continuity
5: An Introduction to Continuity

Broadly speaking, a function is continuous if there is no interruption in the curve when its graph is drawn. Explore the three conditions that must be met for continuity-along with applications of associated ideas, such as the greatest integer function and the intermediate value theorem.

31 min
Infinite Limits and Limits at Infinity
6: Infinite Limits and Limits at Infinity

Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it. Learn how to analyze these functions, and try some examples from relativity theory and biology.

31 min
The Derivative and the Tangent Line Problem
7: The Derivative and the Tangent Line Problem

Building on what you have learned about limits and continuity, investigate derivatives, which are the foundation of differential calculus. Develop the formula for defining a derivative, and survey the history of the concept and its different forms of notation.

31 min
Basic Differentiation Rules
8: Basic Differentiation Rules

Practice several techniques that make finding derivatives relatively easy: the power rule, the constant multiple rule, sum and difference rules, plus a shortcut to use when sine and cosine functions are involved. Then see how derivatives are the key to determining the rate of change in problems involving objects in motion.

30 min
Product and Quotient Rules
9: Product and Quotient Rules

Learn the formulas for finding derivatives of products and quotients of functions. Then use the quotient rule to derive formulas for the trigonometric functions not covered in the previous lecture. Also investigate higher-order derivatives, differential equations, and horizontal tangents.

31 min
The Chain Rule
10: The Chain Rule

Discover one of the most useful of the differentiation rules, the chain rule, which allows you to find the derivative of a composite of two functions. Explore different examples of this technique, including a problem from physics that involves the motion of a pendulum.

31 min
Implicit Differentiation and Related Rates
11: Implicit Differentiation and Related Rates

Conquer the final strategy for finding derivatives: implicit differentiation, used when it's difficult to solve a function for y. Apply this rule to problems in related rates-for example, the rate at which a camera must move to track the space shuttle at a specified time after launch....

31 min
Extrema on an Interval
12: Extrema on an Interval

Having covered the rules for finding derivatives, embark on the first of five lectures dealing with applications of these techniques. Derivatives can be used to find the absolute maximum and minimum values of functions, known as extrema, a vital tool for analyzing many real-life situations.

30 min
Increasing and Decreasing Functions
13: Increasing and Decreasing Functions

Use the first derivative to determine where graphs are increasing or decreasing. Next, investigate Rolle's theorem and the mean value theorem, one of whose consequences is that during a car trip, your actual speed must correspond to your average speed during at least one point of your journey.

31 min
Concavity and Points of Inflection
14: Concavity and Points of Inflection

What does the second derivative reveal about a graph? It describes how the curve bends-whether it is concave upward or downward. You determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you use the second derivative.

31 min
Curve Sketching and Linear Approximations
15: Curve Sketching and Linear Approximations

By using calculus, you can be certain that you have discovered all the properties of the graph of a function. After learning how this is done, focus on the tangent line to a graph, which is a convenient approximation for values of the function that lie close to the point of tangency.

32 min
Applications-Optimization Problems, Part 1
16: Applications-Optimization Problems, Part 1

Attack real-life problems in optimization, which requires finding the relative extrema of different functions by differentiation. Calculate the optimum size for a box, and the largest area that can be enclosed by a circle and a square made from a given length of wire.

31 min
Applications-Optimization Problems, Part 2
17: Applications-Optimization Problems, Part 2

Conclude your investigation of differential calculus with additional problems in optimization. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer.

31 min
Antiderivatives and Basic Integration Rules
18: Antiderivatives and Basic Integration Rules

Up until now, you've calculated a derivative based on a given function. Discover how to reverse the procedure and determine the function based on the derivative. This approach is known as obtaining the antiderivative, or integration. Also learn the notation for integration.

31 min
The Area Problem and the Definite Integral
19: The Area Problem and the Definite Integral

One of the classic problems of integral calculus is finding areas bounded by curves. This was solved for simple curves by the ancient Greeks. See how a more powerful method was later developed that produces a number called the definite integral, and learn the relevant notation.

31 min
The Fundamental Theorem of Calculus, Part 1
20: The Fundamental Theorem of Calculus, Part 1

The two essential ideas of this course-derivatives and integrals-are connected by the fundamental theorem of calculus, one of the most important theorems in mathematics. Get an intuitive grasp of this deep relationship by working several problems and surveying a proof.

30 min
The Fundamental Theorem of Calculus, Part 2
21: The Fundamental Theorem of Calculus, Part 2

Try examples using the second fundamental theorem of calculus, which allows you to let the upper limit of integration be a variable. In the process, explore more relationships between differentiation and integration, and discover how they are almost inverses of each other.

31 min
Integration by Substitution
22: Integration by Substitution

Investigate a straightforward technique for finding antiderivatives, called integration by substitution. Based on the chain rule, it enables you to convert a difficult problem into one that's easier to solve by using the variable u to represent a more complicated expression....

31 min
Numerical Integration
23: Numerical Integration

When calculating a definite integral, the first step of finding the antiderivative can be difficult or even impossible. Learn the trapezoid rule, one of several techniques that yield a close approximation to the definite integral. Then do a problem involving a plot of land bounded by a river.

31 min
Natural Logarithmic Function-Differentiation
24: Natural Logarithmic Function-Differentiation

Review the properties of logarithms in base 10. Then see how the so-called natural base for logarithms, e, has important uses in calculus and is one of the most significant numbers in mathematics. Learn how such natural logarithms help to simplify derivative calculations....

31 min
Natural Logarithmic Function-Integration
25: Natural Logarithmic Function-Integration

Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous lecture. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming lectures.

31 min
Exponential Function
26: Exponential Function

The inverse of the natural logarithmic function is the exponential function, perhaps the most important function in all of calculus. Discover that this function has an amazing property: It is its own derivative! Also see the connection between the exponential function and the bell-shaped curve in probability.

31 min
Bases other than e
27: Bases other than e

Extend the use of the logarithmic and exponential functions to bases other than e, exploiting this approach to solve a problem in radioactive decay. Also learn to find the derivatives of such functions, and see how e emerges in other mathematical contexts, including the formula for continuous compound interest....

31 min
Inverse Trigonometric Functions
28: Inverse Trigonometric Functions

Turn to the last set of functions you will need in your study of calculus, inverse trigonometric functions. Practice using some of the formulas for differentiating these functions. Then do an entertaining problem involving how fast the rotating light on a police car sweeps across a wall and whether you can evade it.

31 min
Area of a Region between 2 Curves
29: Area of a Region between 2 Curves

Revisit the area problem and discover how to find the area of a region bounded by two curves. First imagine that the region is divided into representative rectangles. Then add up an infinite number of these rectangles, which corresponds to a definite integral.

30 min
Volume-The Disk Method
30: Volume-The Disk Method

Learn how to calculate the volume of a solid of revolution-an object that is symmetrical around its axis of rotation. As in the area problem in the previous lecture, you imagine adding up an infinite number of slices-in this case, of disks rather than rectangles-which yields a definite integral.

30 min
Volume-The Shell Method
31: Volume-The Shell Method

Apply the shell method for measuring volumes, comparing it with the disk method on the same shape. Then find the volume of a doughnut-shaped object called a torus, along with the volume for a figure called Gabriel's Horn, which is infinitely long but has finite volume.

31 min
Applications-Arc Length and Surface Area
32: Applications-Arc Length and Surface Area

Investigate two applications of calculus that are at the heart of engineering: measuring arc length and surface area. One of your problems is to determine the length of a cable hung between two towers, a shape known as a catenary. Then examine a peculiar paradox of Gabriel's Horn.

32 min
Basic Integration Rules
33: Basic Integration Rules

Review integration formulas studied so far, and see how to apply them in various examples. Then explore cases in which a calculator gives different answers from the ones obtained by hand calculation, learning why this occurs. Finally, Professor Edwards gives advice on how to succeed in introductory calculus.

30 min
Other Techniques of Integration
34: Other Techniques of Integration

Closing your study of integration techniques, explore a powerful method for finding antiderivatives: integration by parts, which is based on the product rule for derivatives. Use this technique to calculate area and volume. Then focus on integrals involving products of trigonometric functions.

31 min
Differential Equations and Slope Fields
35: Differential Equations and Slope Fields

Explore slope fields as a method for getting a picture of possible solutions to a differential equation without having to solve it, examining several problems of the type that appear on the Advanced Placement exam. Also look at a solution technique for differential equations called separation of variables.

30 min
Applications of Differential Equations
36: Applications of Differential Equations

Use your calculus skills in three applications of differential equations: first, calculate the radioactive decay of a quantity of plutonium; second, determine the initial population of a colony of fruit flies; and third, solve one of Professor Edwards's favorite problems by using Newton's law of cooling to predict the cooling time for a cup of coffee.

31 min
Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.

ALMA MATER

Dartmouth College

INSTITUTION

University of Florida

About Bruce H. Edwards

Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.

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