Understanding Calculus: Problems, Solutions and Tips

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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....

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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....

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.

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....

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.

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.

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....

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.

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.

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.

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.

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.

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.

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.

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.

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.