Description

21-120 Differentiation and Integration provides an introduction to the fundamental concepts of calculus. The course assumes no prior experience with calculus, but does require a strong background in algebra, trigonometry and pre-calculus skills. The course begins with a discussion of limits, then proceeds to the definition and applications of the derivative before concluding with an introduction to integration.

The concept of a limit is simple: L is the limit of f(x) as x approaches a if f(x) is close to L when the difference between x and a is small. The difficulty is in making precise the meaning of "close" (the ε in the definition) and "small" (the corresponding δ). Although this definition may be a conceptual challenge, it is fundamental to all of calculus.

The derivative of a function may be though of as an instantaneous rate of change in the value of the function, or as the slope of a tangent line to the graph. When viewed as a rate of change, it is computed by first finding the average rate of change over an interval, then taking a limit as the length of the interval approaches zero. When viewed as a slope, a similar limiting process is used.

Differentiation is used in a wide variety of disciplines in the physical sciences, social sciences and, of course, in mathematics. We will consider a wide variety of applications to both reinforce the concept of the derivative and demonstrate its usefulness.

We will interpret integration as a way to find the area under a curve, though it has may related applications (which are considered in 21-122). This definition of the integral also involves a limit. We approximate the area under the graph of a function by the sum of the areas of some number of rectangles. The definite integral of the function is the limit of this sum as the rectangle's widths approaches zero (and the number of rectangles approaches ∞).

Learning Objectives

After completing this course, you should be able to

1. Use the ε-δ definition to find the limit of a function.
2. Find limits by using theorems and facts about limits of continuous functions.
3. Find derivatives of functions using the definition.
4. Find derivatives of functions using the product rule, quotient rule and chain rule.
5. Interpret the derivative of a function as a slope of a curve or as a rate of change, as appropriate.
6. Apply the technique of implicit differentiation.
7. Use derivatives to solve a variety of related rates problems.
8. Compute derivatives of exponential, logarithmic, trigonometric and hyperbolic functions.
9. Use differentiation to sketch graphs of curves.
10. Use differentiation to find maximum and minimum values of a function
11. Use differentiation to find approximate solutions to equations using Newton's Method.
12. Find the antiderivative of a function, i.e. the indefinite integral.
13. Understand Riemann sums and the concept of a definite integral.
14. Use the Fundamental Theorem of Calculus to evaluate definite integrals via antidifferentiation.
15. Apply the substitution rule for integration.

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