Introduction to Calculus

Sifan Hassan, an Ethiopian-born Dutch runner, has dominated distance running for several years. She became the first runner to win both the 1500 and 10,000 meter races at a World Championship. During the Tokyo Olympics, she joined only one other runner in history when she medaled in the rarely attempted distance triple: the 1500, 5000, and 10,000 meter races. She won the gold medal in both the 5000 and 10,000 meter races and the bronze in the 1500.

Hassan’s signature racing style is to stay at the back of the pack for much of the race, and then move up during the final laps. Hassan does not run at her top speed at every instant. How then, do we approximate her speed at any given instant? We will find the answer to this and many related questions in this chapter.

Sections

• Finding Limits: Numerical and Graphical Approaches — understand limit notation, including left- and right-hand limits, and find a limit from a graph or from a table of values.
• Finding Limits: Properties of Limits — find the limit of a sum, a difference, and a product, of a polynomial, of a power or a root, and of a quotient by factoring, finding a common denominator, or rationalizing with a conjugate.
• Continuity — determine whether a function is continuous at a number using the three conditions of continuity, classify jump and removable discontinuities, and find where a piecewise function is discontinuous.
• Derivatives — find the derivative of a function from the definition, find instantaneous rates of change, write the equation of a tangent line, and find the instantaneous velocity of a particle.

The sections above are adapted from Precalculus 2e, Chapter 12 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. The chapter opener’s photograph of Sifan Hassan passing the lead competitors is omitted; its introduction is reproduced above.