Introduction
757
- 11.1Vector Functions and Their Derivatives
- 11.2Unit Tangent and Principal Unit Normal Vectors; Arc Length
- 11.3Arc Length as Parameter; Curvature
- 11.4Motion Along a Curve
- 11.5Integrals of Vector Functions; Projectile Motion
- 11.6Application: Kepler's Laws of Planetary Motion
- Chapter Review
- Chapter Project
How to Design a Safe Road
The Innerbelt Curve is a stretch of road in downtown Cleveland that has been given the ominous nickname of “Deadman's Curve”. For decades Ohio transportation department officials have struggled to find ways to help drivers safely negotiate the sharp 90-degree curve, which has a posted speed limit of 35 mph, as they connect from one 50 mph roadway to another. Improvements have included changing the speed limit, posting warning signs, introducing rumble strips to create more friction, and modifying the bank of the curve, but accidents are still common.
Though perhaps among the more famous, Cleveland's Deadman's Curve is hardly the only dangerous curve among the country's interstates and highways. And while transportation officials and engineers can look at statistical evidence to identify places where accidents are abnormally frequent, engineers who are designing new exit ramps or roadways that curve around geographical features need to understand the forces that act on an automobile of a given size and moving at a given speed so that it can safely travel the road.
Chapter 11 PROJECT
In the Chapter Project on page 807, we examine four different scenarios in which forces act on a vehicle to keep it on the road.
Until now, the domain and range of a function have both been subsets of real numbers. Such functions are referred to as real-valued functions of a real variable or, simply, as real functions. In this chapter, we discuss functions whose domains remain a subset of real numbers, but whose range consists of vectors in the plane or vectors in space. Such functions are called vector-valued functions of a real variable or, simply, vector functions.
Vector functions are particularly useful for describing the motion of a particle as it moves through space. As a result, vector functions have diverse applications from describing Kepler's laws of planetary motion to modeling DNA to producing lifelike, animated films with computer graphics.