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EXAMPLE 1 Graph the vector function r(t) = (cos z)i + (sin )j + tk.

EXAMPLE 1 Graph the vector function r(t) = (cos z)i + (sin )j + tk.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 97E
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Exercise 1. Read Example 1 (p. 764). What type of curve is considered in this example, and what 
does it look like?

The calculus of curves follows the same rules as single-variable calculus, so it’s best to refresh yourself 
on those computations. The difference is that the computation must be done for each component. For 
example, to take a derivative of a curve is the same as taking the derivatives of the components, each of 
which is a single-variable function. There are no new techniques needed to compute derivatives, since 
the curve will have either two or three components.

764 1 = 2m Chapter 13 Vector-Valued Functions and Motion in Space (1,0,0) 1=0 r x² + y² = 1 Chapter 13 Vector-Valued Functions and Motion in Space FIGURE 13.3 The upper half of the helix r(t) = (cost)i + (sin t)j + tk (Example 1). r(t) = (sin 3)(cost)i + (sin 3r)(sin/)j + k r(r) (cost)i + (sin!)j + (sin 2r)k EXAMPLE 1 (b) FIGURE 13.2 Space curves are defined by the position vectors r(?). Graph the vector function r(t) = (4 + sin 20r)(cost)i + (4 + sin 201)(sin)j + (cos20r)k (c) r(1) = (cost)i + (sin 1)j + tk. Solution This vector function r(t) is defined for all real values of t. The curve traced by r winds around the circular cylinder x² + y² = 1 (Figure 13.3). The curve lies on the cyl- inder because the i- and j-components of r, being the .x- and y-coordinates of the tip of r. satisfy the cylinder's equation: x² + y² = (cos 1)² + (sin t)² = 1. The curve rises as the k-component z = t increases. Each time t increases by 27, the curve completes one turn around the cylinder. The curve is called a helix (from an old Greek word for "spiral"). The equations x = cos 1, y = sin 1, z = 1 parametrize the helix. The domain is the largest set of points for which all three equations are defined, or -∞ < t <∞ for this example. Figure 13.4 shows more helices.
Transcribed Image Text:764 1 = 2m Chapter 13 Vector-Valued Functions and Motion in Space (1,0,0) 1=0 r x² + y² = 1 Chapter 13 Vector-Valued Functions and Motion in Space FIGURE 13.3 The upper half of the helix r(t) = (cost)i + (sin t)j + tk (Example 1). r(t) = (sin 3)(cost)i + (sin 3r)(sin/)j + k r(r) (cost)i + (sin!)j + (sin 2r)k EXAMPLE 1 (b) FIGURE 13.2 Space curves are defined by the position vectors r(?). Graph the vector function r(t) = (4 + sin 20r)(cost)i + (4 + sin 201)(sin)j + (cos20r)k (c) r(1) = (cost)i + (sin 1)j + tk. Solution This vector function r(t) is defined for all real values of t. The curve traced by r winds around the circular cylinder x² + y² = 1 (Figure 13.3). The curve lies on the cyl- inder because the i- and j-components of r, being the .x- and y-coordinates of the tip of r. satisfy the cylinder's equation: x² + y² = (cos 1)² + (sin t)² = 1. The curve rises as the k-component z = t increases. Each time t increases by 27, the curve completes one turn around the cylinder. The curve is called a helix (from an old Greek word for "spiral"). The equations x = cos 1, y = sin 1, z = 1 parametrize the helix. The domain is the largest set of points for which all three equations are defined, or -∞ < t <∞ for this example. Figure 13.4 shows more helices.
Oog r(t) = (cos r)i FIGURE 13.4 y + (sin r)j + rk r(t) = (cos 1)i + (sin r)j + 0.3rk Helices spiral upward around a cylinder, like coiled springs. r(t) = (cos 5t)i + (sin 51)j + 1k Limits and Continuity The way we define limits of vector-valued functions is similar to the way we define limits of real-valued functions.
Transcribed Image Text:Oog r(t) = (cos r)i FIGURE 13.4 y + (sin r)j + rk r(t) = (cos 1)i + (sin r)j + 0.3rk Helices spiral upward around a cylinder, like coiled springs. r(t) = (cos 5t)i + (sin 51)j + 1k Limits and Continuity The way we define limits of vector-valued functions is similar to the way we define limits of real-valued functions.
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