2. CHAPTER 11 Vector-Valued Functions Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11.1VECTOR-VALUED FUNCTIONS 11.2THE CALCULUS OF VECTOR-VALUED FUNCTIONS 11.3MOTION IN SPACE 11.4CURVATURE 11.5TANGENT AND NORMAL VECTORS 11.6PARAMETRIC SURFACES

3. 11.4CURVATURE Unit Tangent Vector Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For the smooth curve C traced out by the endpoint of the vector-valued function r(t), recall that for each t, v(t) = r'(t) can be thought of as both the velocity vector and a tangent vector, pointing in the direction of motion (i.e., the orientation of C). Notice that is also a tangent vector, but has length one. We call T(t) the unit tangent vector to the curve C. That is, for each t, T(t) is a tangent vector of length one pointing in the direction of the orientation of C.

4. EXAMPLE 11.4CURVATURE 4.2Finding a Unit Tangent Vector Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the unit tangent vector to the curve determined by r(t) =  t 2 + 1, t .

5. EXAMPLE Solution 11.4CURVATURE 4.2Finding a Unit Tangent Vector Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6. 11.4CURVATURE Curvature Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The rate of change of the unit tangent vectors with respect to arc length along the curve suggests a measure of "sharpness" or curvature.

7. DEFINITION 11.4CURVATURE 4.1 Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8. 11.4CURVATURE Curvature – Alternative Form Slide 8 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Observe that by the chain rule,

9. 11.4CURVATURE Curvature – Alternative Form Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10. EXAMPLE 11.4CURVATURE 4.3Finding the Curvature of a Straight Line Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the curvature of a straight line.

11. EXAMPLE Solution 11.4CURVATURE 4.3Finding the Curvature of a Straight Line Slide 11 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12. EXAMPLE 11.4CURVATURE 4.4Finding the Curvature of a Circle Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the curvature for a circle of radius a > 0.

13. EXAMPLE Solution 11.4CURVATURE 4.4Finding the Curvature of a Circle Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. THEOREM 11.4CURVATURE 4.1 Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The curvature of the smooth curve traced out by the vector-valued function r(t) is given by

15. EXAMPLE 11.4CURVATURE 4.5Finding the Curvature of a Helix Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the curvature of the helix traced out by r(t) =  2 sin t, 2 cos t, 4t .

16. EXAMPLE Solution 11.4CURVATURE 4.5Finding the Curvature of a Helix Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17. EXAMPLE Solution 11.4CURVATURE 4.5Finding the Curvature of a Helix Slide 17 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18. 11.4CURVATURE Curvature of a Plane Curve Slide 18 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Note that a plane curve is traced out by the vector-valued function r(t) =  t, f (t), 0 , where the third component is 0.

19. EXAMPLE 11.4CURVATURE 4.6Finding the Curvature of a Parabola Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the curvature of the parabola y = ax 2 + bx + c. Also, find the limiting value of the curvature as x → ∞.

20. EXAMPLE Solution 11.4CURVATURE 4.6Finding the Curvature of a Parabola Slide 20 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21. EXAMPLE Solution 11.4CURVATURE 4.6Finding the Curvature of a Parabola Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In other words, as x → ∞, the parabola straightens out, which you’ve likely observed in the graphs of parabolas. It is a straightforward exercise to show that the maximum curvature occurs at the vertex of the parabola (x = −b/2a).