1. AP Calculus Parametric/Vector Equations (1.4/11.) Arc Length (8.4) Created by: Bill Scott Modified by: Jen Letourneau 1

2. Arc length Finding the length of the curve is an integral expression that accumulates the lengths line segments in the plane as the distance between the endpoints approaches zero: ArcLength40.ggb 2

3. Arc length There are two arc length formulas, depending on the nature of the function given. Your function with endpoints (a,c) and (b,d) must be continuous and differentiable. For most curves, a calculator is required to evaluate the length of a curve. ArcLength_v2.ggb For y as a function of x (smooth) For x as a function of y (smooth) 3

4. Arc length Example #1 Write an integral expression that equals the length of the curve of between x = a and x = b. 4

5. Arc length Example #1: Answer Write an integral expression that equals the length of the curve of between x = a and x = b. 5

6. Arc length Example #2 Write an integral expression that equals the length of the curve of between x = -8 and x = 8. Remember: the curve must be smooth… 6

7. Arc length Example #2: Answer Write an integral expression that equals the length of the curve of between x = -8 and x = 8. 7

8. Arc length Example #3 Write an integral expression that equals the length of the curve of between x = -4 and x = 4. Remember: the curve must be smooth… 8

9. Arc length Example #3: Answer Write an integral expression that equals the length of the curve of between x = -4 and x = -4. 9

10. Intro: Parametric Equations Imagine a particle moving in the plane along a curve C during a time interval. In the figure to the right, we can not model this curve using y = f(x) because C fails the vertical line test. We can, however, let time be the independent variable and model the x- and y-coordinates as functions of time. 10

11. Parametric Equations We can represent parametric equations in several ways. The important idea is that the independent variable is called the parameter and the x- and y-coordinates are functions of the parameter. Often, but not always, the parameter is time t. 11

12. Example 1 Consider the curve given by the parametric equations which concerns a particle moving in the plane: What domain makes sense for this problem? Can you picture the curve? Parametrics_sin_cos.ggb 12

13. Example 1 (continued) Now, if we only want half the circle, what should our domain be? How can we reverse the direction of motion? 13

14. Example 2 Consider the curve given by the parametric equations which concerns a particle moving in the plane during a six second time interval What are the precalculus questions that are worth asking? Parametrics.ggb 14

15. Example 2: My questions With a table of values, answer these questions:  What is the starting point when t = -2?  What is the ending point when t = 4?  At what times t does the particle pass through the coordinate axes? With a graph, answer these questions:  At what times t is the particle moving to the left? to the right?  At what times t is the particle moving upward? downward?  At what time t is the particle moving vertically? Other questions:  What is an equation of the graph as a function in one variable?  How would we adjust the parametric equations to have the particle retrace its steps? 15

16. Example 2: Creating a table of paired values time-201234 Position(8,-1)(3,0)(0,1)(-1,2)(0,3)(3,4)(8,5) 16

17. Example 2: Creating a graph with dynamics 17

18. Example 2: Changing the time interval timexy 001 12 203 334 485 18

19. Example 2: Removing the parameter 19

20. The Calculus of Parametrics Now we apply the ideas of calculus to the parametric curves. In particular, we might want to explore:  Slopes of tangents to the curve  The length of the curve (i.e. arc length) 20

21. Arc length Example Write an integral expression that equals the length of the curve of Hint: sketch the curve 21

22. Arc length Example Write an integral expression that equals the length of the curve of 22

23. Slopes along parametric curves 23

24. Vertical and horizontal tangents 24

25. Tangents to parametrics 25

26. Parametric Example The Zorro Curve Parametrics3.ggb 26

27. Vectors and Vector-Valued Functions MY OPINION The precalculus of vectors is great mathematics but it is not especially important to review for the BC exam beyond the basics. MY OPINION What IS important is a clear understanding of parametric equations because vector-valued functions are nothing more than parametric equations with arrows. 27

28. Vector basics There are two important features of a vector  Direction: there is little talk on the AP exam about this. We use the angle formed by the vector when in standard position.  Magnitude 28

29. More vector basics Two vectors are equal if the vectors point in the same direction and have the same magnitude. Scalar multiplication by a positive constant changes the length of the vector but does not change its direction. Multiplication by -1 changes the direction (opposite direction) but does not change the length of the vector. 29

30. More vector basics Vectors can be written in component form (named by the x and y coordinates of their terminal point (head) when the initial point (tail) is at the origin…standard position) Vectors can also be written as linear combinations of the unit vectors i and j (length of 1).  i_j_notation.ggb i_j_notation.ggb 30

31. More vector stuff Vector addition (head to tail) Vector subtraction (head minus tail) (handout) vecor_basics_slide24.ggb 31

32. Precalculus of vectors 32

33. Vector-valued functions Consider the vector-valued function given by This is nothing more exotic that the parametric equations except with some arrows attached. VelocityAccelerationVectors1.ggb 33

34. Vector-valued functions IMPORTANT IDEAS  The velocity vector is given by  The acceleration vector is given by BIG DEAL The magnitude of the velocity vector equals speed at any time t. 34

35. Rate of Change of Arc length = Speed On the BC exam, the exam writers have sometimes asked for the “rate of change of arc length” rather than ask for speed. These are equivalent ideas. 35

36. Vector-valued functions: Example #1 Consider the vector-valued function given by Find the velocity and acceleration vectors at any time, t. Find the velocity and acceleration vector at t=2. Find the speed at t=2. VelocityAccelerationVectors1.ggb 36

37. Vector-valued functions The position vector is a vector in standard position (attached to the pole) that terminates at the point P(t) = (f(t), g(t)). The velocity vector is typically positioned so that its starting point is at the terminal point of the position vector. The acceleration vector is typically positioned so that its starting point is at the terminal point of the position vector. 37

38. Vector-valued functions Vector-valued functions take all the terms from linear motion (position, displacement, velocity, acceleration) and now ask us to give answers in 2 dimensions.  Total Distance = length of the curve  Position = (x-position, y-position)  Displacement = (x-displacement, y-displacement) This is NOT more difficult, it is just more information! 38