Parametric Equations, Tangent Lines, & Arc Length

SUGGESTED REFERENCE MATERIAL:

As you work through the problems listed below, you should reference Chapter 10.1 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.

EXPECTED SKILLS:

• Be able to sketch a parametric curve by eliminating the parameter, and indicate the orientation of the curve.

• Given a curve and an orientation, know how to ﬁnd parametric equations that generate the curve. dy d2y • Without eliminating the parameter, be able to ﬁnd and at a given point on a dx dx2 parametric curve.

• Be able to ﬁnd the arc length of a smooth curve in the plane described parametrically.

PRACTICE PROBLEMS: For problems 1-5, sketch the curve by eliminating the parameter. Indicate the direction of increasing t.   x = 2t + 3 1. y = 3t − 4  0 ≤ t ≤ 3

3 17 y = x − from (3, −4) to (9, 5) 2 2

1   x = 2 cos t 2. y = 3 sin t  π ≤ t ≤ 2π

x2 y2 + = 1 from (−2, 0) to (2, 0) 4 9

 x = t − 5  √ 3. y = t  0 ≤ t ≤ 9

√ y = x + 5 from (−5, 0) to (4, 3)

  x = sec t  y = tan2 t 4. π  0 ≤ t <  2

y = x2 − 1 for x ≥ 1

2   x = sin t  y = cos (2t) 5. π π  − ≤ t ≤  2 2

y = 1 − 2x2 from (−1, −1) to (1, −1)

For problems 6-10, ﬁnd parametric equations for the given curve. (For each, there are many correct answers; only one is provided.)

6. A horizontal line which intersects the y-axis at y = 2 and is oriented rightward from (−1, 2) to (1, 2).   x = t y = 2  −1 ≤ t ≤ 1

7. A circle or radius 4 centered at the origin, oriented clockwise.   x = 4 sin t y = 4 cos t  0 ≤ t ≤ 2π

8. A circle of radius 5 centered at (1, −2), oriented counter-clockwise.   x = 5 cos t + 1 y = 5 sin t − 2 ; Detailed Solution: Here  0 ≤ t ≤ 2π

9. The portion of y = x3 from (−1, −1) to (2, 8), oriented upward.   x = t y = t3  −1 ≤ t ≤ 2

3 x2 y2 10. The ellipse + = 1, oriented counter-clockwise. 4 16   x = 2 cos t y = 4 sin t  0 ≤ t ≤ 2π

dy d2y For problems 11-13, ﬁnd and at the given point without eliminating the dx dx2 parameter.   x = 3 sin (3t) 11. The curve y = cos (3t) at t = π  0 < t < 2π

2 dy d y 1 = 0; 2 = dx t=π dx t=π 9

 2  x = t 12. The curve y = 3t − 2 at t = 1  t ≥ 0

2 dy 3 d y 3 = ; 2 = − dx t=1 2 dx t=1 4   x = 2 tan t  y = sec t π 13. The curve π at t =  0 ≤ t ≤ 4  3 √ √ 2 dy 2 d y 2 = , 2 = ; Detailed Solution: Here dx t=π/4 4 dx t=π/4 16  √  x = √t 14. Consider the curve described parametrically by y = 3 t + 1  t ≥ 0

dy (a) Compute without eliminating the parameter. dx t=64

dy 1 = dx t=64 3

4 (b) Eliminate the parameter and verify your answer for part (a) using techniques from diﬀerential calculus. The curve is equivalent to y = x2/3 + 1, x ≥ 0. And, t = 64 corresponds to x = 8.

dy dy 1 Thus, = = dx t=64 dx x=8 3 (c) Compute an equation of the line which is tangent to the curve at the point cor- responding to t = 64. 1 y − 5 = (x − 8) 3   x = 2 cos t 15. Consider the curve described parametrically by y = 4 sin t  0 ≤ t ≤ 2π

dy (a) Compute without eliminating the parameter. dx t=π/4

dy = −2 dx t=π/4 (b) Eliminate the parameter and verify your answer for part (a) using techniques from diﬀerential calculus. x2 y2 π The curve is equivalent to the ellipse + = 1. And, t = corresponds √ √ 4 16 4 to the point (x, y) = ( 2, 2 2). Thus, you can use implicit diﬀerentiation and dy dy = = −2 √ √ dx t=π/4 dx (x,y)=( 2,2 2) (c) Compute an equation of the line which is tangent to the curve at the point cor- π responding to t = . 4 √ √ y − 2 2 = −2 x − 2

(d) At which value(s) of t will the tangent line to the curve be horizontal? π 3π t = and t = 2 2 For problems 16-18, compute the length of the given parametric curve.  x = t    2 16. The curve described by y = t3/2  3    0 ≤ t ≤ 4

5 √ 2 10 5 − + 3 3

 x = et    2 17. The curve described by y = e3t/2  3    ln 2 ≤ t ≤ ln 3 √ 16 −2 3 + ; Detailed Solution: Here 3  1  x = t2   2   1 18. The curve described by y = t3  3   √  0 ≤ t ≤ 3 7 3 19. Compute the lengths of the following two curves:

   x = cos t  x = cos (3t) C1(t) = y = sin t C2(t) = y = sin (3t)  0 ≤ t ≤ 2π  0 ≤ t ≤ 2π

Explain why the lengths are not equal even though both curves coincide with the unit circle.

The length of C1(t) is 2π and the length of C2(t) = 6π. Notice that C2(t) is the just curve C1(t) traversed three times.

20. This problem describes how you can ﬁnd the area between a parametrically deﬁned curve and the x-axis.

The Main Idea: Recall that if y = f(x) ≥ 0, then the area between the curve and Z b Z b the x-axis on the interval [a, b] is f(x)dx = y dx. Now, suppose that the same a a curve is described parametrically by x = x(t), y = y(t) for t0 ≤ t ≤ t1 and that the Z b Z t1 curve is traversed exactly once on this interval. Then, A = y dx = y(t)x0(t) dt. a t0

6   x = sin t  y = cos (2t) Consider the curve π π  − ≤ t ≤  4 4 (a) Compute the area between the graph of the given curve and the x-axis by evalu- Z t1 ating A = y(t)x0(t) dt. t0 √ Z π/4 2 2 A = cos (2t) cos t dt = −π/4 3 (b) After eliminating the parameter to express the curve as an explicitly deﬁned Z b function (y = f(x)), calculate the area by evaluating A = f(x) dx. a √ √ Z 2/2 2 2 2 A = √ 1 − 2x dx = − 2/2 3