calculus hw 4

MA 225 C HW 4

Due Wednesday Sept. 30 at 12 pm via

Gradescope

1. Sketch the surface with equation x2 + z2 = 9.

2. Sketch the surface with equation x2 + y2 + 2z2 − 1 = 0.

3. Sketch the surface with equation x2 = y2 + z2.

4. Sketch the vector-valued function r(t) = 〈cos(t), 0, sin(t)〉.

5. Sketch the vector-valued function r(t) = 〈cos(t), t, sin(t)〉.

6. Motion around a circle of radius a is described by the 2D vector-valued function r(t) = 〈a cos(t),a sin(t)〉. Find the derivative r′(t) and the unit tangent vector T(t), and verify that the tangent vector to r(t) is always perpendicular to r(t).

7. Motion around an ellipse is described by the 2D vector-valued function r(t) = 〈a cos(t),b sin(t)〉 , where a and b are positive constants. Assume that a 6= b, so that the ellipse is not a circle. Find all the points on the ellipse where the tangent vector is perpendicular to r(t).

8. Let r(t) = ⟨ t2 − 2t, sin(πt/2)

⟩ . Find r(t), and then find all values of t for

which r(t) is smooth.

9. An object’s trajectory in 2D has velocity vector v(t) = 〈1, t〉 and initial position r(0) = 〈0, 0〉. Find r(t).

10. An object’s trajectory in 2D has acceleration vector a(t) = 〈1, 1〉, initial velocity v(0) = 〈0, 1〉, and initial position 〈0, 0〉. Find r(t).

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