MA225 HW5

MA 225 C HW 5

Due Wednesday Oct. 7 at 12 pm via Gradescope

1. Find the length of the curve r(t) = ⟨ t2, t3

⟩ for 0 ≤ t ≤ 4.

2. For the curve r(t) = ⟨ 2t2, 3t2 + 1

⟩ , find the arc length parameter s. Use

it to create a new vector-valued function r(s) which describes the same curve as r(t). Verify that |r′(s)| = 1 for all s.

3. Find the domain of the function ln(x2 + y2).

4. Find the domain of the function √ y −x2.

5. Sketch the graph of the function z = 1 −x−y.

6. Sketch the graph of the function z = √

4 −x2 −y2.

7. Graph several level curves of the function z = x2 + y2/4. Use them to sketch the graph of the function.

8. Graph several level curves of the function z = 1/(x2 + y2 + 1). Use them to sketch the graph of the function.

9. Evaluate lim(x,y)→(5,−5) x2−y2 x+y

, or else explain why it does not exist.

10. Evaluate lim(x,y)→(0,0) xy

x2+y2 , or else explain why it does not exist.

11. Let f(x, y) be the function

f(x, y) =

{ x2y2

x2+y2 , (x, y) 6= (0, 0)

0, (x, y) = (0, 0).

Is f(x, y) continuous at (0, 0)?

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