Calculus II, Fall 2013

Quiz 3

April 18, 2013 Show all details.

1. DERIVE the arclength formula for a polar curve r = f(θ), a ≤ θ ≤ b.

2. In spherical system. ρ > 0, 0 ≤ φ ≤ π, 0 ≤ θ < 2π, what is the Cartesian coordinate corresponding to the point ρ = 1, ϕ = π/2 ,π/3? What is the cylindrical coordinate for this point?

3. True or False? Explain.

If for each θ ∈ [0, 2π], g(t) = f(t cos θ, t sin θ) is continuous at t = 0. Then f(x, y) is continuous at (0, 0).

4. Let f(x, y) = x + 2y + x^{2} + y^{2}, find a linear function L(x, y) = ax + by + c such that lim_{(x,y)→(0,0)} \frac{f(x, y) − L(x, y)}{\sqrt{x^{2} + y^{2}}} = 0. First, find a, b, c and then show that the limit above is actually true using the ϵ − δ argument.

5. Evaluate \frac{d}{dy} ∫_{1}^{2+sin(y)} \frac{cos(xy)}{x}dx

Calculus II, Fall 2013

Quiz 3

April 18, 2013 Show all details.

1. DERIVE the arclength formula for a polar curve r = f(θ), a ≤ θ ≤ b.

2. In spherical system. ρ > 0, 0 ≤ φ ≤ π, 0 ≤ θ < 2π, what is the Cartesian coordinate corresponding to the point ρ = 1, ϕ = π/2 ,π/3? What is the cylindrical coordinate for this point?

3. True or False? Explain.

If for each θ ∈ [0, 2π], g(t) = f(t cos θ, t sin θ) is continuous at t = 0. Then f(x, y) is continuous at (0, 0).

4. Let f(x, y) = x + 2y + x^{2} + y^{2}, find a linear function L(x, y) = ax + by + c such that lim_{(x,y)→(0,0)} \frac{f(x, y) − L(x, y)}{\sqrt{x^{2} + y^{2}}} = 0. First, find a, b, c and then show that the limit above is actually true using the ϵ − δ argument.

5. Evaluate \frac{d}{dy} ∫_{1}^{2+sin(y)} \frac{cos(xy)}{x}dx