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**1. Homework Statement**

1) Consider the paramerric curve given by x = t^2 + 3t and y = 4 - t^2

a) Find an equation of the tangent lne to the curve at the point (x,y) = (0,-5)

b) Determine the equation of every vertical tangent line to this parametric curve.

2)For each of the following definite integrals, determine its value if it convereges, otherwise, explain why it diverges.

a) integral (e to infinity) [dx/(x(ln(x))^(3/2))]

b) integral (-2 to -3) [dx/(sqrt(x^2 - 4))]

**3. The Attempt at a Solution**

1)

a) dy/dx = (-2t)/(4-t^2)

since x = t^2 + 3t and y = 4 - t^2 and (x,y) = (0,-5)

t = -3

dy/dx = (-2*-3)/(4 - (-3)^2) = -(6/5)

y = -6/5(x) -5

b) 4 - t^2 = 0

t = -2, 2

(x,y) = (-2, 0) and (10, 0)

I can figure out the equation if these points are correct

2)

a) integral (e to infinity) [dx/(x(ln(x))^(3/2))]

I get

-2 lim (t -> inf) [ 1 / ln(x)^1/2 ] ( e - t)

b) integral (-3 to -2) [dx/(sqrt(x^2 - 4))]

I got as far as lim ( t -> -2) [ln | x + sqrt(x^2 - 4) | - ln 2 ] ( -3 to t)

any support is much appreciated.

Thanks