Problems

In Problems 1-12 find the derivative.

In Problems 17-26, evaluate the limit.

In Problems 27-34 use the first and second derivatives and limits to sketch the curve.

In Problems 35-50 evaluate the integral.

51             Find the volume generated by rotating the region under the curve y = e^x, 0 ≤ x ≤ 1, about (a) the x-axis, (b) the y-axis.

52            Find the volume generated by rotating the region under the curve y = e^-x, 0 ≤ x < ∞, about (a) the x-axis, (b) the y-axis.

53            Find the length of the curve x = e^t cos t, y = e^t sin f, 0 ≤ t ≤ 2π.

54            A snail grows in the shape of an exponential spiral, r = e^aθ in polar coordinates.

(a)    Find tan ψ the angle between a radius and the curve at θ.

(b)    Sketch the curve for a = 1 and a = 1/√3

(c)    Find the length of the curve where - x < θ ≤ b.

(d)    Find the area of the snail where - x < θ ≤ b. (To avoid overlap, one should integrate from b - 2π to b.)