1. Compute y′ and simplify.

(a) y = ln(x)(x^{7} − \frac{4}{\sqrt{x}})

(b) y = (3^{2x} + 2)^{3}

(c) y = \frac{x + 1}{3x^{4} − 1}

(d) y = \frac{sin(x)}{1 + cos(x)}

(e) y = arctan(e^{5x} − 2^{x})

(f) xy^{2} − 3y^{3} + 2x^{4} = 2

(g) y = cos(x^{2})^{x^{3}}

(a) Evaluate the following limits:

(a) lim_{x→0} \frac{1 − cos x}{x^{2}}

(b) lim_{x→∞} \frac{ln(e^{x} + 1)}{x}

(b) Both parts of this question refer to the function f(x) = 4x^{3} − x^{4}.

(a) Find and classify the critical point(s) of f.

(b) Find the absolute maximum and minimum values of f on the interval [1, 4].

(c) A spherical balloon is being inflated at 20 in^{3}/sec. When the radius is 6 inches, at what rate is the radius increasing? (The volume of a sphere of radius r is V = \frac{4}{3}πr^{3}.)

(d) Let a > 0. A rectangle is inscribed under the curve y = \sqrt{x}:

x y = \sqrt{x}  a

Find the value of x, with 0 < x < a, for which the area of the rectangle is largest. Show all work for full credit.

(e) [10 points] Find the inflection point(s) of the function f(x) = x^{2/3} − \frac{1}{5}x^{5/3}.