Mathematics 131 – Calculus for Physical and Life Sciences 1 Sample Exam 3 November 30, 2004

I. Sketch the following parametric curves: A) x = 2 + t, y = 3 − 2t, 0 ≤ t ≤ 1. B) x = cos(2t), y = sin(2t), 0 ≤ t ≤ π/2. C) x = 3 sin(t), y = 4 cos(t), 0 ≤ t ≤ 2π.

II. A) Differentiate implicitly, and find the equation of the tangent line to the curve

−x^{3} + 2y^{3} − xy = 0

at (1, 1). B) Boyle’s Law states that the pressure P and volume V of an ideal gas are related by

an equation P V = c, where c is constant. If a sample of gas has P = 1 atm, V = 3 liters and P is changing at a rate of .2 atm/min, how is V changing?

III. Compute the following limits: A)

lim_{x→0} \frac{sin(4x)}{tan(3x)}

B)

lim_{x→∞} \frac{3 ln(x) + 2}{x}

IV. All parts of this question refer to the function

f(x) = x^{2}e^{−2x}

A) Find all critical points of f(x) and classify them as local maxima, local minima, or

neither. B) Find all inflection points of f(x). C) Sketch the graph y = f(x) accurately, using the information from parts A and C. D) What are the (global) maximum and minimum values of f(x) for −1 ≤ x ≤ 2?

V. 10000 feet of fencing will be used to enclose a large rectangular area and subdivide it into six smaller rectangular playing fields in two rows of three fields each. What dimensions for the overall rectangle will maximize the total area enclosed?