Mathematics 133 – Intensive Calculus for Science 1 Sample Exam III – November 22, 2005

I. Find derivatives of each of the following functions. Any correct form of the derivative is OK here.

A) f(x) = ln(x) + 2^{4x}

B) g(x) = sin(x) cos(x^{2})

C) h(x) = arctan(x^{2} + e^{2x} + 1)

D) Find \frac{dy}{dx} if x^{2}y^{2} − 2xy + x^{3} = 1.

II. All parts of this question refer to the function

f(x) = \frac{x}{x^{2} + 4}

A) Compute f′(x). Simplify and factor the numerator.

B) Using your f′(x), find all critical points of f and classify each as a local maximum, local minimum, or neither using the First Derivative Test.

C) Compute f′′(x). Simplify and factor the numerator again.

D) Using your f′′(x), at which x are the inflection points on the graph y = f(x) located?

III. The hypotenuse of a right triangle has one end at (0, 0) and the other at a point on y = x^{2}e^{−3x} with x ≥ 0. One of the other two sides lies along the x-axis and the other is parallel to the y-axis. Find the maximum area of such a triangle.

IV. A chemical storage tank is an “upside-down” cone. The depth is 12 meters and the top radius is 5 meters. When the depth of the chemical in the tank is 1 meter the level is falling at the rate of 1 meter per minute. How fast is the volume of chemical changing then?