Sample Problems
These are some problems that we have gone over in our recent tutoring sessions. Typically, we color coordinate the problems, so a different color represents a particular concept of the problem.
Classical Mechanics
This question dealt with finding the resultant force from three force vectors, F1, F2, and F3. We split these force vectors into x and y components, determined the magnitude and direction of the resultant force, and wrote out the resultant force in vector notion (or component form). Other asides (such as inverse functions) were to clarify student understanding.
Organic Chemistry
This question dealt with a dehydration reaction. Heat was supplied, thus favoring elimination (versus substitution). An E1 mechanism was favored due to the secondary carbocation intermediate (as well as being in polar protic conditions). Furthermore, an E1 (versus an E2) provided an opportunity for a methyl shift to relieve ring strain and form a more stable product. The reaction finished through the deprotonation of an acidic proton with the conjugate base of sulfuric acid. The major product favored the Zaitsev conformation for stability.
This question dealt with solving the triple integral over T, where T was a solid tetrahedron with the noted vertices. A three-dimensional sketch aided in determining integral bounds. Integral bounds were carefully selected to be in terms of the variables that followed the antiderivative of the current variable (i.e. x-y for the upper z bound). The problem continued to be solved using the fundamental theorem of calculus, where F(x) on bounds from a to b equals F(b) - F(a).
Multivariable Calculus
Statics
This problem dealt with determining the stress and strain that a girder (attached to a concrete mount, holding up a sign) could undergo. Young's Modulus, the stress, the strain, and the elongation of the girder were determined. Force and elongation were discussed via a diagram to aid in understanding of the elastic and plastic regions as well as the breaking point.
