Exam I---Review guide
Required topics: everything
we discussed. Focus on
understanding the topics.
Here are just a few topics not
to be missed.
The proofs you need to know for Exam 1 are listed below.
Algebra: induction, the sum of geometric and arithmetic
Real numbers: Be able to state
(and use) the theorem that a bounded set has a least upper bound - supremum, and a greatest lower
bound - infimum.
Limits of functions: know the epsilon-delta definition of a
limit, and be able to use it (if asked to, or need be).
Calculation of limits, including limits of rational functions at
Proofs to know: show
that lim (f+g)=lim(f)+lim(g), and that lim(cf(x))=c
Continuity: State the definition. Justify why a given function
is continuous, or discontinuous at some point.
Be able to state (and use) the theorems in A.3. be able to prove
Theorem 5 in A.3.
Derivative of a function: state
the definition, use it in any notation (h, delta x, etc)
Be able to use the definition in order to calculate a derivative, if
Be able to use the definition to prove that (f+g)'=f'+g', that
(cf)'=cf', that (fg)'=f'g+fg'
Know that elementry functions are differentiable wherever defined,
except for the absolute value
(corner at 0) and radicals (derivatives unbounded at 0, vertical
Know the rules of calculation of derivatives and use them to claculate
the derivative of any elemantary function.
Implicit differentiation. Higher order derivatives.
In A.4: be able to state (and use) the theorems 1, 2, 3, 4 and prove
Theorems 3 and 4.
Applications: equation of the
tangent line, of the normal line,
intervals of increase, decrease, local extrema, concavity, inflexion,
Velocity, acceleration, applied max/min.