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 progressions.

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 infinity.
Proofs to know: show that  lim (f+g)=lim(f)+lim(g),  and that  lim(cf(x))=c lim(f(x)).
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 asked to.
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 tangent)
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, graphs.
Absolute extrema.
Velocity, acceleration, applied max/min.