Missing from today's "nutshell" review: the intermediate value theorem and approximations.

In the "nutshell" part I think I omitted the intermediate value theorem.  But I covered it in the example before the break.  You must know this.

Furthermore, the essentials on approximations:

• The derivative, obviously, and the differential.
  • Example: Some exam problems have asked to find the tangent line for the particular solution through (t₀,x₀) of a differential equation. Inserting for the point finds the derivative (without solving).
  • You can calculate (u(x)^v(x))' by now? 
     
• Derivatives/differentials for functions given implicitly by a level curve
   
• Derivatives/differentials in eq. systems. 
  • Calculate differentials,
  • identify as linear eq. system in the differentials of the dependent variables,
  • solve for what is asked.  (You are allowed to "invert the matrix without checking invertibility".) 
    Beware whether you are asked for a general expression (depending on "letters") or in a point.
     
• And as above also for elasticities.  And you need to know the elasticity of substitution.
   
• Lagrange multipliers as derivatives of the value function; this you are allowed to know and use without justification. 
   
• The envelope theorem.  (Good idea to refer to it; except when it is the Lagrange multiplier, as mentioned in the previous point, that you are allowed to know, and you need not point out it is because of the envelope theorem.)
   
• Higher-order approximations:
  • Second-order in one variable.  And the part on two variables that I wrote on the board that you need to know (write as single variable).
  • k^th order in one variable: You should know that we divide by the factorial j!, i.e. 24 for the fourth order term.