Midterm 2
CURVE
This exam was easier than the first midterm, but the overall
student performance was also better: good work!
The MEDIAN score was 80 (compared to a final,
curved median of 72 on the first midterm). In light of the better student
performance (by my estimation) on this exam, the curve has been set so that
the median is one grade higher
than on the first midterm (B+ instead of B),
and the other letter grade ranges have also
been adjusted similarly.
- A: 91 points and up
- A-: 85 points and up
- B+: 80 points and up
- B: 77 points and up
- B-: 74 points and up
- C+: 69 points and up
- C: 63 points and up
- C-: 48 points and up
- D range: 42 points and up
- F: below 42 points
As with the first midterm, it is your point total that goes into determining
your final grade. So missing a letter grade cut-off by one point doesn't
matter much.
The TAs will hand back the exams on Tuesday in section meetings.
A Little General Information:
The exam will cover Chapter 12.
Assignment of partial credit will depend on the problem. Some problems (e.g.
true/false) may be "all or nothing" i.e. with no possible partial credit).
The problems will be mostly independent of one another, so that if you can't solve
one problem, it should not affect your ability to get full credit on
other problems.
I will not use a fixed grading scale (such as "90 = A, 80 = B etc."). I
will set a curve based on the overall difficulty of the exam and the performance
of the class (in particular, if everyone gets a nearly-perfect score on
every exam, I would be very happy to give all As!). This will be announced after
the exam on the course web page.
Summary of concepts
Section 12.2: Functions of several variables
- Function of two or more variables
- Graph of a function of two variables
- Level curves and surfaces
Section 12.3: Limits and continuity
- Definition and intuition of the limit of a function of several variables
- Continuity and limits
Section 12.4: Partial derivatives
- Definition of partial derivatives, various notations
- Geometric interpretation as slopes of tangents to curves
- Interpretation as rates of change
- Computation of tangent planes to surfaces z = f(x,y)
- Higher-order partial derivatives
Section 12.6: Increments and linear approximation
- Differential of a function of several variables
- Linear approximation to a function of several variables
- Application to error estimates (Section 12.6 problem 33)
- Linear approximation using the gradient
Section 12.7: Multivariable chain rule
- Chain rule for functions of several variables
- How to organize your calculation using dependency diagrams
- Implicit partial differentiation
Section 12.8: Directional derivatives and gradients
- Gradient of a function of several variables
- Directional derivatives
- Directional derivatives in terms of the gradient (dot product)
- Geometric interpretation of the directional derivative
- Partial derivatives as directional derivatives
- Geometric interpretation of the gradient (direction of maximal increase)
- The gradient as a normal vector, application to computation of tangent planes
- The differential also encodes the normal vector to a level surface
Section 12.5: Multivariable optimization problems
- Theorem 1, p. 881, on existence of extreme values
- Definitions of local and absolute maxima and minima
- Necessary conditions: vanishing partial derivatives OR an interior point where
one of the partial derivatives doesn't exist OR a boundary point (Theorem 3,
p. 883)
- Definition of "critical point" (one of the first two kinds above!)
Section 12.9: Lagrange multipliers and constrained optimization
- Why solve constrained optimization problems?
- The "points on the boundary" case of Theorem 3, p. 883, leads us to
such problems
- Method of Lagrange multipliers, encapsulated in Theorem 1, p. 920
- Geometry behind the method
- Using the method to narrow down the search for maxima and minima
Section 12.10: Critical points of functions of two variables (aka "Second derivative test")
- Model pictures for the second derivative test in two variables
- Matrix of second partial derivatives and how it looks in the model cases
- Discriminant (capital Delta)
- The test itself: Theorem 1, p. 929
- Local maxima, local minima, and saddle points
- When does the second derivative test give NO information?
Typical computational tasks
- Evaluating limits (not many homework problems were about this! it was not strongly emphasized)
- Computing partial derivatives of first and second order (12.4)
- Computing derivatives (partial and ordinary)
by implicit differentiation (12.7)
- Computing tangent plane to surfaces of the form z = f(x,y) (12.4)
- Using differentials to compute approximate values of
a given function f(x,y) near a given point (a,b) (12.4)
- Using differentials to estimate the effect of small changes in the
variables to the value of a function (12.6) (Note: this was not strongly
emphasized, if you can do the homework about this you will be fine!)
- Computing derivatives via the multi-variable chain rule (12.7)
- Computing gradients and directional derivatives (12.8)
- Computing tangent planes to surfaces of the form F(x,y,z)=k (12.8)
- Finding local/global maxima/minima of
a function of several variables using the first derivative test
(12.5)
- Classifying critical points using the second derivative test (12.10)
- Finding maxima/minima with constraints by Lagrange multiplier
method (12.9)
- Application to optimization problems (12.5, 12.9)
Things to remember:
- Any formulas you need for the kinds of calculations mentioned above. It's a
good idea to prepare a list of all the formulas you will need, and then
memorize those. Of course, you shouldn't bring that list to the exam.
Main course page.