Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Math 11
Fall 2012
Practice Exam I
This practice exam is intended to give you an idea of the kinds of problems we consider
putting on an exam, and of the possible length of a midterm exam. Any topic that appeared
on a homework problem may appear on the exa
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Math 11 Section 1
September 26, 2012
Sample Solutions
f
(x0 , y0 ) is the vertical slope of the intersection of the graph
x
f
of f with the plane y = y0 , so that 1, 0,
(x0 , y0 ) is a vector tangent to the graph of
x
f
f at the point (x0 , y0 , f (x0 , y
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Math 11 Section 1
September 21, 2012
Sample Solutions
Today we are going to talk, among other things, about limits of functions of two or more
variables. Well look at some examples, and then talk about the denition.
(1.) First we will consider the functio
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Mathematics 11
Practice Problems on New Material for the Final:
Sample Solutions
The nal exam is cumulative, but will concentrate on new material not covered on the
two midterm exams. These problems are only on the new material.
1. Consider the paraboloid
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Math 11 Section 1
September 19, 2012
Sample Solutions
For this set of questions, let
r(t) = cos(t), sin(t), t
be the position at time t of a moving object. All the questions are about this specic example.
(1.) Compute the velocity, the speed, and the unit
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Mathematics 11
Practice Exam 2
Sample Solutions
1. Consider the function
f (x, y ) = 2xy + x2 .
(a) Find all critical points of f .
Solution: f = fx , fy = 2y + 2x, 2x , so the only critical point of the function is (0, 0).
(b) For each critical point, de
Multivariable Calculus for TwoTerm Advanced Placement FirstYear Students.
MATH 011

Fall 2011
Math 11 Section 1
September 28, 2012
Sample Solutions
Now we know what dierentiable means for a function f : Rn R.
The function f is dierentiable at the point r0 if and only if there is a linear
function L : Rn R whose graph is tangent to the graph of f a