EXAM1 Review
The exam will cover Chapters 1.2–1.8 and Chapters 2.1–2.9. Chapters 2.2 and 2.6
are omitted. Some of the questions will test your skills, others your understanding of the
material. Here is a selection of topics which we have discussed, along with some sample
problems showing the kind of question that I could ask for each. This list of topics and
questions is NOT exhaustive, and you should be prepared to work any problem of the sorts
that appear on HW1–5.
Graphing Surfaces:
• Be able to graph quadratic functions like f (x, y) = 1 + 2x2 + 3y 2 , g(x, y) = y 2 − x2 ,
h(x, y) = 12 − x2 − y 2 , and linear functions like k(x, y) = 3x + 2y + 4. Be able to recognize
the equation of a generalized cylinder like y = x2 or x + y = 1 and be able to graph them.
• Be able to draw the contour diagrams of functions like f (x, y) = 2x2 +3y 2 , g(x, y) =
x2 − y 2 , h(x, y) = 4 − x2 − y 2 , k(x, y) = 3x + 2y + 1.
• Understand how the same surface can be a graph of a function of 2 variables and
a level surface of a function of 3 variables. Given a quadratic equation be able to name
the type of level surface (ellipse, hyperbola, parabola).
• Write an equation whose graph in space is a cone. Write an equation whose graph
in space is a hyperboloid with two sheets. Write an equation whose graph in space is a
hyperboloid with one sheet.
Lines, Planes, and Hyperplanes:
• Know the definition of a linear vs affine linear function. Be able to find the equation
of a plane given three points on the plane. Be able to find the equation of a line in space
given a point on the line and a direction parallel to the line. Be able to convert from the
vector equation to the standard equation for a line in space. Be able to find the equation
for a plane in space given a normal direction and one point on the plane.
Dot Product and Cross Product:
• Know the algebraic and geometric definitions of dot product. Know the basic
properties of dot product (commutative, distributive over vector addition, etc). Know and
understand how to use dot product to get the equation of a plane, given a normal vector to
the plane and a point on the plane. Be able to use the dot product to find work W = F · d.
• Know the algebraic and geometric definitions of cross product, and be able to
compute the cross product of two vectors in space. Be able to use cross product to find
the equation of a plane through three points in space, or the equation of the line at the
intersection of two planes in space. Be able to compute the area of the triangle with
vertices at three given points, or the area of the parallelogram spanned by two vectors. Be
able to compute torque τ = d × F. Be able to apply the right-hand rule!
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Parametrized Curves:
• Given a parametrized curve in space, compute the velocity and acceleration vectors
at any point on the curve. Compute the speed at any point on the curve. Find the equation
for the line tangent to any given point on the curve. Using F = ma, compute the force
which acts on a particle with mass m moving along the given trajectory.
Partial Derivatives:
• Know the definition of the partial derivative, and know how to estimate a partial
derivative from numerical data in a table. You should be able to calculate the partial
derivatives of any elementary function (polynomial functions, trig functions, exponentials,
logs, quotients).
Linear Approximation, Tangent Planes, and the Differential:
• Know the formula for the tangent plane to the graph of f at the point (a, b, f (a, b)),
know the formula for the local linearization of f at the point (a, b), be able to use the local
linearization to approximate f .
• We approximate a function f by its local linearization; the graph of the linearization of f at (a, b) is the tangent plane to the graph of f at (a, b, f (a, b)). Make sure that
you understand this. Here are some sample problems to test your understanding of this
connection.
1) The tangent plane of f at the point (3, 4, 43) is z − 2x − 5y = 17. What is f (3, 4)?
∂f
What is ∂f
∂x (3, 4) and ∂y (3, 4)? What is the local linearization of f at (3, 4)?
2) If the tangent plane of f at (3, 4, 43) is horizontal, what is the local linearization of f
at (3, 4)?
3) Explain why the x-slope of the tangent plane at (a, b, f (a, b)) is fx (a, b).
4) Write down the formula for the differential, and explain how it is different from the
linearization. Explain why we can use it to approximate the change in a function. What
do we use to approximate the percent change in a function? Why?
Be able to use the differential to estimate the change in a function or the error in
estimating it, and the percent change. Remember that df is the approximate change in f,
while df
f is the approximate percent change in f .
Here are two sample problems which use the local linearization.
1) The following table gives the amount of beef B consumed in pounds per household
per week at various prices P and income levels I in thousands of dollars.
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I = 20
I = 40
I = 60
P = $3
2.65
4.14
5.11
P = $3.50
2.59
4.05
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P = $4
2.51
3.94
4.97
a) Use the table to find approximate values for
∂B ∂B
∂I , ∂P
P = $4.50
2.43
3.88
4.84
when I = 40 and p = $4.
b) Using the table and your answers to (a) give the local linearization for B when I = 40
and p = $4.
c) Use the local linearization to approximate B when I=45 and P = $4.10.
2) The volume of a rectangular parallelpiped with a square bottom is given by
V (l, h) = l2 h
a) What is the local linearization of V at (3, 2)?
b) What is the equation of the tangent plane to the graph of V at (3, 2, 18)?
c) What is the differential of V at (3, 2)? At (l, h)?
d) Use the differential of V at (3, 2) to approximate the change in V , if the change in l is
.05 and the change in h is .04.
e) Use the differential of V at (l, h) to approximate the percent change in V if the percent
change in l is 5% and the percent change in h is 4%.
Chain Rule:
• Be able to apply the Chain Rule to solve problems like the following.
1) The pressure P of an ideal gas is given by
P =
nRT
V
where n is the number of moles of gas, T is the temperature in degrees Kelvin, and R is
the gas constant 8.32 joule/mole-K◦ . P is measured in N/m2 .
∂P ∂P
a) Find ∂V
, ∂T
∂P
b) Find ∂V when V = 100 liters and T = 300◦ K, n = 1. What is the physical meaning
of this number?
c) Suppose when V = 100 liters and T = 300◦ K, n = 1, V is decreasing at a rate of
10 liters/sec and T is increasing at a rate of 10 degrees/sec. Use the Chain Rule to find
the rate of change of P with respect to time at this moment.
2) The position of a moving particle is given by (x(t), y(t)) = (R cos 3t, R sin 3t).
a) What is the velocity of the particle at t = 0?
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b) Suppose f (x, y) = ln(x/R + y/R). What is the rate of change of f with respect to t at
t = 0? How can this problem be solved with the Chain Rule?
3) Suppose we have a function f (r, s, t) with continuous partial derivatives, and we define
u(x, y, z) = f (x − y, y − z, z − x). Show that
∂u
∂u
∂u
+
+
= 0.
∂x
∂y
∂z
Gradients and Directional Derivatives:
• The partial derivative of a function with respect to x is just a special case of a
directional derivative; it’s just f~i . So any question I can ask about partial derivatives, I
can also ask about directional derivatives. To be more specific, you should: Know the
definition of the directional derivative. Know how to calculate directional derivatives from
the formula
Du f (a, b) = ∇f (a, b) · u
where the direction u is a unit vector.
You should spend time meditating on the following properties of the gradient:
1) If ~u is the unit vector in the direction of the gradient at (a, b) then ~u is the direction
of greatest increase of f at (a, b).
2) Not only does the gradient of f at (a, b) give us the direction of greatest increase
of f at (a, b), but the rate of change of f in this direction is |∇f (a, b)|.
3) The gradient of f is always orthogonal to the level sets of f .
• Given a contour diagram you should be able to sketch the direction of the gradient
at each point.
Be able to answer:
Explain why the direction ~u at (a, b) which gives the greatest rate of change
of f is the unit vector in the direction of the gradient.
Explain why the gradient of f at (a, b) is perpendicular to the level curve
of f at (a, b).
Be able to solve problems like:
1.) A developer is building a ski resort at the town of Twin Peaks. The height above
sea level at points in the development is given roughly by h(x, y) = .3 − 0.01y 2 + .02x2 −
.001x4 miles.
a) At the point (x,y)= (2,1), assuming water takes the steepest path down, which way
does it flow? (Your answer should be a unit vector.)
b) How steep is the slope at (2,1)?
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c) In which directions at (2,1) is the height constant? (Your answer should be a unit
vector.)
2.) The temperature of a plate is given by T (x, y) = 30/(1 + 2x2 + 4y 2 ).
a) What is the gradient of T at (3, 2)?
b) If you are at position (3, 2), does the temperature increase or decrease toward the
southeast?
c) What is the direction in which temperature falls the fastest at (3, 2)?
d) What is the greatest rate at which the temperature can increase at (3, 2)?
e) An isotherm is a curve along which the temperature is constant. What is the direction
of the isotherm at (3, 2).
Gradients and Level Sets:
• Be able to find the equation of the tangent plane of a level surface or the tangent
line of a level curve using the gradient. Here is a sample problem of this type.
1a) Find the equation of the tangent plane to the graph of f (x, y) = 2x + 4y + x2 − y 3 at
the point (−1, 1, 2).
1b) Find the equation of the tangent plane to the surface with equation
x2 /2 − y 2 /2 − xy = z 2 − z − 2
at the point (−2, 2, 3).
2.) The surfaces with equations x2 + y 2 + z 2 = 12 and y 3 − x2 + 2z 2 = 12 intersect at
(2, 2, 2). What is the angle they intersect at?
More practice with gradient problems can be found on a separate review sheet.
Parametrized Surfaces:
• Be able to parametrize planes and surfaces. Be able to determine the parametrized
tangent plane at a regular point on a parametrized surface. Given a parametrization, be
able to determine whether or not the parametrization is regular at a point p.
In the following problems, you are given an equation which defines a surface in R3 .
Find a differentiable parametrization of the surface (x, y, z) = r(u, v) which is regular at
the given point p. Use your parametrization to give a parametrization of the tangent plane
at p.
1.) z = x2 + y 2 ,
2.) y = x2 ,
p = (1, 1, 2).
p = (1, 1, 2).
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3.) x2 + z 2 = 9,
p = ( √32 , 1, √32 ).
Local Extrema and Critical Points:
• You should be able to find the critical points for a function f (x, y) of two variables.
You should also be able to classify those critical points according to type (max/min/saddle)
using the Second-Derivative Test.
What do the level curves and gradient vectors of f (x, y) look like near a
local maximum? A local minimum? A saddle?
Given a contour diagram of a function f (x, y), with gradient vectors included, you
should be able to find and label each critical point of f according to type.
Here are two sample problems involving critical points.
1A.) Find the critical points of f (x, y) = 6x − 2x3 + 3y 3 − 9y 2 . Use the second derivative test to tell if the critical points are local maximums, minimums or saddle points.
(Answer: Points are (1, 0), (1, 2), (-1, 0), (-1, 2), types in order are: max,saddle,saddle,
min.)
1B.) Find and classify the critical points of f (x, y) = 31 x3 + xy 2 − 9x.
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