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MATH 251, Section 516 : REVIEW
1st Midterm Exam : October, 7th, 2014, 2 :20-3 :35 p.m
Policy for exam :
Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
Electronic devices are prohibited (cellphones, calculators, laptop or computer,. . .).
Advice : Read the whole exam before starting to answer, beginning by the questions which seem easiest
to you. Make sure you read the questions carefully.
No paper is necessary, you will be able compose on the same paper. I will also give you
scratch paper. I accept the use of pencil for drawings, but please, answer questions with a pen.
The following is a summaryze of the course, specified exercises are not exhaustive, those are just an
idea of the kind of practice which you should be able to do.
CHAPTER 11. :
1. Be familiarized with the objects of the 3-dimensional space : axis, points (coordinates), vectors in the
canonical basis (~i, ~j, ~k).
2. Exercices : 3,11, 29-32.
– Find a vector from a starting point and with a given direction or from 2 points and give its direction
numbers. Find a unit vector.
– Properties of vectors and operations.
– The dot product and its applications (for example to find a normal vector to others)
– Scalar projection and vector projection.
3. Exercices : 7-9, 13,15,19. The cross product and its applications (for example to find a normal vector
to others or verify that two vectors are collinear, application to Section 11.4 too).
4. Exercices : 3,5,14,22,23,35,49.
– Determine parametric equations of a line passing through a point with direction numbers.
– Determine symmetric equations of a line.
– Determine equation of a plane (for example, from 2 vectors, from a normal vector to the plane and
a point in a plane, from 3 points).
5. Exercices : 17-24, 25-34. Classify quadric surfaces, determine the orientation of such a surface in the
axis, know how to sketch them.
6. Exercices : 1-6. How to describe a curve in space and elements which allow to recognize it.
7. Exercices : 1-4. What a smooth curve is. Compute the arc length of a curve described by a vector
function on an interval.
CHAPTER 12. :
1. Exercices : 17, 18, 22, 24, 53-56. Domain of a function of 2 or 3 variables, its range, its graph (for
only a function of 2 variables).
2. Exercices : 7,8,9, 13,16. Compute a limit if this limit exists or say if this one doesn’t exist and explain
why. Application to the continuity of a function of 2 or 3 variables. Recognize classical functions
(cosine, sine, polynomial functions) to evaluate the continuity.
3. Exercices : 7,8,10.
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– Know how to compute the first partial derivatives (or the partial derivatives of the first order) for
a function of 2 or 3 variables.
– Know how to compute the second partial derivatives (or the partial derivatives of the second order)
for a function of 2 or 3 variables.
– Application of Clairaut’s Theorem.
4. Exercices : 5,6,17,18, 25-28.
– How to determine the tangent plane to the surface of equation z = f (x, y) at a point.
– Compute the differential of functions of 2 or 3 variables and its applications (to approximation).
5. Exercices : 5,7,13,27. Use the chain rule to differentiate the composition of a function of several
variables.
6. Exercices : 4,6,13,16,26,37.
– Directional derivatives for a differentiable function of 2 or 3 variables.
– Gradient and its geometrical interpretation (to determine an equation of the tangent plane to a
surface z = f (x, y) when f is differentiable).
7. Exercices : 1, 3, 7,. . .(1-16). Maximum and Minimum values. Know how to solve a system of 2 equations with 2 unknown variables to determine the critical points and use the method of the Determinant
to classify these points (minimum, maximum, saddle points). Know how to determine the maximum
and the minimum points on a compact set of R2 .
A review section is available at the end of each Chapter, take a look to practice more.
End of the exercise seen in class
√
1
Use differentials to approximate ( 99 + (124) 3 )4 .
√
1
Let f (x, y) = ( x + y 3 )4 with (a, b) = (100, 125) and f (a + ∆x, b + ∆y) = f (99, 124) with
∆x = 99 − 100 = −1 and ∆y = 124 − 125 = −1.
So f (99, 124) ≈ f (100, 125) + dz where dz is the differential of f .
dz = fx (a, b).∆x + fy (a, b).∆y.
1
1
2 √
4 √
where fx (x, y) = √ ( x + y 3 )3 and fy (x, y) =
x + y 3 )3 .
2 (
x
3y 3
So, fx (100, 125) = 225 and fy (100, 125) = 180.
dz = −225 − 180.
So f (99, 124) ≈ f (100, 125) + dz ≈ 2970.