Math 241 F1H

Midterm Exam 1 Review Sheet

Practice exams

Note that our exam syllabus does not include 11.7 (quadratic surfaces) and 11.8 (cylindrical/spherical coordinates), which is covered in some of the earlier exams, but does include most of Chapter 12, which is covered in Exams 2 below. Also, you can ignore questions on binormal vectors, since we did not cover those in the homework problems and it is not covered in our text (I briefly mentioned this in class).

• Exam 1, Fall 2000.
  Solutions
• Exam 1, Fall 2004.
  Solutions
• Exam 1, Fall 2005.
  Solutions
• Exam 2, Fall 2000.
  Solutions
• Exam 2, Fall 2004.
  Solutions
• Exam 2, Fall 2005.
  Solutions

General Information

• Special Open House: In addition to the regular Wednesday Open House (5 - 6 pm, 241 Altgeld), I will hold a special Open House, Sunday, 9/23, 5 pm, 241 Altgeld, for last-minute exam questions.
• Exam date, time, and location: The exam will be given during the regular class time, Wednesday, 9/26/2006, 2 - 2:50 pm, in the usual room, 142 Henry.
• Exam rules: No calculators, closed books/notes, no formula sheets, and, of course, no cheating. The problems in the exam will be such that a calculator is not needed and would not be of any help. You will need to memorize the basic formulas, such as those for curvature, acceleration, equation of a tangent plane, etc; see below for more.
• Missed Exam policy. I don't give make-up exams, but if you miss the exam with a valid excuse (e.g., illness), the exam will be counted as "excused". (See the Course Information Sheet for an explanation of an "excused" exam score.) The absence must be documented by a letter from the Dean's Office at 300 Student Services Building, 610 East John St., phone 333-0050; other documentation (e.g., note from McKinley) is not sufficient.

Exam content

The majority of the exam problems will be comparable to an average homework problem and fall into one of the types listed below under "Typical Tasks" (or possibly a combination of these). Some questions may be in multiple choice or in true/false format.

Some of the problems may be less routine, comparable to some of the more difficult homework problems. While there will likely not be an explicit "proof problem", keep in mind that most of the problems in the exercises that use the word "prove" or "show" are really exercises in applying particular techniques, rules, or formulas, and could have been stated as computational problems ("derive a formula for ..."). These problems are quite instructive, and you need to be familiar with the underlying techniques.

Things you should know

An excellent idea is to use the list below to prepare a "cheat sheet" containing the formulas you need to know for the exam. Just looking up formulas in the book or in your lecture notes and writing those formulas down on a sheet of paper helps you committing those formulas to memory. Of course, you shouldn't bring such a formula sheet to the exam.

11.1/11.2 Vectors

• vectors in 2 and 3 dimensions; magnitude (length), direction, coordinates, vectors versus scalars, right-hand/screw-driver rule
• position vector of a point
• addition of vectors (geometric and algebraic interpretation)
• multiplication of vectors by scalar (geometric and algebraic interpretation)
• unit vector; finding a unit vector in a given direction (normalizing a given vector)
• standard basis unit vectors

11.2: Dot products

• algebraic and geometric definitions
• dot products of perpendicular vectors
• angle between two vectors
• direction angles
• properties of dot product
• projection of one vector onto another (component of one vector along another, comp_ab)

11.3: Cross products

• algebraic and geometric definitions, computation of 3x3 determinants, screw-driver (right-hand) rule
• cross products of parallel vectors
• cross products of basic unit vectors
• basic properties of cross product
• scalar triple product
• area of a parallelogram or triangle
• volume of a parallelepiped or pyramid

11.4: Lines and planes in space

• vector and parametric equations of a line (skip the symmetric equations)
• vector, scalar, and linear equations of a plane
• scalar and linear equation of a plane
• normal vector of a plane

11.5: Vector-valued functions and curves and motion in space

• vector functions and space curves
• derivatives of vector functions: definition (componentwise diff.) and geometric interpretation (tangent vector)
• rules for derivatives
• integrals of vector functions
• motion is space: position, velocity, speed, acceleration

11.6: Curvature and acceleration

• arc length of a space curve
• unit tangent vector
• unit normal vector
• curvature
• tangential and normal components of acceleration (two formulas each)

12.2: Functions of several variables

• Function of two or more variables
• Graph of a function of two variables
• Level curves (for functions of two variables)
• Level surfaces (for functions of three variables)

12.4: Partial derivatives

• Definition of partial derivatives, and notations
• Geometric interpretation
• Interpretation as rates of change
• Higher order partial derivatives

12.6: Differentials and linear approximation

• Differential of a function of several variables
• Linear approximation to a function of several variables
• Application to error estimates
• Application to tangent planes

12.7: The multivariable chain rule

• Dependency diagram
• Multivariable chain rule
• Implicit partial differentiation
• Matrix version of the chain rule

12.8: Directional derivatives and gradients

• Gradient of a function of several variables
• Geometric properties of the gradient
• Relation between the gradient and level curves/surfaces
• Directional derivative
• Rate-of-change interpretation of directional derivative
• Partial derivatives as directional derivatives
• Application: Computation of tangent planes to surfaces of the form F(x,y,z)=k

Things you need not know

• Formula (16), p. 794, for a triple cross product
• Symmetric equations of a line
• The equations (22) and (23) for the motion of a projectile. (You should not memorize these equations, but you should be familiar with the underlying method, and be able to derive equations of this type on your own.)
• The formulas (12) and (13) for the curvature of a plane curve.
• The definitions/formulas for T, N, aT, aN, and K, in terms of the arclength s, e.g., the first two formulas in (19), p. 823, or (11), p. 821. (You should, however, know the other formulas for aT, a_N, K, e.g., (23), (24), (26), (27), (28).)

Typical computational tasks

1. Check if two vectors are (a) parallel, (b) perpendicular. (11.2)
2. Compute the angle between two vectors. (11.2)
3. Given a vector, find a unit vector with the same direction. (11.1)
4. Compute the dot product and the cross product of two vectors (11.2, 11.3)
5. Given two vectors, find the component (projection) of one along the other (11.2)
6. Find a vector that is perpendicular to two given vectors. (11.3)
7. Compute the area of a parallelogram/triangle determined by two vectors (or 3 points). (11.3)
8. Compute the volume of a parallelepiped/pyramid determined by three vectors (or 4 points). (11.3)
9. Determine whether three vectors are in the same plane ("coplanar") (11.3)
10. Given a point and a direction vector, or two points, find the vector and parametric equations of the line going through these points. (11.4)
11. Find the equation of a plane determined by (a) a point and a normal vector, (b) three points, (c) a point and two vectors in the plane. (11.4)
12. Given the equation of a plane, find a normal vector to that plane. (11.4)
13. Given two lines, determine whether they are skew, parallel, or intersect. If they intersect, determine the point of intersection. (11.4)
14. Given a line and a plane, determine whether they intersect in a single point (and find the intersection point in this case), or are parallel. (11.4)
15. Given two planes determine whether they coincide, are parallel, or intersect. In the latter case, find the line of intersection and the angle between the planes. (11.4)
16. Find the distance between a point and a plane. (Do not memorize the formula for the distance, but know the technique.)
17. Given a vector function (with explicitly given component functions), find the derivative and the integral of this function (11.5)
18. Find the derivative of a vector function, using the rules for derivatives (product rule for dot/cross products, etc.) (11.5)
19. Given the position function r(t) of a particle, find its velocity, speed, acceleration. (11.5)
20. Given the force acting on a particle or its acceleration, and its initial position and initial velocity, find its position function r(t). (11.5)
21. Find the arc length of a space curve. (11.6)
22. Find the curvature of a given curve. (11.6) (know both formulas for K))
23. Given a curve (described by a vector function r(t)) find (a) the unit tangent vector, (b) the unit normal vector, (c) the curvature, and (d) the tangential and normal components of the acceleration, at a given point on the curve. (11.6)
24. Given the velocity and acceleration of a particle, find the tangential and normal components of its acceleration. (11.6)
25. Find the tangential and normal components of the acceleration if curvature and speed are given. (11.6)
26. Compute partial derivatives of first and second order. (12.4)
27. Compute derivatives (partial and ordinary) by implicit differentiation. (12.7)
28. Compute tangent plane to surfaces of the form z = f(x,y). (12.4)
29. Use differentials to compute approximate values of a given function f(x,y) near a given point (a,b). (12.4)
30. Use differentials to estimate the effect of small changes in the variables to the value of a function. (12.6)
31. Compute derivatives via the multi-variable chain rule. (12.7)
32. Compute gradients and directional derivatives. (12.8)
33. Compute tangent planes to surfaces of the form F(x,y,z)=k. (12.8)