M-1000-5     Winter 2003

1. Basic algebra.

a)  Brackets, parentheses.

Their omission leads to mistakes:

b)  Applying a function termwise

The only class of functions that possess the property

   f(x + y) = f(x)  +  f(y)         (*₊)
or
   f(x - y) = f(x)  +  f(y)         (*_-)

are   linear homogeneous functions     f(x)  =   C  x ,     where  C  is a constant.   In all other cases the "identities" (*) are a mistake.   Mistakes of this type took the highest averall toll on the class' marks. Examples:

2. Algebraic identities.

a)  Trigonometric identities.

The only question where a trigonometric identity played a decisive role was #9a. It couldn't be solved without recalling the formula
    tan²x + 1 = sec² x  .
Another question where a similar identity could be recalled to simplify the answer was #5c.   (See comment below in sect. Miscellaneous (c) ).   Is it possible to memorize all such identities and not to confuse them when the time comes? Perhaps, but I don't find it particularly useful. I remember just the basic identity  (trig form of the Pithagorean theorem)
    sin²x + cos²x = 1
and the definitions
    tan x= sin x / cos x ,     cot x= cos x / sin x ,     sec x= 1 / cos x ,     csc x= 1 / sin x .
Any relations between squares of trig functions follows from these.

3. Limits.

Class' performance on these questions was statistically quite good. Still there were wrongdoings.

a)  Mistreatment of forms of type (0/0)

In ##1a and esp. 1c some people stopped at noticing that the expression has a form (0/0) and alledged that the limit does not exist. Wrong! Identifying the type of indeterminancy is a place to begin analysis.

b)  Arithmetic and algebra

Wrong factorization in 1a, problems with parentheses and cancelation in 1c and 2a. See examples in sect. Basic Algebra.

c)  Sign of infinity

In #1b, the numerator 3x is negative when x is close to -1. The denominator remains positive because it is a complete square. Therefore, in terms of signs, the fraction is (-)/(+) and both right and left limits are equal to negative infinity.

4. Continuity.

a)  Continuity requires a defined value.

In #3a students had not only to find that right and left limits of the function at zero are equal, but also that  f(x)   is defined at  x=0  and its value coinsides with the found limit.

b)  Discontinuities.

In #3b there were all kinds of mistakes, here are the most common.

  b1).   Failure to identify possible points of discontinuity. They are seen from the denominators:   x=2   or   x=-2.   (Another candidate point   x=0  where the definition changes has been investigated in #3a.)

  b2).   Saying that "limit as x>2 is undefined (indeterminate) form 0/0, therefore the discontinuity at x=2 is not removable".   Wrong: the limit exists and therefore the discontinuity is removable

  b3).  Saying that "limit as x>-2 is infinity (so it exists), therefore the discontinuity at x=-2 is removable".   Wrong: existence of an infinite limit implies non-existence of the (usual) limit, therefore the discontinuity at x=-2 is not removable. (The infinite limit is a notion that characterizes the behaviour of the function, but it is not a value.)

c)  Misuse of terminology.

There were several misuses, which were not heavily punished, yet they should be fixed:  
   "continuity is (is not) removable"   (say: discontinuity);
   "limit is continuous (discontinuous)"   (say: function).

5. Differentiation techniques

a) 

b) 

6. Integration

a) 

7. Problems on Related Rates and Optimization

a) 

8. Applications (Tangent line, Moving particle)

a) 

9. Analysis of behaviour of a function (#8)

a) 

10. Graph sketching (#9)

a) 

Miscellaneous.

a)  Consistency of results.

Question 2-a asked to find the derivative of a function by definition. On the other hand, a part of solution of 2-b was finding the derivative of the same function (in any way). Most students did it using differentiation rules. At this point, it was possible (and very helpful) to compare the result with that of (a). Those who made a mistake in (a) could detect and correct it, or at least report the discrepancy.

b)  Wrong interpretation of distance.

In #4 the given distance 26 km was the distance from the radar antenna to the plane. On the drawing, it should be the hypotenuse of the right triangle rather than its base.

c)  Lack of attention (or what?)

In #5c, which was    y-x = tan y ,   after obtaining the correct equation
   dy/dx -   sec² y  dy/dx = 1  ,
quite a few students deduced something like
    dy/dx = -1 / sec² y  ,      instead of correct     dy/dx = 1 / (1-sec² y)  =-1/tan² y =-cot² y ,
or even worse
    dy/dx = -1 / sec² x  .