Many are using the following symbols as if they meant the same thing: (a) The symbol “ = ” means equals, and nothing else. It connects expressions, sets, quantities, or any objects of the same nature, but not statements. It is not an all-purpose connector, nor does it open a paragraph. (b) The symbol “ ⇒ ” means implies. It connects sentences or logical statements; it does not connect values or expressions. For instance: sin2 x + cos2 x = 1 ⇒ 2cos2 x + 3sin2 x = 2 + sin2 x is correct, an equality being a particular kind of statement. sin2 x + cos2 x ⇒ 1 does not make sense: “sin2 x + cos2 x” and “1” are not statements. (c) The symbol “ → ” means tends to, as in “has limit”. It cannot be substituted either to “ ⇒ ” or “ = ”.
2. (a) The lim symbol is never used by itself. So “lim = − 1” is not valid; you must specify t →0 limit of what. (b) The lim symbol takes precedence over algebraic operators. So lim x2 − x + 1
is read as “(lim x2 ) − x + 1”, which is the function − x + 2, not at all the same as x→1
lim (x2 − x + 1)
which is the number 1. Always use parentheses following the limit symbol when taking the limit of compound expressions. 3.
Consider the function 2 f (x) = x − 1 . x−1
Find its limit as x → 1: x2 − 1 = x + 1 → 2 as x → 1. x−1
-2The limit, 2, is a fixed number and does not depend on x . In our use of notation, we carefully distinguish fixed quantities from functions of some variable. Therefore the following is correct: 2 lim x − 1 = lim (1 + x) x→1 x − 1 x→1
=2 (fixed number = fixed number = fixed number), whereas in the following, both “ = ” signs are wrong: 2 lim x − 1 = 1 + x x→1 x − 1
=2 (fixed number = function of x = fixed number). 4.
Never write 0/0, a/0, 1/∞, a/∞, this style.
∞/∞, 0 × ∞, ln(0), 00 or any other variant of
Distinguish between the expression of a function and its value at a particular point. If I found that g(x) = 1 + x and I will need its value at x = 1, I don’t write g(x) = 1 + x = 2 but I write two separate statements: g(x) = 1 + x g(1) = 2 . This is one more reason to name all functions, even if the data of the problem does not name them. If no name was assigned, it is still possible to distinguish the function and a particular value: (1 + x)|x = 1 = 2 . The underlying principle is the same as in note 3: functions are one kind of object, fixed values, another.
As with any language, a solution in mathematics is a sequence of sentences, separated by periods. The sentences are often, but not always, themselves sequences of equalities (or inequalities). They may sometime contain words (“since the function is continuous, the hypothesis of such theorem holds…”). A “solution” which consists of a heap of disconnected expressions will be considered nonsensical, and may not get any credit.
(Adventures with square root). Negative numbers are not the square root of their square. The “identity” x ≡ √ x2 is not correct. The correct one is
√ x2 ≡ |x | . 8.
Use of parentheses. Algebraic symbols + , − , ⋅ , × etc… are not allowed to collide and must be separated by parentheses. To multiply x by −y, we write x(−y) . “x ⋅ −y” or “ x × −y” is incorrect, even if you reduce the size of the minus sign, raise it and stick it very close to y . Also, x − y and x + (−y) are correct. “ x + −y ” is not, even if you reduce the size of the minus sign, raise it and stick it very close to y .
Distinguish between exact and numerical (approximate) value. π is an exact value. 3.1416, or π expanded to the thousandth decimal digit, if you mean it to stand for π, is not. As stated in the exam template, you are to provide exact values (this holds practically for any problem involving limits), not numerical approximations, unless the problem explicitly requires the latter. I also notice than when using your calculator to deal with trigonometric functions (when for the most part, you should not use the calculator at all), there is a tendency to forget to use radian mode.
10. We never use the integral symbol without the differential symbol. you are integrating with respect to x, you must write ∫ f (x) dx .
∫ f (x)
is incorrect. If
11. We never equate differential expressions with scalars: d(cosθ) = −sin(θ)dθ is correct (on the right, scalar times differential = differential). d(cosθ) = −sin(θ) is not, and will steadily drain you of precious points on tests. 12. The comparison properties of the integral are the one thing you are required still to remember when you will have forgotten everything else about the integral. They are found at the end of §5.2 of Stewart. Upshots of these properties are: (a) If f (x) ≥ 0 on an interval, then ∫ f (x) dx on that interval cannot be negative. b
(b) If f (x) ≤ M on the interval (a, b), then ∫ f (x) dx cannot exceed M(b − a) . a
These facts hold not only for the exact integral, but for the numerical integration schemes as well. This means that if, for instance, the values of f at the integration node points do not b
exceed M, then the numerical approximation of ∫ f dx cannot exceed M(b − a) . a
13. The “indefinite integral” always refers to a family of functions, not a single function.
dx = ln| x | + K x
Omitting the additive constant (which does not have to have name C) counts as a serious mistake.
Some common corrections Read carefully. Making these mistakes will be penalised at the rate of 1.5 points for each instance (in addition to any other point penalty). 1.