Here we will list some of the most common mistakes. If you can avoid these, then at least your mistakes will be
uncommon. Most of the mistakes that occur repeatedly involve algebra, rather
than calculus. They can be avoided by being careful and checking your work.
Others involve common misunderstandings about various aspects of calculus.
- (x+y)2 = x2 + y2. MISTAKE!
Powers don't behave this way. The correct way to expand this expression gives
- 1/(x+y) = 1/x + 1/y . MISTAKE!
The rule for adding fractions gives
1/x + 1/y = (x+y)/xy
- 1/(x+y) = 1/x + y. MISTAKE!
This very common error comes from carelessness about what's in the denominator.
Can be avoided by careful handwriting or frequent use of parentheses.
- √(x+y) = √x + √y. MISTAKE!
There is no simplified way to write √(x+y). You just have to live with
it as is.
- x < y so kx < ky where k is a constant.
MISTAKE!
This is true when k is a POSITIVE constant. If k is negative you need to
reverse the inequality. If k is zero all bets are off. For example, if x < y
then -x > -y.
- ax = bx therefore a = b. MISTAKE!
This is a more subtle mistake. The cancellation
is correct IF x is not 0. For example 2x = 3x forces x = 0. You cannot
cancel the x and conclude that 2 = 3. Not in this universe, anyway.
- sin 2x / x = sin 2 . MISTAKE!
You can only cancel terms in the numerator and denominator of a fraction if they
are not inside anything else and are just multiplying the rest of the numerator
and denominator. The function sin2x is NOT sin2 multiplied by x. If the
fraction had been written as
it would be harder to make such an error.
- Forgetting to simplify fractions in limits: MISTAKE!
It is not correct to say limx → 1 (x2-1)/(x-1) = 0/0
and therefore the limit is undefined. Even worse would be to cancel the zeroes
and say the limit equals one.
Any time you get 0/0 for a limit, it is a BIG WARNING SIGN that says
YOU HAVE MORE WORK TO DO! In this case,
|
lim
x → 1
|
|
x2-1 x-1
|
= |
lim
x → 1
|
|
(x-1)(x+1) x-1
|
= |
lim
x → 1
|
x+1 = 2. |
|
- d/dx [2x] = x 2x-1 MISTAKE!
The correct answer is 2x (ln2). The power rule only applies if the base
is a variable and the exponent is a constant, as in x3.
- d/dx [sin(x2+1)] = cos2x. MISTAKE!
This is a typical example of the kind of mistakes made when applying the chain
rule. The correct answer is
|
d dx
|
[sin(x2+1)] = (cos (x2+1 )) (2x). |
|
- d/dx [sin(x2+1)] = cos(x2+1) + sin2x. MISTAKE!
Another common way in which the chain rule is misapplied. This time the
product rule has been used in a setting where the chain rule was the way to go.
- d/dx [cosx] = sinx . MISTAKE!
The answer should be -sinx . Extremely common error costing students over
10 million points a year on exams around the world.
- d/dx [f/g] = (fg′-gf′)/( g2).
MISTAKE!
This is backwards! It should be
|
d dx
|
( |
f g
|
) = |
gf′-f g′ g2
|
. |
|
- d/dx [ln3] = 1/3. MISTAKE!
The quantity ln3 is a constant, so d/ dx [ln3] = 0.
The same is true for ALL constants. So
d/ dx [e] = 0 and d/ dx [sin(π/2)] = 0 as well.
- ∫1/x dx = x0 / 0 + C. MISTAKE!
The power rule for integration does not apply to x-1.
Instead,
| ⌠ ⌡ |
1/x dx = ln(x) + C1 (per x>0) |
| |
| ⌠ ⌡ |
1/x dx = ln(-x) + C2 (per x<0) |
|
- ∫tanx dx = sec2 x + C. MISTAKE!
It's the other way around. d/ dx [tanx] = sec2 x.
The correct answer is
| ⌠ ⌡ |
tanx dx = ln|sec x| + C |
|
as can be found by u-substitution
with u = cosx.
- Forgetting to simplify: MISTAKE!
For example
is easy if you notice that x √x = x3/2 and then apply the
power rule for integration. But if you try to do it using
integration by parts or
substitution, you will find yourself in outer space without a
space suit.
- Not substituting back to the original variable: MISTAKE!
does not equal u → eu + C. It equals x → ex2 + C.
- Misreading the problem: MISTAKE!
If asked to find an area, don't find a
volume. If asked to find a derivative, don't find an integral. If asked to use
calculus to solve a problem, don't do it in your head using algebra. Although
it seems silly to include this item in our list, billions of points have
been taken off exams for mistakes of this type. After you finish a problem on
the exam, go back and read the question again. Check to make sure you answered
the question that was asked.
- Thinking you're prepared when you're not. MISTAKE!
This mistake is perhaps the most important, so we'll put it in even though it pushes us over
the 20 mistakes limit. The worst mistake many students make is to think they know the material better than
they really do. It's easy to fool yourself into thinking you can solve a problem when you're
looking at a worked out solution. Test your knowledge by trying
problems under "exam conditions".
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