Integrals, limits and series · Calculus and linear algebra refresher

11 cards3 min readCalculus

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Idea: Quick questions, standard moves
Integrals, $\frac{0}{0}$ limits and infinite series each have a move that turns them into arithmetic.
- Integral: run the power rule backwards
- Limit: cancel the hidden factor or compare degrees
- Series: find the first term and the ratio
Formula: Power rule backwards
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$
For n ≠ −1. The integral from a to b is F(b) − F(a) for any antiderivative F.
Example: Average value
What is the average value of $3 x^{2} - 2 x + 1$ on $[0, 2]$ ?
1. Antiderivative: $F (x) = x^{3} - x^{2} + x$
2. $F (2) = 8 - 4 + 2 = 6$ and $F (0) = 0$ , so the integral is $6$
3. Divide by the length $2$ : $\frac{6}{2} = 3$
Formula: A 0/0 limit is a slope
$p (c) = 0 \Rightarrow lim \frac{p ( x )}{x - c} = p^{'} (c)$
Getting 0/0 means a common factor hides: cancel it, or take the derivative of the top.
Quick check: Cancel the hidden factor
Find the limit of $\frac{x ^{2} + 2 x - 15}{x - 3}$ as $x \to 3$ .
Show the answer
Answer: $8$
$x^{2} + 2 x - 15 = (x - 3) (x + 5)$ ; cancel, and $3 + 5 = 8$ .
Idea: Limits at infinity
For a ratio of polynomials as $x \to \infty$ , only the highest powers matter.
- Equal degrees: the ratio of leading coefficients
- Lower degree on top: the limit is $0$
- Higher degree on top: no finite limit
Common mistake: Reading off the constants
For $\frac{6 x ^{2} + x - 4}{3 x ^{2} - 7 x + 2}$ the constants give $- \frac{4}{2} = - 2$ : the value at $x = 0$ , not at infinity.
Instead As $x$ grows only the $x^{2}$ terms matter: $\frac{6}{3} = 2$ .
Formula: Geometric series
$a + a r + a r^{2} + \dots = \frac{a}{1 - r}$
Needs |r| < 1. If |r| ≥ 1 the terms do not shrink and there is no finite sum.
Example: Halving forever
Sum $5 + \frac{5}{2} + \frac{5}{4} + \dots$ .
1. First term $5$ , ratio $\frac{1}{2}$ , and $∣ r ∣< 1$
2. Sum: $\frac{5}{1 - \frac{1}{2}} = 10$
Quick check: A series weighted by position
What is $\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \dots$ ?
Show the answer
Answer: $2$
$\sum k r^{k} = \frac{r}{( 1 - r ) ^{2}}$ , so $\frac{\frac{1}{2}}{\frac{1}{4}} = 2$ : the mean number of flips to a first head.
Idea: Name the move first
Say which move you use before you compute.
- Integral: name the antiderivative, then plug in
- $\frac{0}{0}$ : say which factor cancels; at infinity, compare degrees
- Series: first term, ratio, check $∣ r ∣< 1$ , then the sum

Idea: Quick questions, standard moves

Formula: Power rule backwards

Example: Average value

Formula: A 0/0 limit is a slope

Quick check: Cancel the hidden factor

Idea: Limits at infinity

Common mistake: Reading off the constants

Formula: Geometric series

Example: Halving forever

Quick check: A series weighted by position

Idea: Name the move first