# CTMUA

> Source: https://ollybritton.com/notes/a-level/general/ctmua/ · Updated: 2025-04-30 · Tags: school

## Flashcards
##### $$\frac{24\pi (t - 1)}{1 + \sqrt{t}}$$ How can you integrate this??
Rewrite as
$$
\frac{24\pi (\sqrt{t} - 1)(\sqrt{t} + 1)}{\sqrt{t} + 1} = 24\pi (\sqrt{t} - 1)
$$

##### When checking if something is a counterexample, what should you do??
Make sure that the counterexample follows the definition of what is valid.

##### $$\log(ab^2c) = 7$$ If you have 3 sets of equations of something like this, how can you rewrite it??
As a set of simultaneous equations in the form $\log a + 2\log b + \log c = 7$

##### What is the general form of a geometric sequence for term $n$??
$$
ar^(n-1)
$$

##### What is the formula for the sum of the first $n$ terms of a geometric sequence $ar^n$??
$$
\frac{a(r^n - 1)}{(r - 1)}
$$

##### What is the formula for the trapezium rule with width $h$ and $y$ values $y_0$, $y_1$, $y_2$, etc??
$$
\frac{1}{2}h(y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n)
$$

##### What does it mean for a curve $f(x)$ to be concave??
$$
f''(x) \le 0
$$

##### What does it mean for a curve $f(x)$ to be convex??
$$
f''(x) \ge 0
$$

##### When does the trapezium rule produce an overestimate in terms of concavity??
When the curve is convex.

##### When does the trapezium rule produce an overestimate in terms of derivatives??
When $f''(x) \ge 0$ on the interval being integrated.

##### When does the trapezium rule produce an underestimate in terms of concavity??
When the curve is concave.

##### When does the trapezium rule produce an underestimate in terms of derivatives??
When $f''(x) \le 0$ on the interval being integrated.

##### The trapezium rule has produced an overestimate. In terms of concavity, what is true??
The curve is convex.

##### The trapezium rule has produced an overestimate. In terms of derivatives, what is true??
$$
f''(x) \ge 0
$$

##### The trapezium rule has produced an underestimate. In terms of concavity, what is true??
The curve is concave.

##### The trapezium rule has produced an underestimate. In terms of derivatives, what is true??
$$
f''(x) \le 0
$$

##### What's another way of saying concave up??
Convex.

##### What's another way of saying convex??
Concave up.

##### What's another way of saying concave down??
Just "concave".

##### What's another way of saying concave??
Concave down.

### 2021-07-01
##### As a consequence of the factor theorem, for what polynomial in terms of $f(x)$ is $x - t$ a factor??
$$
f(x) - f(t)
$$

### 2021-07-03
##### You're stuck trying to solve a tricky equation. What's very likely the next step??
Make a substitution!

##### What should be the slogan of the CTMUA??
Make a substitution.

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