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Flashcards
Integration and logarithms
\[\frac{24\pi (t - 1)}{1 + \sqrt{t}}\]
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 $\log a + 2\log b + \log c = 7$
Geometric sequences
What is the general form of a geometric sequence for term $n$?
What is the formula for the sum of the first $n$ terms of a geometric sequence $ar^n$?
The trapezium rule
What is the formula for the trapezium rule with width $h$ and $y$ values $y _ 0$, $y _ 1$, $y _ 2$, etc?
Concavity and convexity
Trapezium rule error and concavity
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?
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?
Concave and convex terminology
The factor theorem and substitution
As a consequence of the factor theorem, for what polynomial in terms of $f(x)$ is $x - t$ a factor?
You’re stuck trying to solve a tricky equation. What’s very likely the next step?
Make a substitution!