Flashcards · 7357 · Topic 1

Overarching themes flashcards

17 cards on Topic 1, subtopics OT1 to OT3. Pick a subtopic or study them all.

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Equation vs identity: An equation is true only for some values (you solve it). An identity is true for all values and uses ≡, for example (x + 2)² − 1 ≡ x² + 4x + 3.
What do ⇒, ⇐ and ⇔ mean?: ⇒: implies. ⇐: is implied by. ⇔: if and only if; both directions must be true.
Find a, b and c if 3x² − 6x + 8 ≡ a(x + b)² + c.: Compare coefficients: x²: a = 3; x: 2ab = −6, so b = −1; constant: ab² + c = 8, so c = 5. 3(x − 1)² + 5.
Term, coefficient, constant and expression in 5x³ − 2x + 9: Terms: 5x³, −2x and 9. Coefficients: 5 and −2. Constant: 9. It is an expression because it has no equals sign.
Set notation: what do ∈, ∪, ∩ and ∅ mean?: ∈: is an element of. A ∪ B: union (in A or B or both). A ∩ B: intersection (in both A and B). ∅: the empty set.
The converse of "P ⇒ Q", and why it matters: The converse is "Q ⇒ P". It is not automatically true: x = 4 ⇒ x² = 16, but x² = 16 does not imply x = 4.

The four stages of the problem-solving cycle: 1. Specify the problem. 2. Collect information. 3. Process and represent. 4. Interpret the results (which may send you round the cycle again).
How do you start an unstructured problem?: Underline what is wanted and what is fixed; name the variables with units; draw a diagram; spot the structure (optimise, rate of change, equilibrium, …); write one equation in one variable and note its domain.
Resolve a force of 10 N at 40° to the horizontal.: Horizontal: 10 cos 40° = 7.66 N. Vertical: 10 sin 40° = 6.43 N.
How are bearings written?: As three figures, measured clockwise from north (for example 045°).
How do you check an answer at the end of a problem?: Answer the original question in a sentence, with units. Test the sign, size and units, and check that the value lies in the allowed domain.
When should you round in a multi-step calculation?: Only at the end. Carry full calculator accuracy through every intermediate step, or the final answer may be wrong.

The stages of the modelling cycle: 1. Translate the situation into mathematics, stating simplifying assumptions. 2. Use the model with chosen inputs. 3. Interpret the outputs in context. 4. Compare with reality, judge the model and refine it.
Give three common simplifying assumptions in mechanics models.: Treat a body as a particle; treat a string as light and inextensible; treat a surface as smooth (no friction) or ignore air resistance.
A taxi model is cost = 8 + 1.6d pounds for d km. A 20 km fare was actually £44. Evaluate the model.: The model predicts 8 + 1.6 × 20 = £40, so it underestimates. Refine it by adding a missing feature (such as waiting time or a minimum charge) and test it on new data.
What should you consider when criticising a model?: Whether its assumptions are realistic, whether the inputs used are in a sensible domain, how well its predictions match real data, and a specific refinement.
Why must model outputs be interpreted in context?: A numerical answer may be impossible in reality (negative, too large, not a whole number). State what it means with units, and whether it is sensible.