Flashcards

Flashcards

Only what earns marks: exact values, the standard results the formulae booklet does not print, definitions in precise wording, and the slips that lose marks. Grade yourself and the deck schedules each card for just before you would otherwise forget it.

Progress lives in this browser, on this device only, with no accounts, just like the revision checklist. These starter decks will grow with the library.

Exact trig values 14 cards

The degree-and-radian values the paper expects without a calculator.

Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

sin 30° = ?
sin 30° = 12

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

cos 30° = ?
cos 30° = √32

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

tan 30° = ?
tan 30° = 1√3

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

sin 45° = ?
sin 45° = 1√2

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

cos 45° = ?
cos 45° = 1√2

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

tan 45° = ?
tan 45° = 1

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

sin 60° = ?
sin 60° = √32

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

cos 60° = ?
cos 60° = 12

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Result Exact trig values — Exact value

STANDARD RESULT — LEARN IT

Exact value:

tan 60° = ?
tan 60° = √3

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Result Exact trig values — Degrees to radians

STANDARD RESULT — LEARN IT

Degrees to radians:

30° = ? rad
30° = π6

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Result Exact trig values — Degrees to radians

STANDARD RESULT — LEARN IT

Degrees to radians:

45° = ? rad
45° = π4

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Result Exact trig values — Degrees to radians

STANDARD RESULT — LEARN IT

Degrees to radians:

60° = ? rad
60° = π3

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Result Exact trig values — Degrees to radians

STANDARD RESULT — LEARN IT

Degrees to radians:

90° = ? rad
90° = π2

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Check Exact trig values

CHECK YOURSELF

Without a calculator, state the exact value of cos 30° × tan 60°.

cos 30° = √3/2 and tan 60° = √3, so the product is 3/2.

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Differentiation: standard results 9 cards

The derivatives the formulae booklet does not hand you.

Result Differentiation: standard results — Power rule

STANDARD RESULT — LEARN IT

Power rule:

ddx(xn) = ?
ddx(xn) = nxn−1

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Result Differentiation: standard results — Exponential

STANDARD RESULT — LEARN IT

Exponential:

ddx(ekx) = ?
ddx(ekx) = kekx

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Result Differentiation: standard results — Natural log

STANDARD RESULT — LEARN IT

Natural log:

ddx(ln x) = ?
ddx(ln x) = 1x

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Result Differentiation: standard results — Sine (x in radians)

STANDARD RESULT — LEARN IT

Sine (x in radians):

ddx(sin x) = ?
ddx(sin x) = cos x

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Result Differentiation: standard results — Cosine (x in radians)

STANDARD RESULT — LEARN IT

Cosine (x in radians):

ddx(cos x) = ?
ddx(cos x) = −sin x

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Result Differentiation: standard results — General exponential

STANDARD RESULT — LEARN IT

General exponential:

ddx(akx) = ?
ddx(akx) = kakx ln a

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Define Differentiation: standard results

DEFINE

Define: differentiation from first principles.
The derivative of f(x) is the limit as h tends to zero of (f(x + h) − f(x))/h, provided the limit exists.

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Check Differentiation: standard results

CHECK YOURSELF

Differentiate y = 3 ln x − cos 2x.

dy/dx = 3/x + 2 sin 2x.

The 2 appears from the chain rule on cos 2x, and the sign flips because the derivative of cos is minus sin.

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Spot the error Differentiation: standard results

SPOT THE ERROR

Say what is wrong with this claim:

The derivative of cos x is sin x.
It is −sin x. Sine's derivative is cos x with no sign change; cosine's picks up the minus. Sketch both curves: cos x is falling just after zero, so its gradient there must be negative.

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Integration: standard results 7 cards

The integrals to produce on sight, constant of integration included.

Result Integration: standard results — Power rule (n ≠ −1)

STANDARD RESULT — LEARN IT

Power rule (n ≠ −1):

∫ xn dx = ?
∫ xn dx = xn+1n+1 + c

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Result Integration: standard results — The n = −1 case

STANDARD RESULT — LEARN IT

The n = −1 case:

∫ 1x dx = ?
∫ 1x dx = ln|x| + c

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Result Integration: standard results — Exponential

STANDARD RESULT — LEARN IT

Exponential:

∫ ekx dx = ?
∫ ekx dx = 1kekx + c

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Result Integration: standard results — Sine (x in radians)

STANDARD RESULT — LEARN IT

Sine (x in radians):

∫ sin x dx = ?
∫ sin x dx = −cos x + c

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Result Integration: standard results — Cosine (x in radians)

STANDARD RESULT — LEARN IT

Cosine (x in radians):

∫ cos x dx = ?
∫ cos x dx = sin x + c

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Check Integration: standard results

CHECK YOURSELF

Find the integral of 6e^(3x) with respect to x.

2e^(3x) + c: divide by the 3 from the chain rule, and never forget the constant on an indefinite integral.

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Spot the error Integration: standard results

SPOT THE ERROR

Say what is wrong with this claim:

The integral of sin x is cos x.
It is −cos x + c. Check by differentiating your answer: the derivative of −cos x is sin x, which recovers the integrand. Differentiate-to-check catches every sign slip of this kind.

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Indices and log laws 6 cards

The rewriting rules everything in algebra leans on.

Result Indices and log laws — Product law

STANDARD RESULT — LEARN IT

Product law:

loga x + loga y = ?
loga x + loga y = loga (xy)

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Result Indices and log laws — Quotient law

STANDARD RESULT — LEARN IT

Quotient law:

loga x − loga y = ?
loga x − loga y = loga xy

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Result Indices and log laws — Power law

STANDARD RESULT — LEARN IT

Power law:

k loga x = ?
k loga x = loga (xk)

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Define Indices and log laws

DEFINE

Define: logarithm.
log to base a of x is the power to which a must be raised to give x: if a^y = x then y equals log base a of x, for a > 0, a ≠ 1 and x > 0.

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Check Indices and log laws

CHECK YOURSELF

Write 2 log 3 + log 4 as a single logarithm.

2 log 3 = log 9, and log 9 + log 4 = log 36.

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Spot the error Indices and log laws

SPOT THE ERROR

Say what is wrong with this claim:

log(x + y) is the same as log x + log y.
The product law says log x + log y = log(xy): logs turn products into sums, not sums into sums. log(x + y) does not simplify. Try x = y = 10: log 20 is about 1.3, but log 10 + log 10 = 2.

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Key definitions 7 cards

Wording that earns marks, term by term.

Define Key definitions

DEFINE

Define: function.
A rule that assigns to each element of its domain exactly one element of its range.

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Define Key definitions

DEFINE

Define: increasing function on an interval.
A function whose derivative is greater than or equal to zero throughout the interval, so its value never falls as x increases.

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Define Key definitions

DEFINE

Define: arithmetic sequence.
A sequence in which the difference between consecutive terms is constant, the common difference d.

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Define Key definitions

DEFINE

Define: geometric sequence.
A sequence in which the ratio between consecutive terms is constant, the common ratio r.

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Define Key definitions

DEFINE

Define: radian.
The angle subtended at the centre of a circle by an arc equal in length to the radius.

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Define Key definitions

DEFINE

Define: counterexample.
A single specific case for which a general statement fails, which is enough to disprove the statement.

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Check Key definitions

CHECK YOURSELF

A geometric series converges. State the condition on the common ratio, and the sum to infinity.

Convergence needs |r| < 1, and then the sum to infinity is a/(1 − r).

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Spot the error 6 cards

The classic slips, and why each one is wrong.

Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

√(a² + b²) simplifies to a + b.
Square roots do not distribute over addition. Try a = 3, b = 4: the left side is 5, the right side is 7. Only √(a²b²) = ab behaves like that, because the inside is a product.

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Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

(x + 3)² = x² + 9.
Squaring a sum needs the cross term: (x + 3)² = x² + 6x + 9. The slip drops 2 × x × 3. Expanding the bracket term by term makes the middle term impossible to lose.

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Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

In (x² + 5x)/x², the x² terms cancel to leave 1 + 5x.
Only common factors of the whole numerator and denominator cancel. Factorise first: x(x + 5)/x² = (x + 5)/x, which is 1 + 5/x. Cancelling terms that are added, rather than factors that are multiplied, is the single most common algebra slip.

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Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

If sin x = 0.5 then x = 30° is the complete answer.
Sine repeats: in 0° to 360° alone, x = 30° and x = 150° both work, and the general solution adds full turns to each. Always read the given interval and use the sine curve or CAST to find every solution inside it.

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Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

Dividing both sides of x² = 4x by x gives x = 4, done.
Dividing by x silently assumes x ≠ 0 and throws away the solution x = 0. Move everything to one side and factorise: x(x − 4) = 0, so x = 0 or x = 4.

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Spot the error Spot the error

SPOT THE ERROR

Say what is wrong with this claim:

ln(a)/ln(b) simplifies to ln(a/b).
The quotient law is ln a − ln b = ln(a/b): a difference of logs, not a quotient of logs. ln(a)/ln(b) is the change-of-base form of log base b of a, and it does not simplify further.

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