Pancyprian Advanced Math common mistakes
Pancyprian Advanced Math · Understand the traps and how to avoid them.
When differentiating a composite function, the candidate differentiates only the outer function and writes, for example, (sin 3x)' = cos 3x or (e^{x^2})' = e^{x^2}, omitting the inner derivative entirely.
Chain rule is learned as a formula but practised mostly on single-layer examples, so under exam pressure the brain pattern-matches to the familiar derivative table entry and stops. Nested expressions written without brackets (sin 3x rather than sin(3x)) make the inner function visually invisible.
What to do instead
Adopt a written ritual: before differentiating, explicitly label u = inner expression and write dy/dx = (dy/du)(du/dx) on the script. Train on deliberately nested items such as ln(cos 2x), sqrt(1 - x^2), e^{sin^2 x}, and always rewrite trig arguments with brackets first. Mark inner derivatives in a different colour during practice until the step becomes automatic.
The candidate substitutes the limit point into a rational or radical expression, obtains 0/0, and either declares the limit does not exist or stops, instead of factorising, rationalising or cancelling the common factor.
0/0 is treated as an answer rather than as a signal that algebraic work is required. Weak factorisation fluency (especially of cubics and of differences of roots) makes cancelling feel risky, so the candidate avoids it.
What to do instead
Drill the rule: 0/0 means 'indeterminate, do more algebra'. Build a short decision list - factorise numerator and denominator, multiply by the conjugate for surds, combine fractions over a common denominator, or use standard limits for trig/exponential forms. Practise twenty limits of each type and always state the cancelled factor and the domain restriction (x ≠ a) explicitly.
In integration by parts the candidate assigns u and dv arbitrarily, or applies the formula with the wrong sign, writing ∫u dv = uv + ∫v du, or forgetting to evaluate the second integral.
The formula is memorised as a shape rather than derived, and the choice of u is made without a criterion. Sign errors arise because the minus sign is buried inside the formula and is easily lost when substituting limits.
What to do instead
Memorise the selection rule LIATE (logarithmic, inverse trig, algebraic, trig, exponential) for u, and verify by differentiating the proposed antiderivative. Derive the formula once from the product rule so the minus sign has a reason. On definite integrals, write the uv term with limits in square brackets separately from the remaining integral before substituting.
When finding the area between a curve and the x-axis, the candidate integrates f(x) from a to b without checking where f changes sign, so regions below the axis subtract from regions above and a wrong (sometimes negative) area is produced.
Area is treated as a single definite integral by reflex, and sign change is not linked to the geometry. Sketching is skipped to save time, removing the only cue that the curve crosses the axis.
What to do instead
Make a sketch mandatory for every area question. Solve f(x) = 0 first and split the integral at each root, taking the absolute value of each piece: A = ∫|f(x)| dx. Check plausibility - area must be positive and roughly comparable to a bounding rectangle. Practise a curve such as y = x^3 - x over [-1, 1] where naive integration gives zero.
In a binomial expansion, the candidate writes the general term as C(n,r) a^n b^r with the powers swapped, or forgets the binomial coefficient when asked for a specific coefficient, or uses n instead of n - 1 for the number of terms.
The general term is memorised as a blob and the symmetry between a and b is not understood. Coefficient questions are read as 'find the term' and the combinatorial factor is dropped.
What to do instead
Anchor the general term by writing the first three terms of (a + b)^n explicitly every time: a^n, n a^{n-1} b, C(n,2) a^{n-2} b^2. For a coefficient question, set up the general term with the unknown r, equate the exponent to the required power, solve for r, then substitute into the full term including the coefficient. Verify with a small case such as (1 + x)^3.
With complex numbers, the candidate adds arguments when multiplying but forgets to multiply the moduli, or writes arg(z1 z2) = arg z1 + arg z2 without adjusting for the principal range, and mishandles division by not using the conjugate.
Polar form is learned as two separate rules and their interaction is not practised. The principal argument range (-π, π] is ignored because most textbook examples happen to fall inside it.
What to do instead
Write multiplication and division in modulus-argument form as a single block: |z1 z2| = |z1||z2| and arg(z1 z2) = arg z1 + arg z2, then explicitly reduce the result into (-π, π] and state the adjustment. For division, always multiply numerator and denominator by the conjugate of the denominator and show the denominator becoming |z|^2. Verify by converting back to Cartesian form for one numerical example.
The candidate computes a vector dot product to test perpendicularity but uses it to test parallelism, or confuses the scalar product with the vector (cross) product, and in 3D writes the equation of a line without a direction vector.
Both products are taught close together and the geometric meanings are memorised as labels rather than derived. The line equation is recalled as 'a point and something' without checking which something.
What to do instead
Build a one-page comparison table: dot product gives a scalar, zero means perpendicular, used for angles and projections; cross product gives a vector, zero means parallel, used for normals and areas. For lines, always write r = a + t d and label a as position and d as direction, then check a known point satisfies it. Test each answer against a simple example with obvious geometry.
For a conic, the candidate completes the square incorrectly (losing the factor of the squared term, e.g. writing 4x^2 + 8x as 4(x + 1)^2 - 1 instead of 4(x + 1)^2 - 4), then reads off the centre, foci or eccentricity from the wrong standard form.
Completing the square is done mechanically without factoring the leading coefficient first, and the constant adjustment is not tracked. The standard forms of ellipse and hyperbola are then matched by shape rather than by derivation.
What to do instead
Force the sequence: group terms, factor out the coefficient of the squared term, halve the linear coefficient inside the bracket, add and subtract its square, then multiply back through the bracket and rebalance the constant on the other side. Always reduce to a canonical form with 1 on the right before extracting a, b, c and eccentricity, and check the sign pattern to distinguish ellipse from hyperbola.
When a sequence is defined recursively, the candidate treats it as arithmetic or geometric, or uses the sum formula for an arithmetic series on a sequence that is not arithmetic, and misapplies the condition for convergence of a geometric series (using |r| > 1).
Formula recall dominates over checking the defining property. The convergence condition is remembered as a symbol string without the reasoning that terms must shrink.
What to do instead
Before using any formula, write the first four terms from the definition and test whether the difference or ratio is constant. For geometric series, state explicitly S = a/(1 - r) only when |r| < 1 and give a one-line justification (terms tend to zero). For recursive sequences, compute several terms numerically and look for a pattern before guessing a closed form, then prove it by induction.
In probability, the candidate adds probabilities of overlapping events instead of using P(A ∪ B) = P(A) + P(B) - P(A ∩ B), or assumes independence without justification when computing P(A ∩ B) = P(A)P(B).
The addition rule is learned in the disjoint case and the subtraction term is forgotten; independence is assumed because it makes the arithmetic easy, not because the problem states it.
What to do instead
Draw a Venn diagram or two-way table for every union/intersection question and shade the region asked for. Write the general addition rule before substituting, and only replace P(A ∩ B) by P(A)P(B) after explicitly writing 'A and B are independent because …' or verifying P(A|B) = P(A). Where conditional probability appears, use the definition P(A|B) = P(A ∩ B)/P(B) rather than intuition.