Kaoyan Mathematics III common mistakes
Kaoyan Mathematics III · Understand the traps and how to avoid them.
When differentiating a composite function, the candidate differentiates only the outer function and drops the inner derivative - writing d/dx sin(3x²) = cos(3x²) or d/dx e^(x²) = e^(x²).
Chain rule is learned as a pattern ('derivative of sin is cos') and under timed conditions the candidate pattern-matches the outer function and stops, treating the argument as a single variable. Nested functions with polynomial or exponential inner parts are where the omission is most likely, because the inner derivative is not visually salient.
What to do instead
Before writing any answer, explicitly name the inner function u and write the chain as outer′(u)·u′ on the page, even for one-line answers. Drill a mixed set of 20 composites (trig, exponential, log, power, inverse trig) and mark each inner derivative in a different colour until the step becomes automatic. On the exam, do a final pass over every derivative checking that each composite has a visible inner-derivative factor.
The candidate computes a definite integral or area but substitutes the limits in the wrong order or forgets to change limits after substitution, producing a sign-flipped or wrong-valued answer, e.g. evaluating ∫₀² with F(0) − F(2).
The upper-minus-lower convention competes in memory with the substitution rule's 'evaluate at old limits', and under pressure the candidate writes the antiderivative values in the order they appear on the page. When using u-substitution the candidate often back-substitutes inconsistently or plugs original limits into the u-antiderivative.
What to do instead
Adopt a fixed written template: bracket the antiderivative, write '= [F(x)]_a^b = F(b) − F(a)', and physically copy b above a before evaluating. For substitution, always either convert the limits to u-values in the same line or write 'back-substitute to x, then apply original limits' - never mix. Practise five substitution integrals where the limits are converted, checking that a positive integrand gives a positive result.
When finding eigenvectors, the candidate solves det(A − λI) = 0 for λ correctly but then solves (A − λI)x = 0 by row-reducing A itself rather than A − λI, or picks a single eigenvector for a repeated eigenvalue and misses the second independent one.
Row-reduction of A is a well-practised reflex, and substituting the numeric λ into every diagonal entry is error-prone mental arithmetic. For repeated eigenvalues the candidate stops once one free variable yields a vector, not checking the nullity of A − λI.
What to do instead
Write A − λI explicitly with λ substituted before any row operation, and compute its nullity to know in advance how many independent eigenvectors to expect (nullity = multiplicity for diagonalisable cases). For each eigenvalue, state the dimension of the eigenspace before choosing vectors. Verify each candidate v by computing Av and confirming it equals λv - a 30-second check that catches both errors.
The candidate computes a cross product or scalar triple product and reports a magnitude or a signed value when a vector (or the opposite orientation) was required, or takes the dot product where the cross product is needed for a normal vector.
Area, volume and normal-vector problems all invoke 'cross product' in the candidate's notes, and the operations get conflated. Sign conventions in the triple product (a·(b×c) versus (a×b)·c) and the direction of a×b are also easy to reverse under time pressure.
What to do instead
Write the geometric quantity required first - 'normal vector =', 'area = ½|a×b|', 'volume = |a·(b×c)|' - then perform the operation that matches it. Compute cross products with the determinant expansion written out in full (i, j, k rows) rather than mentally, and check orientation with the right-hand rule or by confirming a·(a×b) = 0 and b·(a×b) = 0.
The candidate computes P(A|B) as P(A)P(B), or uses P(A|B) = P(A∩B)/P(B) but substitutes P(A) for P(A∩B) when events are not independent, and applies the multiplication rule for independent events to dependent sampling without replacement.
The formula P(A∩B) = P(A)P(B) is over-learned from independence examples and gets applied as a default. Condition notation is also read backwards: candidates compute P(B|A) when P(A|B) is asked, especially in Bayes-type word problems where the given and target events are described in prose.
What to do instead
Before computing, write the two events in words and label which is given ('given B') and which is the target. Use the general rule P(A∩B) = P(A|B)P(B) unless independence is explicitly justified, and for without-replacement problems draw the tree with changing denominators. For Bayes problems, write the partition and total probability denominator explicitly before dividing.
The candidate carries out a hypothesis test and states the conclusion as 'the null hypothesis is true' or 'proven', or interprets a p-value as the probability that H₀ is true, rather than reporting a decision about evidence at the stated significance level.
Everyday language treats 'not rejected' as 'confirmed', and the p-value's definition (probability of data at least as extreme, given H₀) is easily inverted in memory to 'probability H₀ is true'. The candidate also forgets to state the test statistic, degrees of freedom and the decision rule in context.
What to do instead
Use a fixed conclusion sentence: 'At the 5% significance level there is (insufficient/sufficient) evidence to reject H₀; we cannot conclude H₀ is true, only that it is not contradicted.' Write the p-value definition once in the margin as a reminder. Always report the test statistic, its distribution and degrees of freedom, and phrase the conclusion in terms of the original variables.
The candidate sets up a modelling problem by substituting numbers immediately, producing an equation in the wrong variables or with mismatched units, and never states the assumptions or checks the solution against the physical constraints (non-negativity, boundary values, limiting behaviour).
Exam pressure pushes toward computation, and word problems invite premature substitution before the relationship between quantities is defined. Units (per hour versus per minute, percentage versus proportion) are silently mixed, and the resulting answer is accepted without a plausibility test.
What to do instead
Write three lines before any algebra: define every variable with its units, state the governing relationship or balance law (rate in − rate out, conservation, proportionality), and list the assumptions. Keep symbols until the final step, then substitute with units carried through. Finish by checking the solution against boundary conditions and limits (e.g. t → 0 and t → ∞) and confirming it lies in the physically admissible range.
When asked to discuss the system AX=B for a parameter a, the candidate computes det(A), sets it to zero, finds the critical values, and then stops, declaring the system 'incompatible' at those values without ever computing the ranks of A and of the augmented matrix for each critical value.
The scheme 'det=0 means no solution' is memorised from drills where the coefficient matrix was square and the critical case always happened to be incompatible. It works often enough in practice that the habit is never corrected, and computing ranks at the critical values requires extra work that feels redundant once a determinant has been found.
What to do instead
Adopt a fixed three-branch script: (1) det(A)≠0 gives a unique solution via Cramer or inverse; (2) for each value with det(A)=0, write out the numerical matrices A and [A|B] with that value substituted and compute ranks by Gaussian elimination; (3) conclude rank(A)=rank([A|B])=n unique, =rank([A|B])<n infinitely many with n−rank parameters, <rank([A|B]) incompatible. Practise at least five systems where the critical case is compatible and gives infinitely many solutions, so the branch is not a surprise.
In a matrix problem, the candidate finds a matrix X satisfying an equation such as AX+B=C by writing X=A⁻¹C−B, i.e. multiplying on the wrong side or subtracting before multiplying by the inverse.
Scalar algebra is so automatic that the non-commutativity of matrix multiplication is acknowledged in theory but not applied in the manipulation. Candidates also forget that B must be moved first, so the inverse multiplies the whole right-hand side, and they treat (C−B) as if order did not matter.
What to do instead
Before touching symbols, write the target equation and isolate X with explicit side labels: AX=C−B, then left-multiply both sides by A⁻¹ to get X=A⁻¹(C−B). Check invertibility (det≠0) explicitly, and verify the answer by substituting back into the original equation. As a rule, never write a product of matrices in an order different from the one derived.
In an area-between-curves problem, the candidate integrates f−g over the interval between the two given x-values without locating the interior intersection points, so regions where the curves cross are counted with the wrong sign and partially cancel.
Textbook exercises often have one curve above the other throughout, so the formula ∫(f−g) is applied as a reflex. The possibility of an interior crossing is not checked because finding it requires solving f(x)=g(x), an extra step that does not look like part of 'the area formula'.
What to do instead
Always begin by solving f(x)=g(x) on the interval and splitting the integral at every interior root, taking |f−g| on each subinterval (or identifying which function is on top by testing a midpoint). For areas involving the y-axis or curves given as x=g(y), decide the variable of integration before setting up. Sketch the region and mark the crossing points on the sketch.
In a volume-of-revolution problem, the candidate uses the disk formula π∫f(x)²dx when the axis of revolution is not the x-axis and the region is not adjacent to it, omitting the offset (R(x)−r(x)) or using the wrong radius.
The formula is memorised for rotation about the x-axis with the region bounded by the curve and the axis. When the axis is y=−1 or x=2, the distance from a typical point to the axis changes, and the candidate substitutes f(x) directly instead of the actual radius.
What to do instead
Draw the region and the axis, then mark a typical slice and write the distance from the slice to the axis explicitly: radius = |f(x) − axis|. For a region between two curves use π∫(R²−r²)dx with R and r measured from the axis. For rotation about a vertical axis, integrate in y or use shells; state which method and why before computing.