Ehime Entrance Mathematics common mistakes
Ehime Entrance Mathematics · Understand the traps and how to avoid them.
Differentiating a composite function such as log(sin x) or e^{x^2} and writing only the outer derivative, e.g. 1/sin x or e^{x^2}, without the inner factor cos x or 2x.
The chain rule is remembered as a table of outer derivatives, and under time pressure the inner derivative is dropped because the expression looks familiar.
What to do instead
Before differentiating, name the inner function u and write dy/dx = dy/du · du/dx explicitly. After simplifying, test one simple case such as d/dx e^{3x} to confirm the inner factor appears.
Finding the area between two curves by integrating f(x)-g(x) over the interval between only the first and last intersection points, without splitting where the curves cross or where f-g changes sign.
The candidate assumes one curve stays above the other and skips solving f(x)=g(x) for all intersections inside the interval.
What to do instead
First solve f(x)=g(x) and list all intersection x-values. On each subinterval, test one point to determine which curve is above, or integrate |f-g|; then split the integral at every crossing.
In a proof by induction, proving the algebraic step from n=k to n=k+1 but never checking the base case, or substituting n=k+1 into the target formula without using the assumption for n=k.
Induction is treated as a direct algebra exercise rather than as proving an implication, and the base case seems too trivial to write.
What to do instead
Write the base case explicitly, state the induction assumption P(k), then transform the left side of P(k+1) so that the expression from P(k) appears and is replaced. Finish by writing the conclusion.
Computing an angle between vectors with cos θ = a·b instead of a·b/(|a||b|), or treating the dot product a·b as a vector when writing line or plane equations.
The scalar-versus-vector distinction blurs, and the dot-product formula is recalled without its normalization denominator.
What to do instead
Write the full formula cos θ = (a·b)/(|a||b|) every time and label the result as a scalar. Check the right-angle case: if a·b=0, the formula must give cos θ=0.
Solving sin 2x = 1/2 by giving only 2x = π/6 + 2nπ and missing 2x = 5π/6 + 2nπ, or solving a logarithmic inequality without first imposing that the logarithm's argument is positive.
Inverse trigonometric functions are treated as single-valued, and domain restrictions are handled after algebra rather than before it.
What to do instead
Use the unit circle to write the complete general solution before dividing by 2. For logarithms, state the domain first, solve the inequality, then intersect the solution set with that domain.
Adding probabilities P(A)+P(B) for events that can occur together, or using nPr when the order of selection is irrelevant, such as choosing a committee.
The word 'or' triggers addition automatically, and counting formulas are chosen by keyword rather than by checking whether order matters.
What to do instead
Check disjointness before adding; if A and B can overlap, use inclusion-exclusion. For counting, decide whether swapping two chosen objects produces a different outcome, and verify with a small case.
Writing arg(z1 z2) = arg z1 + arg z2 without reducing modulo 2π, or using |z|^2 = z^2 for a complex number z instead of z times its conjugate.
Real-number intuitions are carried over to complex numbers, and the principal argument branch is ignored.
What to do instead
Keep arguments modulo 2π while multiplying or taking powers, then choose the principal value at the end. For moduli, use z\bar z = |z|^2 and never replace |z|^2 by z^2 unless z is real.
Applying the rational root theorem by listing only integer divisors of the constant term, forgetting possible fractions p/q with q dividing the leading coefficient, or making a sign error in the Euclidean algorithm remainder and losing the gcd.
The theorem is misremembered as an integer-root test, and division steps are done quickly without preserving the remainder condition.
What to do instead
List all p/q where p divides the constant term and q divides the leading coefficient, then test candidates systematically. In the Euclidean algorithm use remainders 0 ≤ r < |b| and verify the final gcd by back-substitution.
For similar solids, using the linear scale factor for volume, e.g. treating a doubling of all lengths as a doubling of volume rather than an eightfold increase, or confusing the incenter and circumcenter in coordinate geometry.
Length, area and volume ratios are mixed up, and triangle centers are identified by position rather than by their defining construction.
What to do instead
State the scale factor k explicitly and apply k for lengths, k^2 for areas and k^3 for volumes. For triangle centers, use the definition: angle bisectors for the incenter, perpendicular bisectors for the circumcenter.
Finding maxima or minima on a closed interval by solving f'(x)=0 only, ignoring endpoints and points where f is not differentiable.
The candidate equates extrema with stationary points and forgets that a continuous function on a closed interval can attain its extremes at boundary points.
What to do instead
Build the full candidate set: endpoints, stationary points, and points where f' is undefined. Evaluate f at every candidate and compare the values before stating the maximum and minimum.