Kyoto Entrance Mathematics study plan

Derive the derivative from the limit definition for at least six function families (polynomial, rational, radical, trigonometric, exponential, logarithmic) and write each derivation out from memory

Compute tangent lines, normal lines, and rates of change for 15 mixed functions, including composite, product, and quotient forms

Solve 10 maxima/minima word problems (volume, area, cost) and justify each answer with a sign chart or second-derivative test

Compute 12 indefinite integrals by substitution and by parts, then verify each by differentiating the result

Evaluate 8 definite integrals and set up 4 area problems between curves, identifying the correct upper and lower function before integrating

Set up and evaluate volumes of solids of revolution using both disk/washer and cylindrical shell methods for 6 problems, sketching each region first

Compute areas involving curves that cross the x-axis, splitting the integral at intersection points

Handle integrals of trigonometric functions using identities, and integrate rational functions by partial fractions in 5 problems

Solve 5 motion problems (velocity, displacement vs. distance traveled) and 3 optimization problems that require calculus plus algebra

State the Fundamental Theorem of Calculus precisely and explain in writing why the two parts are equivalent

Solve 15 sequence problems distinguishing arithmetic, geometric, and mixed/recurrence-defined sequences, including sums of the first n terms and infinite geometric sums with convergence conditions

Write complete induction proofs for 6 statements (summation formulas, divisibility, inequalities), stating base case, hypothesis, and step explicitly

Perform vector addition, scalar multiplication, dot product, and magnitude calculations in 2D and 3D, and use the dot product to find angles and perpendicularity conditions

Use position vectors to solve 6 geometry problems (collinearity, section points, triangle area via cross product magnitude in 3D)

Set up 4 recurrence relations from word problems and solve them by finding an explicit closed form

Solve 8 plane-vector geometry problems combining vectors with equations of lines and circles, including the condition for a point to lie on a line segment

Represent complex numbers in rectangular, polar, and exponential form and convert fluently among all three for 12 numbers

Compute products, quotients, powers, and roots of complex numbers using De Moivre's theorem, and plot all n-th roots on the complex plane

Solve 6 equations over the complex numbers (quadratic with negative discriminant, polynomial factorization) and interpret solutions geometrically

Use complex numbers to solve 3 locus/rotation problems (rotation about a point, perpendicular bisectors, circles)

Solve 15 counting problems, choosing correctly among permutations, combinations, and cases with repetition or grouping restrictions

Compute probabilities for 10 problems using complementary events, conditional probability, and independence, and verify with a sample-space listing for small cases

Calculate expected values for 8 random variables, including games and decision problems, and interpret the result in context

Apply the Euclidean algorithm to find gcd and lcm of 8 integer pairs, and solve 6 linear Diophantine equations using the extended algorithm

Solve 10 congruence problems, including linear congruences and systems solved by the Chinese remainder theorem

Solve 15 problems on quadratic functions: completing the square, vertex form, discriminant-based condition counting, and max/min with restricted domains

Solve 10 quadratic inequalities and 5 problems mixing quadratics with absolute values or parameters

Derive and apply all addition, double-angle, and half-angle trigonometric identities in 12 simplification and equation problems

Solve 8 trigonometric equations on a specified interval, listing all solutions without missing periodic families

Convert freely among sine law, cosine law, and area formulas in 8 triangle problems, including ambiguous cases

Solve 12 exponential and logarithmic equations and inequalities, tracking domain restrictions and extraneous solutions

Graph exponential and logarithmic functions with correct asymptotes and transformations, and solve 5 problems on maximum/minimum of log expressions

Work 8 problems converting between exponential and logarithmic forms, including applications such as growth/decay and pH-style scales

Review sets, logic, and proof language: negate 10 quantified statements correctly and evaluate 8 valid/invalid argument forms

Rework every problem missed in weeks 1-6 that touches domains 1-4, and classify each error as conceptual, procedural, or careless

Complete two full-length timed mock exams under exam conditions, then score them strictly and record time spent per section

Re-solve every missed mock problem from scratch without looking at the solution, then compare the two attempts in writing

Drill the three weakest domains identified by mock scores with 10 targeted problems each

Memorize and self-test a one-page sheet of formulas across all 11 domains, writing it out from memory twice

Finalize an exam-day time-allocation plan per section based on measured mock timing