My account Contact me

CEGEP • Montreal

Differential Calculus for Social Science, 201-103

Differential Calculus 201-103, often listed as Calculus I in English-language colleges, is the calculus course of the Social Science profile with mathematics. It follows the same progression as the Science program's calculus course, from limits to optimization, but it ends on economics rather than physics. Each chapter has its own set, written on the Social Science course framework and not on that of 201-SN2-RE (formerly 201-NYA), all the way to marginal analysis and elasticity, which close the course.

12 fully corrected exercise sets (142 exercises), 12 revision sheets, and a practice paper. Free, no account needed. The same course in French.

Functions, limits and continuity

  1. 1 Functions and their domain One calculation exercise first: twelve domains to find directly, from a rational function to a piecewise-defined function; then the vertical line test on an unemployment model, the only three restrictions on an algebraic domain, a tiered plan and its point of discontinuity, the domain that shrinks when two functions are divided, the composition of two economic models. Then an exam-level part: demand and its inverse, the three families of linear, exponential and logistic models with their asymptotes, five statements to correct, a child care fee schedule that is progressive in dollars and regressive as a percentage, and a library report to put into equations. Sheet
  2. 2 Limits and continuity One calculation exercise first: twelve limits to compute directly, from substitution to the form infinity minus infinity; then reading one-sided limits from a graph with a hole and a jump, resolving a zero over zero form by factoring, expanding and the conjugate, the dominant term and horizontal asymptotes, two budgets whose absolute gap explodes while the relative gap vanishes, and the recovery rate that makes a benefit schedule continuous. Then an exam-level part: the Intermediate Value Theorem applied to a population threshold, the vertical asymptote of a service counter's waiting time, five statements to correct, the average cost of a survey between its blow-up near zero and its floor at infinity, and a city report that mistakes increasing for unbounded. Sheet

The derivative and its rules

  1. 3 The derivative and differentiation rules Two calculation exercises first: twelve derivatives to compute directly with the formulas and ten tangent lines to write; then the average rate that becomes instantaneous and the hole at h equals zero, the power, product and quotient rules with their two sign traps, the chain rule on a chain of models, the tangent line and the units of a first and second derivative, and the difference between a rise that slows down and a decline. Then an exam-level part: the three ways to lose a derivative, including the corner of a tax bracket schedule, the relative rate of change in percent per year, five statements to correct, a graduation rate fitted from a table of data, and a fundraising campaign in which donors and average gift move in opposite directions. Sheet
  2. 4 Exponential and logarithmic functions One calculation exercise: twelve derivatives to compute directly, from an exponential with any base to a power of a logarithm; then the derivative of the exponential and the factor the base leaves behind, the logarithm whose domain is read before differentiating, the chain rule on nested models, the constant relative rate that signs an exponential, and the doubling time with the economists' rule of 70. Then an exam-level part: the semilog scale where an exponential becomes a straight line, logarithmic differentiation and relative rates that add, five statements to correct, compound interest up to continuous compounding and its effective rate, and Ebbinghaus's forgetting curve. Sheet
  3. 5 Implicit differentiation One calculation exercise first: ten implicit derivatives to compute directly, from the circle to logarithmic differentiation; then the full technique on equilibrium constraints: a labour supply curve that bends backward in which the wage is not a function of hours, the exact spot where the factor everyone forgets is born, the implicit route set against the explicit route when both are possible, the marginal rate of transformation on a production possibilities frontier and logarithmic differentiation on a production function. Then an exam-level part: the marginal rate of substitution proved and then computed at four points, three families of preferences and three behaviours of that rate, five statements to correct, the sacrifice ratio of a Phillips curve, and the consumer's choice, wrapped up. Sheet

Applications of the derivative

  1. 6 Related rates One calculation exercise: ten relations to differentiate directly with respect to time, from the weighted sum to the relative rate of a product of powers; then the five-step method on the processing delay of a public service, the error of substituting before differentiating on an average salary that falls while the payroll rises, a product whose price falls while its volume rises, a stock and its demographic flows, and a three-link chain that runs from investment to waiting time. Then an exam-level part: the rule for relative rates proved, a capped market share whose contribution reverses, five statements to correct, the density of a sprawling city, and an unemployment rate that falls without a single job having been created. Sheet
  2. 7 Differentials and linear approximation One calculation exercise: ten differentials and estimates to compute directly, from the square root to relative errors; then the differential compared with the actual change on a program budget, the linearization that predicts without recomputing, relative errors that add up in a product as in a quotient, the sensitivity of an investment to the interest rate, and the case where linear approximation collapses near an asymptote. Then an exam-level part: the margin of error of an unemployment rate and what a report is allowed to comment on, the inverse problem of the precision to demand, five statements to correct, the price index against the nominal wage, and the choice of the variable on which to focus the measurement effort. Sheet
  3. 8 Curve sketching One calculation exercise first: ten functions whose critical numbers and inflection points are found directly, from the parabola to the function with a root; then the domain and the three types of asymptotes, the oblique one included, the complete variation table of an infrastructure cost per dwelling, the first derivative test on a cubic indicator, concavity with the case where the second derivative test falls silent, and the study of a logistic adoption model. Then an exam-level part: going back from a published derivative to the function, the three kinds of candidates for an absolute extremum, five statements to correct, the complete study of a tax revenue curve, and the critique of a variation table published in a report. Sheet
  4. 9 Applied optimization Splitting a budget between prevention and treatment, two opposing costs that give an optimal number of service points, the sample size that minimizes the total cost of a survey, the optimal length of a training program, and the case where a regulatory cap pushes the optimum to the endpoint. Then an exam-level part: an integer optimum where rounding gets it wrong, return per dollar versus net effect, five statements to correct, the number of places in a subsidized childcare service, and the trade-off between coverage and depth of a program. Sheet

Trigonometric functions

  1. 10 Trigonometric functions and their derivatives One calculation exercise: ten derivatives of sine, cosine and tangent to compute directly, from the composite function to the product of two trigonometric functions; then the circle, radians and identities as a refresher, the limit of sine over x, an indeterminate form of type 0 over 0 from which the whole table comes, the four parameters of a periodic model read off twelve months of real sales data, and the derivatives of sine, cosine and tangent established and then checked with the double-angle identity. Then an exam-level part: the chain rule on a seasonal model and the factor everyone forgets, the month of fastest growth that is never the month of the peak, five statements to correct, emergency room traffic over seven days with its sign chart, and two superimposed cycles reduced to one by the angle addition identity. Sheet

Calculus in economics

  1. 11 Marginal analysis One calculation exercise: ten marginal quantities to compute directly, from marginal cost to marginal revenue equal to marginal cost; then fixed cost, variable cost and marginal cost as a derivative, the demand function and the marginal revenue that falls twice as fast as the price, the rule that marginal revenue equals marginal cost, the minimum average cost where it crosses marginal cost, the derivative as an approximation of the next unit. Then an exam-level part: the two break-even points by a sign study, diminishing returns and the inflection point, five statements to correct, a student co-op worked through from start to finish, and a management article in which every calculation is wrong. Sheet
  2. 12 Elasticity and economic optimization One calculation exercise: ten elasticities to compute directly, from the linear demand to the arc elasticity; then the price elasticity of demand and its point-by-point calculation, the rule that links elasticity and total revenue, income and cross-price elasticities with their classifications of goods, constrained optimization in four steps, and related rates applied to cost and profit. Then an exam-level part: the proof that a profit optimum always lies in the elastic range, a unit tax with its sharing and its maximum revenue, five statements to correct, the pricing decision of a cafeteria worked out scenario by scenario, and a tax-analysis memo to rewrite. Sheet

When the chapters are done

A full paper under exam conditions, on the whole course. Do it once every chapter is behind you: it measures what you can do on a question you have never seen.

Practice exam

Practice final exam, 201-103

Three hours and ten exercises on the whole course: four limits and continuity by extension, the product, quotient and chain rules, exponential growth and the logarithm as diminishing returns, the full study of a cubic with its endpoints, optimizing a container under a volume constraint. Then marginal analysis carried through to average cost, elasticity and the comparison of the revenue-maximizing and profit-maximizing prices, related rates, five statements to correct, and a management committee report that mistakes average cost for marginal cost.

  • 10 questions
  • 100 points
  • 180 minutes
Sit the paper

Need support and not only exercises? CEGEP tutoring in Montreal

Looking for a tutor in Montreal?

Get in touch for a first session. We work on exercises set at the real level of your CEGEP's exams, in English or in French.

Contact me WhatsApp
Site by Studio Squalli