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Concordia University • Montreal

MATH 203 Tutor in Montreal, Concordia Calculus I

In MATH 203, the calculus is rarely where the marks go. They go in the algebra around it: a negative exponent rewritten wrongly before the power rule, a complex fraction left unsimplified in the definition of the derivative, a common factor never pulled out after the chain rule, an identity nobody remembered. The calculator the department allows does none of that for you. Everything on this page is written for MATH 203 as it is taught and examined at Concordia, with that algebra named where it costs.

20 chapters, 220 corrected exercises and 20 revision sheets, plus 2 practice papers. Free, no account needed: the full worked solution is on the page.

Functions and limits

  1. 1 Functions, trigonometric and exponential functions Ten corrected exercises on the review chapter that opens MATH 203, Thomas 1.2, 1.3 and 1.5, built on one habit: rewrite first, read after. Part A: domains of sums, quotients and compositions, shifts and stretches in the right order, radians and the six trigonometric functions, identities and equations on an interval, the general sine function read from a graph. Part B: exponent laws with rational and negative powers, periods of combined functions, five statements to correct, a tide modelled by a sinusoid, and fuel economy as a composition of unit conversions. Every exercise checks your answers as you type them. Sheet
  2. 2 Inverse functions and logarithms Ten corrected exercises on section 1.6 of Thomas for MATH 203 at Concordia, built on one gesture: when the unknown sits in two places, collect it on one side and factor it out, then check the answer against a domain. Part A is the toolkit: one-to-one functions and the horizontal line test, finding inverses of rational, radical, logarithmic and exponential functions, the graph of an inverse and restricted domains, the laws of logarithms and the fake ones, and the cancellation laws with their domains. Part B works at final exam level: a complete inverse with a rejected root, equations with and without extraneous candidates, five statements to correct, the half-life of a medical tracer, decibels and the Richter scale. Sheet
  3. 3 Rates of change and limit laws Ten corrected exercises on the first limit chapter of MATH 203, built on one idea: the secant is the only slope you can compute, and the tangent is what the secants approach. Part A is the toolkit: average rates of change, tangent slopes estimated by secants on a calculator table, limits read on a graph and combined by the limit laws, the form 0 over 0 by factoring, conjugates, fractions inside fractions and negative exponents. Part B works at exam level: the Sandwich Theorem, a calculator table that lies, five statements to correct, a growing culture and a tool tossed up on the Moon. Sheet
  4. 4 One-sided limits Ten exercises on section 2.4 of Thomas, built on one idea: where the formula changes, each side is computed with its own formula. Part A is the toolkit: one-sided limits read on a graph with its endpoints, absolute values opened with the sign of their inside, floor and ceiling functions, the limit of sin over its angle made to match, and the identities that open trigonometric limits. Part B works at final exam level: two junctions and two constants, the angle as a root or a polynomial, five statements to correct, a parking tariff by the hour, and the sine over the fractional part. Sheet
  5. 5 Limits involving infinity and asymptotes Ten corrected exercises on limits involving infinity and asymptotes for MATH 203, built on one observation: the rules are short, and the marks are lost in the algebra that comes before them. Part A: dividing every term by the dominant power, with negative and fractional exponents; the square root of x squared, which is the absolute value of x; infinite limits read from a factored denominator; exponentials and logarithms at the ends of the axis; oblique asymptotes by long division. Part B: constants fixed by asymptotes, graphs built from limits, five statements to correct, average cost and a drug concentration in the long term. No L'Hôpital's Rule. Sheet
  6. 6 Continuity Ten corrected exercises on section 2.5 of Thomas for MATH 203, built on one idea: continuity is decided at a few suspect points, and there the algebra decides. Part A reads the three conditions on a graph, classifies discontinuities after a complete factorization, uses the continuity theorems for quotients and compositions, glues pieces with a conjugate, a complex fraction and a squared parameter, and follows the Intermediate Value Theorem with a bisection table in radians. Part B covers continuous extensions, what a table of values proves, five statements to correct, a room thermostat and two transcendental equations from physics. Sheet

The derivative and its rules

  1. 7 The derivative at a point and as a function Ten corrected exercises on the definition of the derivative in MATH 203, built around the algebra that costs the marks: the bracket around f of a plus h, the complex fraction put over one common denominator, the conjugate, the sign of a minus x. Part A drills the definition at a point and as a function, recognizes limits as derivatives and separates corners, cusps and smooth joins. Part B graphs f' from f, joins two pieces with two parameters, corrects five false statements, and reads the rates of an oven and a draining tank in words, with units. Sheet
  2. 8 Differentiation rules Ten corrected exercises on the differentiation rules of MATH 203, built on one observation: the rule takes one line, and the marks are lost in the algebra around it. Part A: the power rule after rewriting negative and fractional exponents, product and quotient rules followed by the simplification they need, tangent and normal lines, prescribed slopes and unknown constants, and derivatives from a table and a graph. Part B: higher derivatives, where the rules come from, five statements to correct, a camera dolly that stops without turning back, and marginal cost and revenue. Every exercise checks your answers as you type them. Sheet
  3. 9 Derivatives of trigonometric functions Ten corrected exercises on the trigonometric derivatives of MATH 203, built on one observation: the derivative takes one line, and the marks are lost in the algebra around it. Part A uses the six derivatives with the Pythagorean identity before differentiating, after the quotient rule and in complex fractions, then tangent and normal lines, the cycle of four and the equation y'' + y = 0. Part B solves f'(x) = 0 by factoring and reading every quadrant, matches secant, tangent and their derivatives on a graph, corrects five statements including the degree-mode calculator, follows a rider on a Ferris wheel, and ends with a complete final exam question on sec x + tan x. Sheet
  4. 10 The chain rule Ten corrected exercises on the chain rule in MATH 203, section 3.6 of Thomas, built on one thread: name the inside function, differentiate the outside one at it, multiply, and then finish the algebra. Part A: powers of a function rewritten with negative and fractional exponents, three and four layers, a table of values, exponentials in base a through the laws of exponents, and derivatives factored at the lowest power. Part B: second and higher derivatives, tangents to a logistic curve, five statements to correct, a radiosonde balloon in the standard atmosphere and the wind chill index during a storm. Sheet
  5. 11 Implicit differentiation Ten corrected exercises on implicit differentiation for MATH 203, built on one observation: after one line of calculus, what remains is a linear equation in dy/dx, and that is where marks are lost. Part A drills the collecting step and its sign, three slopes with three algebraic traps, horizontal and vertical tangents found by substituting a line into the curve, the second derivative and its complex fraction, and the power rule for rational exponents. Part B works at final exam level: a tangent that meets a cubic again at a double root, orthogonal families of circles, five statements to correct, a cardioid handled as a block, and a conic whose coefficients come from a horizontal tangent. Sheet
  6. 12 Derivatives of inverse functions and logarithms Ten corrected exercises on section 3.8 of Thomas for MATH 203, built on one idea: every derivative of the chapter comes from undoing, so the slope of an inverse is read at the swapped point and the laws of logarithms are applied before differentiating, the true ones only. Part A is the toolkit: the derivative of ln with its domain, expanding a logarithm first, the derivative of an inverse without its formula, other bases and their ln a, and logarithmic differentiation. Part B works at final exam level: a variable in the base and the exponent, e as a limit with a calculator table, five statements to correct, decibels and pH, and a car and a line of credit. Sheet
  7. 13 Inverse trigonometric functions Ten corrected exercises on arcsin, arccos, arctan and arcsec in MATH 203, built on one idea: an inverse trigonometric function returns ONE angle, the one in its range, and that range decides every sign. Part A covers exact values and domains, compositions and the reference triangle, derivatives with the chain rule, the implicit proof for arcsec and its absolute value, and square roots of squares. Part B works at final exam level: arcsec computed from the arctan key, horizontal tangents, five statements to correct, the viewing angle of a painting, and one function with two corners. Sheet

Applications of the derivative

  1. 14 Related rates Ten exercises on related rates in MATH 203, Thomas section 3.10, where the derivative is one line and the marks are lost in the algebra around it. Part A: a ladder whose top slides down, a slick and a melting snowball, a draining cone given by its diameter, a spotlight rolled across a stage, a radar read backwards. Part B: a car on a bridge above a train, resistors and gases without a picture, five statements to correct, a pulley that speeds up a load and a hot-air balloon passed by a cyclist. Every rate is signed, every unit checked, every decimal rounded as asked. Sheet
  2. 15 Linearization and differentials Ten exercises on section 3.11 of Thomas, built on one observation: the calculus of a linearization fits in one line, and the marks are lost in the algebra around it. Part A: the fourth root at 16 and its negative exponents, the right centre for e to the 0.7, tan 44 degrees, arcsin 0.52 and ln 2.7, (1 + x) to the k after factoring, differentials as simplified formulas, and dy against Delta y with Thomas' epsilon. Part B: a ball measured by its circumference, a pendulum clock whose rod expands, five statements to correct, a cylindrical tank with two measurements, and a curve linearized through implicit differentiation. Every estimate is checked against the calculator. Sheet
  3. 16 Indeterminate forms and L'Hôpital's Rule Ten corrected exercises on indeterminate forms and L'Hôpital's Rule in MATH 203, built on one observation: the rule is one line, and the marks are lost in the algebra around it. Part A is the toolkit: the form 0/0 and the complex fraction each round produces, infinity over infinity simplified between rounds, products and differences turned into quotients with the exponents flipped correctly, repeated rounds and the round too many, and the powers 1 to the infinity, 0 to the 0 and infinity to the 0 through the logarithm. Part B works at final exam level: a rule that loops, a calculator table destroyed by roundoff, growth rates that a table gets wrong, five statements to correct, the effective annual rate as compounding gets more frequent, and one function studied through every kind of limit. Sheet
  4. 17 Extreme values of functions Ten corrected exercises on section 4.1 of Thomas for MATH 203, built on one idea: the candidates hide in the algebra of the derivative. Part A is the toolkit: extrema read on a graph with Thomas's endpoint convention, critical points found by rewriting f prime as one factored fraction, the Extreme Value Theorem and what survives when a hypothesis fails, and the closed interval method on polynomials, quotients, roots and fractional exponents. Part B works at final exam level: trigonometric, exponential, logarithmic and inverse trigonometric functions compared on a calculator in radian mode, reasoning with Thomas's two theorems, five statements to correct, the temperature over a day, and a parameter inside the formula. Sheet
  5. 18 Monotonic functions and the First Derivative Test Ten corrected exercises on section 4.3 of Thomas for MATH 203 at Concordia: where a function increases and decreases, and what the First Derivative Test concludes. The thread is the algebra that produces a factored derivative, because that is where the marks go: the minus sign of the quotient rule, the lowest power factored out, a cos x that must never be divided away, the points outside the domain. Part A drills the sign chart on polynomials, quotients, fractional powers, a graph of f' with a jump and a trigonometric function on a closed interval. Part B counts solutions, proves inequalities, corrects five false statements, reads weekly sales after a launch and ends on a logarithmic family with a parameter. Sheet
  6. 19 Concavity and curve sketching Ten corrected exercises on concavity and curve sketching for MATH 203, section 4.4 of Thomas. Part A decides inflection points when the second derivative vanishes without a change of sign or changes sign without vanishing, runs the Second Derivative Test and its silent case, sketches a rational function and a square root, and matches the graphs of f, f prime and f double prime. Part B studies a fractional power with two cusps and an exponential with a flat inflection point, corrects five statements, and closes on two applications: the point of diminishing returns of a fertilizer and the inflection of an epidemic curve. Every solution shows the algebra that makes the sign of the second derivative readable. Sheet
  7. 20 Applied optimization Ten corrected exercises on applied optimization in MATH 203, built on one idea: the calculus is one line, and the marks are lost in the algebra around it. Part A poses the classics with their figures: a garden with two fence prices, a square-based box from a fixed amount of cardboard, a one-litre cup and a can with waste, the closest point of a parabola, and the rectangle in a right triangle. Part B works at final exam level: a Norman window of fixed area, a pipeline across a river, five statements to correct, the price of a poke bowl with an exponential demand, and the largest cone inside a sphere, with the height as the variable that keeps square roots away. Sheet

When the chapters are done

Full papers under exam conditions, with an approved scientific calculator and nothing else: the midterm on the first six lectures, up to implicit differentiation, and the three-hour final on the whole course. None of their questions repeats one already solved in the chapter sets above, so they measure what you can do rather than what you remember reading.

Practice exam

Practice midterm, MATH 203 at Concordia

A ninety minute practice midterm for MATH 203, eight questions and one hundred points, on the eleven chapters before the midterm, with the full solution under every question. Part A asks four short questions: a domain, an inverse and an exponential equation, three limits with two gestures each, every asymptote of two functions, a constant for continuity and the Intermediate Value Theorem with a bisection table. Part B asks four long problems: a derivative from the definition, the power, quotient and trigonometric rules, the product and chain rules with a short applied problem, and implicit differentiation on a cubic curve.

  • 8 questions
  • 100 points
  • 90 minutes
Sit the paper

Practice exam

Practice final, MATH 203 at Concordia

A three hour practice final for MATH 203, twelve questions and one hundred points on the whole course, with the full solution under every question. Part A weighs the techniques: four limits opened by algebra, three derivatives to simplify, a moving exponent and an inverse with no formula. Part B weighs the theorems: two seams and a bisection table, the derivative from its definition, an implicit curve with an exponential in it, a cube root on a closed interval and a complete study with two slant asymptotes. Part C, half the paper, weighs the applications: the hands of a clock, the power of a wind turbine, a wire cut into a square and a triangle, and a CES production function that tends to Cobb-Douglas.

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

What MATH 203 covers

MATH 203 is the first of Concordia’s two connected calculus courses, followed by MATH 205. The calendar lists an overview of functions and limits, the derivative as a rate of change, the derivatives of the elementary functions, the product, quotient and chain rules, implicit differentiation, higher derivatives, linearization and the differential, then related rates, optimization, and the analysis and graphing of functions. Its prerequisite is MATH 201, Elementary Functions, or an equivalent course.

The department’s outline says a great deal about who is in the room. Weekly tutorials review arithmetic and algebra, a pre-test on Moodle checks the background, and chapter 1 of the textbook is announced as review that a student who does not master it should take MATH 201 for first. The course is taken by students who did not do calculus in CEGEP, and that is exactly where they lose marks: not in the rules, which are learned in a week, but in the algebra every rule leads into.

The chapters above follow the department’s outline and its textbook, Thomas’ Calculus: Early Transcendentals, from section 1.2 to section 4.6. The Mean Value Theorem and antiderivatives are not on the outline and are left out. Each chapter has a corrected exercise set, with the full solution written out on the page, and a revision sheet that answers a different question: not what the chapter says, but what loses marks on it.

Where MATH 203 marks are actually lost

Differentiating before rewriting

A root in a denominator is a negative fractional power, and the power rule only applies once it has been written that way. Students who differentiate the expression as it stands invent a quotient rule they do not need, then lose the sign or the exponent inside it. Rewrite first, differentiate second.

Stopping at the complex fraction

The definition of the derivative almost always produces a fraction inside a fraction. The limit cannot be taken until it has been put over a common denominator and the h has cancelled, and that is pure algebra. The same step comes back in implicit differentiation, where the answer is only readable once it has been simplified.

Substituting before differentiating

In a related rates problem, the quantities change with time. Put the values of the instant in after differentiating the relation, never before, or the rate you are looking for disappears from the equation.

An extremum claimed, not justified

A critical point is a candidate, not an answer. On a closed interval the endpoints must be compared with it, and in an optimization problem the maximum must be argued on the whole domain, by the closed interval method or by the First Derivative Test. The marker is grading the argument as much as the value.

Frequently asked questions

What is MATH 203 at Concordia?

MATH 203, Differential and Integral Calculus I, is the first of Concordia University’s two connected calculus courses, followed by MATH 205. The calendar lists an overview of functions and limits, the derivative as a rate of change, the derivatives of power, exponential, logarithmic and trigonometric functions, the product, quotient and chain rules, implicit differentiation, higher derivatives, linearization and the differential, and as applications related rates, optimization and the analysis and graphing of functions. The textbook is Thomas’ Calculus: Early Transcendentals.

Who takes MATH 203?

Students who need differential calculus and did not take it in CEGEP: Extended Credit Program students in engineering and science, students from outside Quebec, mature students and students returning to school. Its prerequisite is MATH 201, Elementary Functions, or an equivalent functions course, and the department itself says that a student who does not know chapter 1 of the textbook well should consider taking MATH 201 first.

Can I use a calculator on MATH 203 exams?

Only a calculator approved by the Department of Mathematics and Statistics, with the sticker that proves it, and the department publishes the list. These are scientific calculators: they do not differentiate, find limits or draw graphs. So the calculator helps with the final number, and every mark before it still comes from the written work, which is why every solution on this page is written out step by step.

How is MATH 203 graded?

According to the department’s outline, 10% for the WeBWorK assignments, 30% for a 90-minute midterm test on the first six lectures, which ends with implicit differentiation, and 60% for a three-hour final on the whole course. If it gives a higher grade, the final counts for 90% instead. There is no make-up midterm and no 100% final option, and the practice papers on this page follow the same split.

Do you offer MATH 203 tutoring in Montreal?

Yes. I tutor MATH 203 and the other first-year Concordia and McGill math courses in Montreal, in person and online. I hold an M.Sc. from Concordia and I work on your own course outline, WeBWorK assignments and past exams rather than on a generic syllabus.

Other first-year courses

The same free corrected material exists for other courses, some of it in English and the rest in French.

COMP 202, Foundations of Programming (McGill)PHYS 101, Introductory Mechanics (McGill)MATH 133, Linear Algebra and Geometry (McGill)MATH 140, Calculus 1 (McGill)MATH 141, Calculus 2 (McGill) Math tutoring in Montreal All revision sheets in English Higher education, in French
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