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McGill University • Montreal

MATH 140 Tutor in Montreal, McGill Calculus 1

MATH 140 is examined without a calculator, and its marks go to the reasoning as much as to the result. The indeterminate form is checked before L’Hospital’s rule is applied, the hypotheses of the Mean Value Theorem are verified before it is invoked, the inner function of a chain rule is named, and a maximum is argued rather than assumed. Most calculus you find by searching ends on a decimal and skips exactly those steps. Everything on this page is written for MATH 140 as it is actually examined.

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, graphs and transformations Ten exercises on the review chapter that opens MATH 140, built on one idea: read a formula from the inside out. Part A is the toolkit: domain and range from a graph and from a formula, piecewise functions and absolute value with the candidate checked against its piece, even and odd functions decided by computation, transformations of a graph point by point, and completing the square to read a transformed parabola, root or hyperbola. Part B works at final exam level: composition and its domain, linear, power and rational models, five statements to correct, income tax by brackets, and a coupon, a discount and free delivery composed. No calculator anywhere, as the course requires. Sheet
  2. 2 Exponential, inverse, log and inverse trig functions Ten exercises on the second chapter of MATH 140, built on one idea: an inverse undoes a function only where it is one-to-one, and every answer must be checked against the domain it came from. Part A is the toolkit: the laws of exponents and an exponential read from two points, one-to-one functions and inverses with their domains, the laws of logarithms and the domain they hide, exponential and logarithmic equations with their rejected candidates, and the inverse trigonometric functions with the triangle. Part B works at final exam level: an inverse from start to finish, arcsin of sin x and the second solution, five statements to correct, radiocarbon dating and the pH scale. No calculator anywhere, as the course requires. Sheet
  3. 3 Limits and the limit laws Ten exercises on the first limit chapter of MATH 140, built on one idea: a limit describes what a function does near a number, never at it, and every limit law needs its permit before it is used. Part A is the toolkit: limits read on a graph, the limit laws and their hypotheses, the form 0 over 0 by factoring, conjugates and common denominators, absolute values and the floor function. Part B works at final exam level: the squeeze theorem, the fundamental limit of sin(x)/x used by rewriting, five statements to correct, constants chosen so that a limit exists, and a function that cannot be drawn. No calculator anywhere, as the course requires. Sheet
  4. 4 Continuity and the Intermediate Value Theorem Ten exercises on section 2.5 of MATH 140, built on one idea: continuity is checked, never assumed. Part A is the toolkit: the three conditions read on a graph, discontinuities located and classified from a formula, the continuity theorems for sums, quotients and compositions, piecewise functions glued with one parameter, two parameters or none possible, and the Intermediate Value Theorem written with its hypotheses. Part B works at final exam level: what the theorem does not say and the sign table it justifies, continuous extensions, five statements to correct, the hiker who climbs on Saturday and comes down on Sunday, and fixed points with the temperature on a ring. No calculator anywhere, as the course requires. Sheet
  5. 5 Limits at infinity and asymptotes Ten exercises on infinite limits, limits at infinity and asymptotes in MATH 140, built on one idea: infinity is not a number you substitute, every limit is decided by a sign and a dominant term. Part A is the toolkit: the sign on each side of a vertical asymptote, and the hole that is not one; rational functions at infinity; the square root of x squared, which is the absolute value of x; exponentials, logarithms and arctangent at the ends of the axis; oblique asymptotes by long division. Part B works at final exam level: asymptotes given and parameters to find, a sign chart turned into limits, five statements to correct, a brine tank and two skydivers approaching terminal velocity. No calculator, no L'Hospital's rule. Sheet

The derivative and its rules

  1. 6 The derivative as a limit Ten exercises on the definition of the derivative in MATH 140, built on one idea: the slope of a tangent is a limit of slopes of secants, a zero over zero form that must be rewritten before h goes to zero. Part A is the toolkit: secants closing in on a tangent, the derivative at a point in both forms of the definition, the derivative as a function with its domain, limits recognized as derivatives, and the four ways a function fails to be differentiable. Part B works at exam level: sketching f' from the graph of f, estimating a rate from data and saying what it means, five statements to correct, a stone dropped from a bridge, and marginal cost in words. No calculator anywhere, as the course requires. Sheet
  2. 7 Differentiation rules Ten exercises on the first differentiation rules of MATH 140, built on one idea: every rule applies to a form, so the work happens before differentiating and after it. Part A is the toolkit: the power rule after rewriting roots and fractions, the product and quotient rules never applied factor by factor, choosing between rewriting and the quotient rule, tangent and normal lines, and tangents through a point off the curve. Part B works at final exam level: higher derivatives and induction, derivatives from a table and from a graph, five statements to correct, a particle whose acceleration changes sign, and a complete study of the tangents to e to the x over x. No calculator anywhere, as the course requires. Sheet
  3. 8 Derivatives of trigonometric functions Ten exercises on the trigonometric derivatives of MATH 140, built on one idea: every formula of the chapter is borrowed from the limit of sin t over t, and inherits its conditions, radians, a variable going to zero, the same quantity under the sine and in the denominator. Part A proves that limit from the unit circle, reduces harder limits to it, derives the six derivatives, applies the product and quotient rules with the Pythagorean identity, and reads derivatives of order 99 off the cycle of four. Part B works at final exam level: horizontal tangents without dividing by sine, tangent and normal lines, five statements to correct, a mass on a spring whose true amplitude comes from the derivative, and a complete study question. No calculator, no chain rule. Sheet
  4. 9 The chain rule Ten exercises on the chain rule in MATH 140, built on one gesture: name the inner function, differentiate the outer one at it, untouched, and multiply by the inner derivative, one factor per layer. Part A is the toolkit: one layer, three layers, the rule read from a table and from a graph with a corner, exponentials in any base, and powers and roots of a quotient in factored form. Part B works at final exam level: second derivatives of composites, tangent lines, five statements to correct, a weekly cost that depends on production that depends on time, and unit conversions from kelvins to degrees. No calculator anywhere, as the course requires. Sheet
  5. 10 Implicit differentiation Ten exercises on implicit differentiation in MATH 140, built on one idea: y is a function you cannot see, so every y-term leaves a factor dy/dx and every slope belongs to a point, not to an x. Part A is the toolkit: the factor dy/dx and the product rule on xy, the folium of Descartes cut three times by one vertical line, horizontal and vertical tangents and the 0/0 that decides nothing, the second derivative simplified by the equation of the curve, and the derivatives of arcsin, arccos and arctan proved from their ranges. Part B works at final exam level: the derivative of an inverse function without its formula, curves that cross at right angles, five statements to correct, the astroid and its tangent of constant length, and a lemniscate with every kind of tangent. No calculator, exact answers throughout. Sheet
  6. 11 Logarithmic differentiation and growth Ten exercises on the derivatives of logarithms and on exponential growth in MATH 140, built on one idea: the logarithm turns products into sums and powers into products, so rewrite with ln before differentiating, and read k as a relative rate. Part A is the toolkit: the derivative of ln with its domain, the laws of logarithms applied first, logarithmic differentiation of heavy quotients, x to the x and its relatives, and the number e read as a derivative. Part B works at final exam level: bacterial growth, radioactive decay and its tangents, five statements to correct, Newton's law of cooling, and continuous compounding with inflation. No calculator anywhere, as the course requires. Sheet
  7. 12 Related rates Ten exercises on related rates in MATH 140, built on one gesture: write the relation that holds at every instant, differentiate it with respect to time, and only then substitute the numbers of the instant. Part A is the toolkit, each with its figure: the sliding ladder, the balloon and the tank, the inverted cone solved by similar triangles, the shadow under a lamp and the rocket tracked by a camera. Part B works at final exam level: two cars whose distance stops changing, a lighthouse beam racing along the shore, five statements to correct, a kite that loses height and a pool with a sloped bottom. Every answer is exact, with its sign and its unit, and no calculator is used. Sheet
  8. 13 Linear approximation and differentials Ten exercises on section 3.10 of Stewart, built on one idea: the tangent line is exact at its point and wrong everywhere else, so an estimate needs its centre, its factor (x - a) and its side. Part A is the toolkit: the square root at 4, cube roots and reciprocals, the small-x rules and the right centre for arctan, dy against Delta y on a figure, and the side of the error read on f'' alone. Part B works at final exam level: the propagated error on the volume of a sphere, which measurement limits g in a pendulum experiment, five statements to correct, the paint on a cube as a differential, and the square root of 26 trapped between a tangent and a chord. No calculator anywhere, as the course requires. Sheet

Applications of the derivative

  1. 14 Maximum and minimum values Ten exercises on section 4.1 of MATH 140, built on one idea: Fermat nominates the candidates, the table of values decides. Part A is the toolkit: absolute and local extrema read on a graph with its endpoints, critical numbers including corners, cusps and points outside the domain, the Extreme Value Theorem and what happens when a hypothesis fails, Fermat's theorem and its false converse, and the closed interval method with ties, rejected candidates and a hidden discontinuity. Part B works at final exam level: exact comparisons without a calculator, proofs with Fermat and the EVT, five statements to correct, the concentration of a drug in the blood, and a piecewise function whose minimum is a corner. Sheet
  2. 15 Rolle's theorem and the Mean Value Theorem Ten exercises on Rolle's theorem and the Mean Value Theorem in MATH 140, built on one idea: the theorem promises a number c only once every hypothesis is checked on the right interval, and its power lies in what it rules out. Part A is the toolkit: the three hypotheses of Rolle and the functions that break them, finding c and rejecting the values outside the interval, zero derivative and the domain that is not an interval, inequalities from a bound on the derivative, and exactly one root by the IVT plus Rolle. Part B works at final exam level: slopes forced by a table of values, the proof from Rolle, five statements to correct, average speed cameras, and a quintic with exactly three roots. No calculator anywhere, as the course requires. Sheet
  3. 16 Shape of a graph Ten exercises on section 4.3 of MATH 140, built on one idea: a zero of the derivative only nominates a candidate, and the sign on each side decides. Part A is the toolkit: the First Derivative Test with stationary points and cusps, concavity and inflection points with the candidates that fail, the Second Derivative Test and its silent case, and the two readings of a graph, f from the graph of f', and f' and f'' from the graph of f. Part B works at final exam level: a cubic rebuilt from its extrema, inequalities proved by monotonicity up to e to the pi against pi to the e, five statements to correct, the point of diminishing returns, and the peak of an epidemic. No calculator anywhere, as the course requires. Sheet
  4. 17 L'Hospital's rule Ten exercises on indeterminate forms and L'Hospital's rule in MATH 140, built on one idea: the rule is a permit checked at every round. Part A is the toolkit: the form 0/0 and its tangent lines, infinity over infinity and the growth ranking, products and differences rewritten as quotients, repeated rounds with the form written each time, 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: the cases where the rule loops, makes things worse or says nothing, constants chosen so that a limit exists, five statements to correct, compound interest as the frequency grows, and one function studied through every kind of limit. No calculator anywhere, as the course requires. Sheet
  5. 18 Curve sketching Ten complete curve sketches for MATH 140, built on one idea: a graph is a synthesis, and every feature of it comes from a computed sign or limit, gathered in one sign table written on the domain. Part A runs the full checklist on five families: a rational function with two vertical asymptotes, a square root with two different horizontal asymptotes, a fractional power with a cusp, the logarithm over x, and three graphs to match with f, its derivative and its second derivative. Part B reads a sign table backwards, studies a trigonometric function on one period, corrects five statements, and closes with two final exam problems: a slant asymptote crossed at the origin, and the number of solutions of an exponential equation read on a graph. Every sketch is in the solution, and no calculator is used anywhere. Sheet
  6. 19 Optimization problems Ten exercises on optimization in MATH 140, built on one idea: a critical number is only a candidate, and the global maximum is proved by the domain. Part A sets up the classics with their figures: a divided pasture along a river, the open box cut from a sheet, the can of fixed volume on an open domain, the closest point of a curve through the squared distance, and the rectangle inscribed in a semicircle. Part B works at final exam level: the Norman window with stained glass, the fastest row-then-walk route, five statements to correct, ticket pricing with a sold-out hall and a profit, and the largest cylinder inside a cone. No calculator anywhere: every answer is exact. Sheet
  7. 20 Antiderivatives Ten exercises on the last chapter of MATH 140, built on two gestures: rewrite before reading the table backwards, then check every answer by differentiating it, and count the constants, one per antidifferentiation and one per interval. Part A is the toolkit: the general antiderivative, products and quotients rewritten first, trigonometric identities, initial value problems up to the second derivative, and the domain with a gap. Part B works at final exam level: sketching an antiderivative from the graph of f, a family of curves and the conditions that pick one member, five statements to correct, a stone thrown up from a bridge, and a braking car whose deceleration builds up. No calculator anywhere, as the course requires. Sheet

When the chapters are done

Full papers under exam conditions, without a calculator. None of their questions repeats one already solved in the chapter sets above: the situations are new, so they measure what you can do rather than what you remember reading.

Practice exam

Practice midterm, MATH 140 at McGill

A ninety minute practice midterm for MATH 140, eight questions and one hundred points, on the first ten chapters of the course, with the full solution under every question. Part A asks four short questions: domains, an inverse and a logarithmic equation, four limits, two constants for continuity and the Intermediate Value Theorem, and every asymptote of two functions. Part B asks four long problems: derivatives from the definition, the product, quotient and trigonometric rules, the chain rule on three layers, and implicit differentiation on a cissoid. No calculator, every answer exact.

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

Practice exam

Practice final exam, MATH 140 at McGill

A three hour practice final for MATH 140, twelve questions and one hundred points on the whole course, weighted like a real cumulative final, with the full solution under every question. Part A weighs the limits: four limits opened by algebra, a derivative read as a limit and a seam made smooth, and four forms for L'Hospital's rule. Part B weighs the rules: three derivatives to simplify, an implicit curve with its vertical tangent, an inverse function, and two logarithmic derivatives. Part C, half the paper, weighs the applications: a trough filling up, tan 46 degrees and an artery that narrows, three theorems with their hypotheses, the complete study of x minus twice its arctangent, a gutter folded at the best angle, and three initial value problems. No calculator, every answer exact.

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

What MATH 140 covers

The official description fits on one line: review of functions and graphs, limits, continuity, derivative, differentiation of elementary functions, antidifferentiation, applications. It is a three-credit course whose only prerequisite is high school calculus, and it is closed to anyone holding the CEGEP differential calculus objective, 00UN, or credit for MATH 139 or MATH 150. That restriction tells you who is in the room: students whose high school calculus ranges from solid to a few weeks of rules learned by heart.

What the line hides is where the difficulty sits. Differentiating is quickly mechanical; what is not mechanical is everything around it. Limits reward algebra done carefully, the definition of the derivative rewards a limit written out in full, related rates and optimization reward a set-up drawn before anything is computed, and the theorems reward hypotheses checked on the page. Students who prepare by drilling derivatives lose their marks in those four places.

The chapters above follow the order of the course and of its textbook, Stewart’s Calculus: Early Transcendentals, chapters 1 to 4. Each one 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. The course that follows is MATH 141, Calculus 2, covered the same way.

Where MATH 140 marks are actually lost

Writing "= 0/0" and moving on

An indeterminate form is a signal to do algebra, factor, multiply by the conjugate, put everything over a common denominator, not a value. And L’Hospital’s rule applied to a limit that was never indeterminate returns a confident, wrong number. Write the form you found before choosing what to do with it.

Losing a layer of the chain rule

Most differentiation errors on a final are not errors of rule but of bookkeeping: an inner derivative forgotten on the third layer of a composition, or a product rule applied where a chain rule was needed. Naming the inner function before differentiating costs one line and saves the question.

Substituting before differentiating

In a related rates problem, the quantities change with time: a ladder’s height at the moment it reaches 3 m is not a constant. Plug in the instant’s values after differentiating the relation, never before, or the rate you are looking for disappears from the equation.

Invoking a theorem without its hypotheses

The Intermediate Value Theorem needs continuity on a closed interval, the Mean Value Theorem needs differentiability on the open one, and an optimization answer needs an argument that the critical point is a global maximum on the whole domain. On these questions, the hypotheses written on the page are what the marker is grading.

Frequently asked questions

What is MATH 140 at McGill?

MATH 140, Calculus 1, is McGill University’s three-credit first course in differential calculus. The official description reads: review of functions and graphs, limits, continuity, derivative, differentiation of elementary functions, antidifferentiation, applications. In practice that means inverse, exponential and logarithmic functions, limits and asymptotes, the derivative from its definition, the differentiation rules and implicit differentiation, related rates and linear approximation, then extreme values, the Mean Value Theorem, L’Hospital’s rule, curve sketching, optimization and antiderivatives.

I took Calculus 1 in CEGEP. Do I need MATH 140?

Usually not. MATH 140 is closed to students who have CEGEP objective 00UN or its equivalent, which is the CEGEP differential calculus course, and to students who have taken MATH 139 or MATH 150. The students in the room are therefore mostly first-year students from outside Quebec, international students and U0 students, with a high school calculus background that varies a great deal from one student to the next.

Can I use a calculator on MATH 140 exams?

Assume you cannot: McGill calculus midterms and finals have typically been written without calculators, so check the rule on your own course outline. That is why every answer on this page is exact, written with fractions, square roots, pi and logarithms rather than decimals, and why every solution names the rule or the theorem it uses. A method the marker can follow earns the marks even when an arithmetic slip spoils the final number.

Which part of MATH 140 is the hardest?

Two places, for different reasons. Related rates and optimization, because the calculus is easy and the set-up is not: most wrong answers come from an equation that was never the right one. And the theorems, the Intermediate Value Theorem, the Mean Value Theorem and L’Hospital’s rule, because each has hypotheses that must be checked on the page before it is invoked, and a correct conclusion without that check earns very little.

Do you offer MATH 140 tutoring in Montreal?

Yes. I tutor MATH 140 and the other first-year McGill math courses in Montreal, in person and online. I am a McGill graduate and I work on your own course outline, WeBWorK assignments and past midterms 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 141, Calculus 2 (McGill)MATH 203, Calculus I (Concordia) Math tutoring in Montreal All revision sheets in English Higher education, in French
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