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CEGEP • Montreal

Differential Calculus, 201-SN2-RE (formerly 201-NYA)

Differential Calculus, 201-SN2-RE in the revised program (formerly 201-NYA), is the first mathematics course of Sciences de la nature, and the one where many students stumble. The move from algebraic manipulation to reasoning about limits catches almost every student arriving from high school by surprise.

16 fully corrected exercise sets (280 exercises), 15 revision sheets, and 3 practice papers. Free, no account needed. The same course in French.

Functions and limits

  1. 1 Functions, domains and operations Two direct calculation exercises first, twelve domains to write in interval notation, then ten compositions and inverses to evaluate; then the opening chapter of differential calculus, the one the rest of the course depends on: the only three restrictions that build a domain, a root in the denominator and its strict inequality, a sum of two absolute values read as a sum of distances, the intersection of domains and the zeros of the denominator to remove, composition and its domain built in two steps, transformations of a graph and the order that makes them non-commutative, inverse functions and the swap of domain and range, the unit circle through reference angles, and two modelling problems where the physical domain decides whether the answer is valid. A Part C adds five classics: the domain by a sign chart, logarithm and exponential, inverse trigonometric functions, the spreading slick and the arched tunnel. Sheet
  2. 2 Limits and continuity Two calculation exercises first: limits to compute directly, twelve at a point and ten at infinity; then reading left-hand and right-hand limits from the graph of a function with a jump and a hole, indeterminate forms resolved by factoring and by the conjugate, limits at infinity with horizontal and oblique asymptotes and the trap of the TWO asymptotes of a square root, continuity of piecewise functions and the classification of discontinuities, the Intermediate Value Theorem with the bisection method, and trigonometric limits with the Squeeze Theorem. Also covered: the limit laws and their false converse, substitution in a limit and the counterexample where the composite fails, the formal epsilon-delta definition with two complete proofs, and the Extreme Value Theorem with each of its three hypotheses shown to be essential. A Part C adds five classics: parameters that make a function continuous, the floor function, limits of transcendental functions, the learning curve and the cost of pollution removal. Sheet
  3. 3 Continuity at a point: practice bank Two banks of functions to settle one at a time, is it continuous at a: six you can decide in your head, then eight with sine over x, the conjugate, the edge of the domain, e to the power one over x and the Squeeze Theorem. Then spotting the suspicious points and stating where a function is continuous, six holes to fill by continuous extension, five parameters to set, one of which admits two values and one none. Exam-level part: the sign of infinite one-sided limits and absolute values removed on the correct side, eight indeterminate forms and their techniques, five statements to correct, the joint of a playground slide where continuity is not enough to remove the corner, and a parking rate charged per hour or part of an hour, with its three-dollar jumps. Questions from a first CEGEP midterm are built in: a sum of cubes and polynomial division, absolute values removed on the correct side, a parameter that decides between continuity, a displaced value and a jump, a squared denominator to cancel twice, and the squeeze whose inequalities reverse on the left.

The derivative

  1. 4 The derivative and its rules Two calculation exercises first: ten derivatives from the difference quotient, at a point and then in general; then the secant line that turns into the tangent and the derivative computed from the definition (with a conjugate for the square root), the power, product and quotient rules proved and then applied to the tangent line, the chain rule from simple cases up to triple compositions, transcendental functions with the logarithmic differentiation of x to the power x, implicit differentiation on the circle and the folium of Descartes, and the three ways to lose a derivative: the corner, the cusp and the vertical tangent. Also covered: higher-order derivatives (the polynomial that stops, the exponential that amplifies, the sine that cycles), equations of tangent and normal lines including the two tangents drawn from an outside point, derivatives of inverse functions (inverse sine, inverse tangent, and an inverse with no explicit formula), and the proofs of the derivative of sine and of the quotient rule, with the trap of circular reasoning. A Part C adds five classics: the rules applied to a table of values, the common tangent, the numerical derivative, the path of a ball and the transition curve between two roads. Sheet
  2. 5 Derivatives of trigonometric, exponential and logarithmic functions One calculation exercise first: twelve derivatives to compute directly, from a composite cotangent to a quotient of secants; then the table of the six trigonometric derivatives rebuilt from just two, with the tangent function that never has a horizontal tangent line, then put to work on composite cotangents, secants and cosecants (written cotg and cosec in French-language Quebec textbooks), a radical rewritten as a power and the trap cos(x²) against cos²x; the exponential with any base and the logarithm factor that gets forgotten, illustrated by the three slopes at zero that define the number e; the logarithm and the domain that comes before the derivative, including the absolute value, the fake extremum of a function that does not exist there, and the trigonometric logarithms that simplify, like ln(sec x + tan x); three-layer compositions; logarithmic differentiation for x to the power x, for x to the power sin x and for a quotient of three factors, with the criterion that says when it is pointless. Part B proves the derivatives of the inverse trigonometric functions by implicit differentiation, explaining where the sign of the square root comes from and why arcsine has a vertical tangent at its endpoints, with the domain of arccos(3/x) and its hidden absolute value; it sets up the hyperbolic functions, the identity cosh squared minus sinh squared equals one, and the true shape of a hanging cable; it ends with carbon-14 dating and the difference between a relative rate and an absolute rate, and with the logistic model whose maximum growth rate is found without the second derivative. A Part C adds five classics: the tangent lines through the origin, the identities of the inverse functions, the extrema of e to the x times sin x, the tide, and the best viewing angle. Sheet
  3. 6 Implicit differentiation One calculation exercise first: eleven implicit derivatives to compute directly, from sums of cubes to trigonometric equations; then the principle set out on the circle, with the tangent perpendicular to the radius and a point-by-point comparison with the explicit method, which has to switch formulas from one half to the other; the five structures that leave a y prime, from a plain y cubed to the exponential of a product, then trigonometric functions of y and of x plus y, and the equation that fixes the product x squared y; the folium of Descartes, its horizontal and vertical tangents, and the symmetry that gives the second one without computation; reading the two zeros of a quotient, including the cusp where numerator and denominator vanish together; the implicit second derivative and the concavity of the circle. Part B proves that the hyperbolas xy equals k and the curves x squared minus y squared equals c always cross at right angles, establishes the point form of the tangent to an ellipse and to a hyperbola and explains why it survives at vertical tangents, then applies the method to the critical point of a van der Waals gas and to the speed of the image formed by a thin lens. A Part C adds five classics: the lemniscate, the curve x to the power y equals y to the power x, estimating with the tangent line, the production isoquant and the reflection property of the ellipse. Sheet

Applications of the derivative

  1. 7 Related rates One calculation exercise first: ten relations to differentiate directly with respect to time, from a polynomial to a tangent; then the chapter where the problem is a text and the difficulty is translating it: the ladder sliding down a wall, the spherical balloon and the error of substituting before differentiating, the cube whose volume rate is known, the diameter given instead of the radius and the cylindrical tank whose level rises at constant speed, the inverted cone where similar triangles eliminate a variable, the street lamp and the difference between the length of the shadow and the position of its tip, the lighthouse sweeping a coastline and why degrees are not allowed, two cars moving apart with Pythagoras then the law of cosines, the V-shaped trough that fills more and more slowly, and the plane whose angle of elevation decreases by thirty radians per hour. A Part C adds five classics: the rope at the dock, the shadow on a wall, the cylinder of constant volume, the piston and connecting rod, and the spherical tank. Sheet
  2. 8 Differentials and linear approximation One calculation exercise first: ten direct calculations, from the differential of a polynomial to estimating sin 0.5 with the tangent line at π/6; then the differential dy = f'(x) dx and how it differs from the actual change Δy, approximating a function by its tangent line while controlling the error, propagation of measurement errors in the lab and in the workshop, the way exponents amplify the relative error, and the two situations where the approximation stops being reliable. Also covered: the table of the five standard approximations with their range of validity (small angles hold up to fourteen degrees), estimating a change in a physics formula and the volume of a pipe wall as area times thickness, the tangent line compared with linear interpolation, which gives a free pair of bounds, and the inverse problem: how precise the instrument must be to guarantee the precision of the result. A Part C adds five classics: a special angle in radians, a height measured by sighting, decibels and pH, thermal expansion and the distance to the horizon. Sheet
  3. 9 Applications of the derivative The sliding ladder and the filling cone (related rates with the five-step method), the full analysis of a curve with both derivatives, optimizing the box and the can with the ratio h = 2r that industry ignores, the Mean Value Theorem and its proof that the average speed is reached as an instantaneous speed, L'Hôpital's rule with its disguised indeterminate forms, and Newton's method, which doubles its correct decimals at every step. Also covered: rectilinear motion, with speeding up and slowing down told apart by the sign of v times a, related rates for a shadow, an angle of elevation and two vehicles, the consequences of the Mean Value Theorem (zero derivative means constant, and two proved inequalities), and a complete synthesis problem on the concentration of a drug, from its maximum to the therapeutic window. A Part C adds five classics: the candidates for an absolute extremum, electric current, the lamp above the table, the Betz limit and the cough. Sheet
  4. 10 Rectilinear motion One calculation exercise first: ten positions to differentiate directly, from a quadratic polynomial to a logarithm; then position, velocity and acceleration of one particle followed from start to finish, with its two turnarounds and the moment it moves backward fastest; the sign chart of the velocity, which gives the direction of motion, and a position function that is not one-to-one and takes the same value three times; the difference between displacement and total distance travelled, with the forty percent error that skipping the splitting costs; the product of velocity and acceleration, the only test that tells whether the particle speeds up or slows down, proved from the derivative of the absolute value; free fall solved with the parabola, then checked without the time. Part B reads a motion off a trapezoidal velocity graph and deduces the apparent weight in an elevator, sets up simple harmonic motion and its phase shift, explains why braking distance grows with the square of the speed, and finally rebuilds the position from a measured acceleration, showing what a sensor can never tell you. A Part C adds five classics: an exponential motion, two runners who overtake each other, reading the position graph, Keller's sprinter and the train between two stations. Sheet
  5. 11 Rolle's theorem and the mean value theorem One calculation exercise first: ten values of c to compute directly, from a quadratic to the cosine; then Rolle's theorem with its three hypotheses separated one by one, and the three counterexamples showing that none of them is decorative, including the reciprocal of the square, which satisfies the most visible hypothesis and fails the one students forget; the mean value theorem, its interpretation in terms of motion, and the auxiliary function that derives it from Rolle's theorem by subtracting the secant line; the two consequences used all year long, a zero derivative means a constant function and two functions with the same derivative, with the trap of a domain in two pieces illustrated on arctan x plus arctan of 1 over x. Part B uses the mean value inequality to bound a square root and a logarithm with a guarantee no calculator display gives, separates existence by the Intermediate Value Theorem from uniqueness by Rolle's theorem, proves the legal validity of an average speed camera, and proves four classic inequalities without studying any difference function. A Part C adds five classics: counting roots with Rolle's theorem, Rolle's theorem applied twice, the addition formulas from a zero derivative, the temperature sensor and Darboux's theorem. Sheet
  6. 12 Bisection, Newton's method and fixed-point iteration One calculation exercise first: ten iterations to compute directly, from bisecting a polynomial to Newton's method applied to the derivative; then the bisection method, with its number of iterations known in advance and a guarantee no other method offers; Newton's method built as the equation of a tangent line solved for its x-intercept, the Babylonian formula recovered as an average, and quadratic convergence proved by an exact identity in which the number of correct decimals doubles at every step; the four ways Newton's method fails, division by zero, a two-value cycle, divergence on the cube root and degeneration into linear convergence at a multiple root; the comparison table and the hybrid strategy of numerical libraries. Part B covers the secant method and its golden-ratio order, shows on a single equation two fixed-point rewrites, one diverging and one converging in a spiral, explains why Newton's method is the choice that makes the derivative of the iteration vanish, then rebuilds the division and the cube root of a processor and computes the internal rate of return of an investment. A Part C adds five classics: Newton's method against fixed-point iteration, locating a minimum, scanning followed by bisection, the sag of a power line and Kepler's equation. Sheet
  7. 13 L'Hôpital's rule and indeterminate forms One calculation exercise first: twelve limits to compute directly with L'Hôpital's rule, from 0/0 forms to the form 0 to the power 0; then the chapter as differential calculus meets it, right after the table of derivatives and right before curve sketching. Sorting a limit into determinate, indeterminate or a derivative in disguise; L'Hôpital's rule with its hypotheses checked at every round, and the circular argument it hides; product and difference forms rewritten as a quotient; transcendental functions at work; the three exponential forms through the logarithm. Then the growth rates that decide the shape of a curve, oblique asymptotes of transcendental functions, five statements to correct, the continuous extension pushed as far as differentiability, and falling with air resistance, whose limit without resistance gives back free fall. Finally the three ways the rule fails, the pocket-sized expansion that measures the accuracy of an approximation, the order of contact of two curves, the catenary against the parabola and continuously compounded interest. Sheet
  8. 14 Function analysis and curve sketching Two calculation exercises first: ten direct calculations of asymptotes, then critical numbers and inflection points; then the long question of the Differential Calculus final exam: domain and symmetry, the three types of asymptotes including the oblique one by long division, extrema by the first derivative test, concavity and inflection points by the second derivative with the counterexample where a zero second derivative is not enough, the complete variation table, and the reverse reading where the function is rebuilt from the graph of its derivative. Also covered: the symmetry and periodicity that shrink the interval of study (with the derivative of an even function shown to be odd), the complete analysis of an exponential function and of a logarithmic one, linked to each other by a change of variable, and the inverse problem where the parameters of a function are recovered from its asymptotes and its table alone. A Part C adds five classics: the corner of an absolute value, two oblique asymptotes, a trigonometric analysis, the Lennard-Jones potential and the speed distribution of a gas. Sheet
  9. 15 Applied optimization The six-step method applied to the long problems of the exam: constraint and objective function in a single variable, a physical domain that is not always closed, the compulsory justification of the maximum, the fence along a barn and the open-top box, marginal cost and revenue in economics, the minimal travel time and the law of refraction recovered by calculus. Also covered: the box cut from a sheet of cardboard with its degenerate critical number, the Norman window whose relation h = r is proved exact, the economic order quantity where ordering cost and holding cost always balance at the optimum, and the undersea cable whose optimality condition gives Snell's law once again. A Part C adds five classics: the rectangle under a parabola, the cone rolled from a disk, the strongest beam, the ladder around an L-shaped corner and the cruising speed of a ship. Sheet

Integration

  1. 16 Elementary integration Two calculation exercises first, antiderivatives to compute directly, twelve from the table and ten by substitution; then the chapter that closes the differential calculus course, handled with the only tools available at that point of the semester: no definite integral, no area, no fundamental theorem, only the table of derivatives read backwards and checking by differentiation. Part A sets up the reversed table and its one empty slot at n = -1, the constant C and the theorem that justifies it, rewriting a product or a quotient that no rule handles, the transcendental lines and their sign errors, then the chain rule played backwards. Part B deals with adjusting a constant factor and the line that a missing x makes you cross, four integrals built on the same quadratic that call for four different decisions, five statements taken from students' tests, and two problems where you work back from an acceleration or a velocity, with an initial condition that is not always given at time zero. 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 exam 1: functions, limits and continuity, 201-SN2-RE

The first exam in differential calculus, the one in week six or seven, calibrated on tests actually given at CEGEP: domain and composition, eight limits that call for eight different techniques, continuity of a piecewise-defined function, continuity on an interval depending on its endpoints, asymptotes versus holes, and a sketch to build under twelve constraints.

  • 10 questions
  • 100 points
  • 150 minutes
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Practice exam

Practice exam 2: the derivative and its rules, 201-SN2-RE

The second test of differential calculus, the midterm, modelled on exams actually given at CEGEP: the derivative from the definition, the complete set of differentiation rules with their disguised-constant pitfalls, transcendental functions, implicit differentiation, higher-order derivatives, tangent lines, differentiability read from a graph and a related rates problem.

  • 10 questions
  • 100 points
  • 150 minutes
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Practice exam

Practice final exam: all of differential calculus, 201-SN2-RE

A three-hour final exam that runs through the whole differential calculus course: limits and continuity, computing derivatives, function analysis, linear approximation, optimization, related rates and modelling. Unlike the chapter-by-chapter series, no question tells you which method to use.

  • 12 questions
  • 100 points
  • 180 minutes
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