Calculus I

Units

1. 1

   Limits and continuity

   10 lessons

   1. Can change occur at an instant? Limits and limit notationUse limit notation to describe instantaneous rates and distinguish a limit from a function value.
   2. Estimating limit values from graphs and tablesEstimate limits from graphs and two-sided tables, and explain when an approaching height does not exist.
   3. Determining limits using algebraic properties of limitsCalculate combined, quotient, and composite limits, and state when each method applies.
   4. Determining limits using algebraic manipulationEvaluate 0/0 limits by rewriting expressions and explain why the rewrite preserves the limit.
   5. Determining limits using the squeeze theoremJustify a limit by trapping a function between two functions approaching the same number.
   6. Selecting procedures for limits and connecting multiple representationsChoose a limit procedure, calculate the limit, and distinguish exact results from graphical or numerical evidence.
   7. Types of discontinuities and continuity at a pointClassify breaks and justify continuity at a point using the function’s value and limits.
   8. Continuity over an interval and removing discontinuitiesIdentify intervals of continuity and choose values that repair joins and holes.
   9. Infinite limits, limits at infinity and asymptotesFind and justify vertical and horizontal asymptotes using one-sided limits and limits at infinity.
   10. Working with the intermediate value theoremJustify that a continuous function reaches a specified value inside an interval.
2. 2

   The derivative and its rules

   8 lessons

   1. Average and instantaneous rates of change at a pointCalculate an average rate over an interval and use a limit to find an instantaneous rate at a point.
   2. Defining the derivative and using derivative notationFind derivatives from limits, interpret derivative notation, and write tangent-line equations.
   3. Estimating derivatives of a function at a pointEstimate a derivative from a table, a drawn tangent, or a calculator model; justify the method and include units.
   4. Connecting differentiability and continuityDetermine where a derivative exists by checking continuity and finite one-sided slopes, and justify the conclusion.
   5. The power rule and the constant, sum, difference and constant multiple rulesDifferentiate sums of powers, including rewritten roots and reciprocals, and find tangent lines.
   6. Derivatives of cos x, sin x, e^x and ln xDifferentiate combinations of these functions and use derivatives to find slopes and tangent lines.
   7. The product rule and the quotient ruleDifferentiate products and quotients, then check whether a simpler rule applies.
   8. Derivatives of tangent, cotangent, secant and cosecantDifferentiate these four functions and use their derivatives in products, quotients and tangent lines.
3. 3

   The chain rule and implicit differentiation

   5 lessons

   1. The chain ruleDifferentiate compositions, including nested and tabulated functions, and use their derivatives to find rates and tangent lines.
   2. Implicit differentiationDifferentiate equations in x and y, then use the derivative to find tangent lines.
   3. Differentiating inverse functionsFind an inverse function’s derivative at a specified input from a formula or table.
   4. Differentiating inverse trigonometric functionsDifferentiate inverse trigonometric functions, including compositions, and use a derivative to find a tangent line.
   5. Selecting procedures for derivatives and higher-order derivativesChoose a differentiation rule from structure, then calculate explicit and implicit second derivatives.
4. 4

   Rates of change and linear approximation

   5 lessons

   1. Interpreting the derivative in context and rates in applied settingsInterpret first and second derivatives with direction and units, and estimate a nearby change.
   2. Straight-line motion: position, velocity and accelerationFind velocity and acceleration from position; justify direction, turns, and speed changes using signs.
   3. Related rates: setting up the relation and differentiating in timeFind a linked rate by differentiating an equation with respect to time, then interpret its sign and units.
   4. Related rates: solving and interpreting (cones, shadows, ladders)Solve complete related-rates problems: remove extra variables with similar triangles, differentiate in time, find the missing value at the instant, substitute last, and state the rate as a sentence with sign and units.
   5. Approximating values using local linearity and linearizationUse a tangent line to estimate a nearby value and justify whether the estimate is high or low.
5. 5

   Maximum and minimum values, and curve sketching

   10 lessons

   1. Extreme value theorem, global and local extrema, and critical pointsFind critical points and justify absolute and interior local extrema on a closed interval.
   2. The mean value theoremCheck the conditions, justify a matching instantaneous rate, and locate it when given a formula.
   3. Increasing and decreasing intervals and the first derivative testUse a graph or formula for f′ to find intervals of increase and decrease and justify relative extrema.
   4. The candidates test for absolute extremaFind and justify absolute extrema on a closed interval, stating each value and where it occurs.
   5. Concavity and the second derivative testJustify concavity, inflection points, and relative extrema using derivatives.
   6. Sketching f' from f and f from f'Sketch f from f′ and justify extrema, concavity, and inflection points using aligned graphs.
   7. Reading f, f' and f'' together: justify extrema, concavity and inflection from a graphRead a graph of f′ and write justified claims about f: relative extrema, concavity, inflection points and absolute extrema, naming the function and the point or interval each time.
   8. L'Hospital's rule for indeterminate formsCheck the form of a limit, apply L'Hospital's rule when permitted, and repeat it when needed.
   9. Optimization problemsModel a quantity with one variable, justify its optimum, and answer with the requested units.
   10. Exploring behaviours of implicit relationsLocate horizontal and vertical tangents and justify extrema on an implicit curve.
6. 6

   Integrals and the Fundamental Theorem of Calculus

   8 lessons

   1. Exploring accumulations of changeUse rate-graph areas to find accumulated change and final amounts, with correct signs and units.
   2. Approximating areas with Riemann sumsCalculate left, right, midpoint, and trapezoidal estimates; justify error direction when the graph’s behavior is known.
   3. Riemann sums, summation notation and definite integral notationConvert between right Riemann-sum limits and definite integrals, identifying the interval, function and variable.
   4. Applying properties of definite integralsCalculate definite integrals from known values and signed geometric areas.
   5. The FTC and accumulation functions: d/dx of an integral, moving upper limitFind accumulated values, derivatives, and behavior from an integrand’s graph, including when the upper limit moves.
   6. Behaviour of an accumulation function from the graph of its rate (FRQ)From the graph of f, find values, extrema, concavity and inflection points of g(x) = ∫ f(t) dt, and justify each claim by naming g′ = f or g″ = f′.
   7. Finding antiderivatives and indefinite integrals: basic rules and notationFind basic indefinite integrals, choose an antiderivative using a given value, and check by differentiation.
   8. The fundamental theorem of calculus and definite integralsEvaluate definite integrals with antiderivatives and use accumulated change to find a final value.
7. 7

   The substitution rule

   2 lessons

   1. Integrating using substitutionChoose a substitution and evaluate indefinite and definite integrals, adjusting constants and limits correctly.
   2. Integrating using long division and completing the squareRewrite rational integrands using division or a completed square, then find their antiderivatives.
8. 8

   Net change and average value

   3 lessons

   1. Position, velocity and acceleration using integralsFind final position, total distance, and final velocity from motion integrals.
   2. Accumulation functions and definite integrals in applied contextsFind an amount from entering and leaving rates, and justify when it is greatest or least.
   3. Average value of a function on an intervalCalculate and interpret average value using a definite integral, and distinguish it from average rate of change.