Fundamental Mathematics II: send your chapter so we can check whether we can help before you book.
Concordia MATH 209, Fundamental Mathematics II, develops calculus for quantitative applications. Its Fall 2026 outline connects limits and derivatives to marginal analysis, demand, graph behaviour and optimization, then introduces integration. The example below separates a marginal estimate from an exact change and checks a production limit before choosing an optimum.
A price rise earns more per unit but can reduce sales. This separate lesson connects elasticity to revenue. It distinguishes a prediction near one price (a local prediction) from the revenue change between two prices (a finite change). It complements the profit-and-capacity example below.
Read the full lesson: one model, two diagrams and three practice questions with explained answers
Try the three questions: percentages, exact change and inverse demand.
Review function notation, factoring, rational expressions, exponents and logarithms. For a function , let be a starting input and its change. Practise expanding : this is the change in the function value, not the new value itself. In a business model, label quantity, price, revenue and cost with their units before doing calculus.
The current Concordia calendar lists MATH 206 or equivalent, not MATH 208, as the prerequisite. Students with credit or exemption for MATH 203 or equivalent cannot take MATH 209 for credit. Other program restrictions apply; confirm eligibility with Concordia.
Check the official course catalogueLet , where is real. Let be a nonzero change in the input from the starting value , so the new input is . Expand and simplify the difference quotient . What number does it approach as approaches zero, and what does that number represent?
1. Evaluate both function values. First, . Since , the new value is . 2. Form the output change before dividing. Subtracting the starting value gives . Because , division is valid and . The quotient is the average change in output per unit change in input between and . 3. Take the limit. As , the term tends to zero, so . This is the derivative , the instantaneous rate of change, or tangent slope, at input . The prime denotes differentiation with respect to . 4. Keep the restriction on the original quotient. The original quotient is undefined at ; its limit describes nearby nonzero values. Cancel the common factor for nonzero changes in the input, then take the limit. Do not substitute zero into the original fraction. As an independent check, the derivative rule gives , hence .
Marginal analysis explains how a model changes locally. Optimization asks for the best feasible decision. This business example shows why those questions are connected but require different checks.
In a simplified weekly production model, is quantity sold and the price per unit is dollars. Total cost is dollars, and capacity limits output to . Treat as continuous for calculus; assume every unit produced is sold at the modelled price. Revenue is quantity times price, not price alone: Profit is revenue minus total cost: The -dollar fixed cost remains in profit even when its derivative is zero.
The prime in means the derivative of total cost with respect to output , measured in dollars per unit. Differentiate term by term: the fixed 600 has derivative zero, contributes 18, and contributes . Let mean an output increase from 40, and define its exact cost change by . The marginal cost is At , the instantaneous cost rate is dollars per additional unit. It estimates the cost of a small increase: . For one whole unit, the exact change is The marginal estimate is close but not identical. It is also not the average cost dollars per unit.
For a differentiable function on a closed interval, compare interior stationary points with endpoints. A stationary point means an interior output where the first derivative is zero. The second derivative, written , describes how the first derivative changes. Here a negative second derivative makes the graph concave down. Marginal profit is Setting it equal to zero gives . Since , this is the maximum of the unconstrained quadratic, but exceeds the capacity of . On the actual interval, , so profit is increasing throughout. The maximum feasible quantity, , therefore maximizes profit. A maximum at a boundary need not have derivative zero.
At , the model gives The other endpoint gives , consistent with the increasing-profit argument. The answer is units at a modelled price of dollars, with weekly profit dollars.
A new model has , , and . Find the quantity, price and maximum profit, checking the endpoints.
Keep the same meanings: is output, is unit price, is total cost, and is profit. Multiplying price by quantity before subtracting cost gives , so . The stationary point is , within the interval, and . Its price is dollars per unit and profit is dollars; endpoint profits are and , confirming the maximum.
In the original model, reduce capacity from to . Find the new optimum and profit.
on , so choose . The price is dollars and profit is dollars. Substitution gives , compared with . The stationary output 90 remains outside the allowed interval.
In the original cost model, estimate the change from to using marginal cost; compare with the exact change and explain the difference.
The estimate is dollars. The exact change is dollars, an underestimate of dollars. Write for an increase in output from 40; in this question . Expanding the quadratic cost formula gives The linear estimate omits the positive quadratic term, which equals when .
On the next application, label the function and units, state the feasible interval, and then differentiate. Keep a separate final check for feasibility, endpoints and the meaning of the reported number.
Official Concordia MATH 209 outline, Fall 2026
Your tutor can separate the model, derivative and interpretation in your attempt to find the step that needs attention. You can then work through a comparable marginal-analysis or optimization problem, discussing units, capacity limits and why a candidate is actually a maximum.
Build mixed sets from the current section outline: limits, derivative rules, applications, graph analysis and integration. Before calculating, name the target quantity and choose a method. Compare marginal estimates with exact changes when both are available; check an antiderivative by differentiating it. Confirm assessment details and permitted aids with your section.
We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.
Connect price, demand and revenue with a worked model, two diagrams and three explained practice questions. Learn when elasticity gives a local prediction and when to compare exact revenues.
Learn why cancelling a factor preserves an excluded input and distinguish a function value from a limit, with a worked example and three self-checks for Concordia MATH 203 and McGill MATH 140.
Ten essential algebra skills with worked examples, common mistakes and self-checks: a practical starting point before university or CEGEP calculus.
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