Review functions, limits, continuity, derivatives and antiderivatives for McGill MATH 140, Calculus 1. Try an explained exercise or get one-to-one help choosing a method and understanding each step. You can start with the question alone.
McGill MATH 140 introduces functions and graphs, limits, continuity, derivatives, antiderivatives and applications. You learn how to differentiate functions and find antiderivatives, building on the function skills used later in integral calculus.
First make a function continuous and find its derivative from the definition. Then combine derivative rules to find a tangent line.
Add antiderivatives and applications if they are on your review list.
Before you start: Allow 15 minutes for both questions, then 15 minutes to review the step that gave you trouble.
For , let , and set . Find the value of that makes continuous at . For that value, find from a difference quotient.
Let for real . Find and the tangent line at .
For a domain or limit error, use the cancelled-factors lesson linked in question 1 and explain which equality needs an excluded input. For a derivative error, use the three-question derivative-rule check linked in question 2.
After the diagnostic, use the open-box worked example on this page. Cover its solution, write the physical domain, then compare the admissible stationary point with the boundary behaviour.
Choose matching questions from the limits-and-derivatives practice linked below. It was organized around Concordia MATH 203 topics; use only the questions matching your McGill outline and add your assigned antiderivative work if it is in scope.
If you would like help with these questions, you can start a lesson with a limit, derivative or application question you could not begin. If you have an attempt, your tutor can review it with you; otherwise, you can work through the first step together and clarify the algebra, definition or derivative rule you need.
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Review high-school calculus, function graphs, factoring, trigonometric identities and exponentials/logarithms. MATH 139 is listed as not offered in the 2026–27 departmental catalogue. Ask McGill advising about a currently available route if your calculus background is limited; MATH 139 and 140 are not a required sequence.
The guidance above is study preparation, not an eligibility decision. Confirm current prerequisites, restrictions and availability with McGill University.
Check the official course catalogueA differentiable function satisfies for every real and . Find .
Before calculating, decide whether the derivative alone determines a single function. What additional information does provide?
Find one term whose derivative is , one whose derivative is and one whose derivative is . A constant disappears when differentiated.
Reverse the power rule term by term. Every antiderivative on the real line has the form , where is a constant.
The derivative does not choose . The initial value does: . Therefore .
Check both requirements. Differentiation recovers ; substituting gives . Checking only the derivative would not detect an incorrect constant.
Find if for every real and . Check the derivative and the given value separately.
An antiderivative is . Now , so . Thus . Its derivative is and . The given function value is not automatically the constant; substitute its input for before solving for .
If checking the derivative is difficult, use the derivative-rule check below. If the derivative is right but the initial value fails, return to the substitution used to find . A tutor can ask you to explain both checks, then change the initial value for a new attempt.
Let for real input , and let be a change from the input 3. Simplify .
Expand the numerator: . Since , cancellation gives . This is the average rate of change of the function over that change in input. Its limit as is , the instantaneous rate of change given by the derivative .
Finding where a derivative is zero is only one step in an optimization problem. This box example uses geometry to find the allowed dimensions, then checks which choice gives the largest volume.
Write for the box’s volume in cubic centimetres when the corner cut has side centimetres. Cut a square of side centimetres from each corner of a sheet and fold up an open box. Its height is , and its base measures by . Therefore The height and both base dimensions must be positive, so the allowed cut sizes form the physical domain .
Expanding gives Setting yields The roots are approximately and . The larger one is outside the physical domain and cannot describe this box.
Write for the remaining allowed cut size where the derivative is zero; the subscript is just a name for this particular cut size. The polynomial model extends continuously to , where both endpoint volumes are zero. There is only one point inside this interval where the derivative is zero, Its volume is positive, so comparing this candidate with the endpoints establishes the global maximum. Equivalently, changes from positive to negative there.
Cut squares of side about . The maximum volume is approximately A length is the decision variable; a volume is the optimized outcome. Keep those units separate and round only after evaluating the exact expression.
Repeat the model for a square sheet measuring . What corner cut maximizes the open box’s volume?
Here with , and The interior candidate is ; is an endpoint where the base dimensions become zero. Comparing with the zero endpoint volumes gives the maximum .
On another optimization problem, write the physical domain before differentiating. Explain how each candidate is checked against boundaries and the original question.
You can start MATH 140 tutoring with a question you have not been able to begin. Your tutor can explain the relevant algebra, show how to read a limit and help you choose a derivative rule. For an application, you can identify the quantities and build the equation together. For optimization, your tutor can help you find the allowed values and justify a maximum or minimum. For an antiderivative, you can check the answer together by differentiating it, then try a similar question with guidance.
Practise derivative selection and interpretation separately. For optimization, compare valid candidates and endpoints. Include basic antiderivatives where your outline requires them.
Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.
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.
Eighteen questions on limits and derivatives, optimization, L’Hôpital’s rule and linearization, with hints and explained solutions. Includes a two-example workshop on tangent estimates and differentials.
Optional application of functions, graphs and derivatives. Check your current section outline for the depth of curve-sketching required.
Supplemental practice only when assigned: check the assumptions behind existence, uniqueness and extreme-value conclusions through two examples and four exercises.
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