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University of Waterloo · Waterloo · sciences route
University of Waterloo MATH127 — Calculus 1 for the Sciences
Review functions, differentiation and the first integral calculations for Waterloo’s science route, with separate limits on the topics supported.
64 original question families · 192 parameter instances
Practise this course →Course alignment and coverage
2026–2027 Undergraduate Studies Academic Calendar, effective September 2026–August 2027, checked on 25 October 2026 through Waterloo’s public catalogue. A catalogue description is not a section syllabus.
Across this collection: 140 distinct original families and 420 parameter instances. Courses share topic banks; translations and shared mappings are not extra questions.
Supported learning objectives
Limits
Evaluate a limit using substitution, factoring or rationalization.
Continuity
Find missing values and compare one-sided limits.
Polynomial derivatives
Apply the power rule and interpret a derivative.
Product rule
Differentiate polynomial products and identify missing terms.
Chain rule
Differentiate composite polynomial functions.
Tangent lines
Connect slope, point and tangent-line equation.
Implicit differentiation
Differentiate a curve and evaluate a local slope.
Related rates
Relate changing lengths, areas, volumes and distances.
Linear approximation
Use a tangent model and estimate a differential.
Optimization
Locate and check a maximum or minimum in a stated domain.
Algebra essentials
Solve linear equations, expand products, factor a quadratic and identify an inequality boundary.
Functions and domains
Evaluate and compose functions, invert a linear function and find a square-root domain endpoint.
Powers and logarithms
Evaluate integer powers and logarithms, apply a doubling model and solve an equal-base equation.
Trigonometry foundations
Convert degrees to radians, use a right-triangle ratio and find a sine period. Numeric answers only.
Polynomial antiderivatives
Reverse the power rule and use a stated constant or initial value. Exact fractional coefficients are supported.
Definite integrals and average value
Evaluate elementary polynomial integrals, distinguish signed accumulation and compute an average value.
What is not covered
Selected single-variable and polynomial foundations only. Trigonometric/logarithmic derivatives, quotient-rule exercises, formal proofs, complete curve sketching and long modelling tasks are not covered. Elementary function evaluation is not equivalent to differentiating those functions. Riemann-sum questions, substitution integrals and general area-between-curves problems are not yet included.
An original worked example
Solve 2x+3=9. Enter x.
Worked solution
Answer: 3
- 2x=6.
- Divide by 2: x=3.
Official course sources
Independent, original, free self-study practice. Not affiliated with or endorsed by this university; these are not official exam questions. Topic alignment does not establish transfer credit, a current section’s weekly syllabus, full course coverage or a grade prediction. Follow your instructor’s rules and use official course materials for assessed work. No live AI chat or paid access is required.