Calculus links local rates of change to accumulated change, and the first difficulty is often choosing a model rather than carrying out a derivative. CLP-1 concentrates on differential calculus with explicit worked examples. Active Calculus asks readers to develop ideas through guided activities and includes an integral-calculus route. Choose the former for a focused repair of limits and differentiation; choose the latter when explaining graphs and models needs as much attention as symbolic manipulation.
What to work on
Compute a derivative from its defining limit and interpret it as a rate.
Use chain and implicit differentiation in an equation with a stated domain.
Find and justify a constrained one-variable optimum, including endpoints.
Use the fundamental theorem of calculus to connect accumulation and differentiation.
A starting text
CLP-1 Differential Calculus — Joel Feldman, Andrew Rechnitzer and Elyse Yeager. Read sections 1.3 and 1.6 on limits and continuity; 2.2-2.3 on the derivative; 2.9 and 2.11 on chain and implicit rules; and 3.5 on optimisation. Use worked examples to identify domains and endpoint checks.
Choosing what to read
Sketch the function and state its domain before calculating. Compare a derivative obtained from the definition with one obtained by rules, and explain the sign in context. In optimization, list endpoints and points where differentiability fails as well as critical points. Work a problem without looking at its hint, then use the hint to identify the missing decision. Keep integral and series preparation on your plan if the next goal is real analysis.
Repair limit, chain-rule, implicit-differentiation and optimization skills.
How it reads
Worked-example route focused on differential calculus.
What it expects
Algebra, functions, trigonometry and graph interpretation.
What to keep in mind
CLP-1 is differential calculus; integration and series require further reading.
Practice
Practice in the separate 2022 combined CLP-1 edition at https://personal.math.ubc.ca/~CLP/CLP1/combined_clp_1.pdf: Public exercises, Hints for Exercises, Answers to Exercises and Solutions to Exercises were inspected in this author-hosted 2022 combined PDF. Sample entries were checked; complete question-to-solution coverage was not audited. Use the exercises within sections 2.2, 2.9, 2.11 and 3.5 of this combined edition; section 2.2's exercise labels begin 2.2.4 and section 2.9's begin 2.9.4. Match those full labels in the hint, answer and solution appendices rather than using page numbers from the 2024 standalone text.
Complete public PDF.
Edition, reading range and source
Read sections 1.3 and 1.6 on limits and continuity; 2.2-2.3 on the derivative; 2.9 and 2.11 on chain and implicit rules; and 3.5 on optimisation. Use worked examples to identify domains and endpoint checks.
Work from graphical and contextual questions toward calculus definitions.
How it reads
Guided activities developing differential and integral calculus.
What it expects
Algebra, functions and trigonometry.
What to keep in mind
Activities need written work and discussion; reading the explanatory prose alone misses their purpose.
Practice
Guided activities are described in the catalogue; public solution coverage was not verified.
AIM source listing with author/download routes; full-text access was not confirmed in the inherited item.
Edition, reading range and source
Use the author's contents to select limits, derivatives and the relevant guided activities, then integral calculus when needed; edition numbering was not checked.
CLP-2 Integral Calculus — Joel Feldman, Andrew Rechnitzer and Elyse Yeager. Sections 1.1-1.4: Definition of the Integral, Basic Properties, Fundamental Theorem of Calculus and Substitution; use 1.7 for integration by parts. Continue from derivatives to accumulation and integration; this selection addresses the fourth objective.
Practice
CLP-1 combined text and exercises, August 23, 2022 PDF. Use the exercises within sections 2.2, 2.9, 2.11 and 3.5 of this combined edition; section 2.2's exercise labels begin 2.2.4 and section 2.9's begin 2.9.4. Match those full labels in the hint, answer and solution appendices rather than using page numbers from the 2024 standalone text.. Public exercises, Hints for Exercises, Answers to Exercises and Solutions to Exercises were inspected in this author-hosted 2022 combined PDF. Sample entries were checked; complete question-to-solution coverage was not audited.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
Derive the derivative of x squared from the difference quotient, then find the tangent line at x = 3.
A hint
Expand (x+h)2 and cancel before letting h tend to zero. Remember that a tangent line needs a point as well as a slope.
Work through the solution
For h=0,
h(x+h)2−x2=2x+h.
Letting h→0 gives f′(x)=2x. At x=3, the point is (3,9) and the slope is 6, so the tangent line is y−9=6(x−3). Cancellation was justified only for h=0; the derivative is then obtained by taking a limit, not by substituting h=0 into the original quotient.
A rectangle has perimeter 20. Maximize its area, specify the allowable side lengths, and explain what happens at the boundary.
A hint
Use the perimeter condition to eliminate one side. Complete the square, or differentiate, but keep the allowable interval in view.
Work through the solution
If the sides are a,b>0, then a+b=10. Thus 0<a<10 and
A(a)=a(10−a)=25−(a−5)2≤25.
Equality holds exactly when a=b=5. At either endpoint of the closed interval [0,10], one side vanishes and the area is zero: those are degenerate rectangles. Including the endpoints for comparison does not change the maximum, which occurs at an interior point.
For F(x) = integral from 0 to x squared of exp(t) dt, find F'(x) and name both rules used.
A hint
First regard the upper endpoint as a separate variable s. Differentiate with respect to s, then compose with s=x2.
Work through the solution
Define G(s)=∫0setdt. The fundamental theorem of calculus applies because et is continuous, and gives G′(s)=es. Since F(x)=G(x2), the chain rule yields
F′(x)=G′(x2)2x=2xex2.
As a check, evaluating the integral first gives F(x)=ex2−1, whose derivative is the same. The endpoint contributes the factor 2x; substituting the endpoint into the integrand alone would miss it.
Already comfortable with this? Skip differential-calculus review if you can complete the first two checks with domains, units where relevant, and endpoint reasoning. Skip the integral continuation only if the accumulation check and substitution are also routine; multivariable calculus remains a separate selection.
Gilbert Strang and Edwin 'Jed' Herman · Textbook · Introductory
An illustrated first course with examples and applications. Useful for computational foundations; a proof-based real analysis text is needed for a mathematics-major transition.
From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.
Calculus — Michael Spivak
Precalculus — David Cohen, Theodore B. Lee and David Sklar
David Guichard · Textbook · Introductory · English · Full-text access unverified
A compact single- and multivariable calculus sequence with computational practice. Useful alongside, rather than as a substitute for, a proof-based analysis course.
Joel Feldman, Andrew Rechnitzer and Elyse Yeager · Textbook · Introductory · English · Full-text access unverified
Four coordinated volumes cover differential, integral, multivariable and vector calculus, with graded exercises and solutions. Counted once, not once per volume.
James Callahan, Kenneth Hoffman, David Cox, Donal O’Shea, Harriet Pollatsek and Lester Senechal · Textbook · Introductory · English · Full-text access unverified
AIM source page with an authorized full-text link.
Develops calculus through scientific models and dynamical systems. Useful alongside a more proof-focused treatment, not a substitute for real analysis.
Preparation: Precalculus, functions and graphs.
Use: Work through examples and exercises before advancing.
Sudhir R. Ghorpade and Balmohan V. Limaye · Textbook · Intermediate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A rigorous multivariable course connecting calculus and analysis. Use it to move from computational recipes to proofs of differentiation and integration results.
Preparation: Single-variable calculus and linear algebra.
Izabella Łaba and Caleb Marshall · Textbook · Intermediate · English · Online text
The author archive offers both full course volumes and separate solutions; both course PDFs returned HTTP 200.
A coordinated honours sequence joins multivariable limits, differentiation, optimization and integration with curves, surfaces, vector calculus and an introduction to differential forms.
Preparation: Single-variable calculus; willingness to learn rigorous proofs.
Use: Bridge computational calculus to proof-oriented multivariable analysis. The two sequential volumes are counted together.