Math I: Calculus exam practice¶

Worked problems from the Math I mock exam and weekly tests, with marks per step.

Each problem gives a short plan, then the solution one step at a time. What you need and Try next sit in the Problem connections panel on the right (below the problem on a phone). Every step is folded: try it yourself, then open it to compare. Steps link to the concept page they use, so a step you cannot do tells you which page to revisit.

These are practice problems. The marks follow the marking schemes of the mock exam and the weekly tests; they do not preview the live exam.

Sets, numbers and sequences¶

Problem	Source	Points
A set identity and a divisibility proof by induction	Week 1 test	7
Limit of a sequence from the definition, squeeze and bounds	Week 2 test	7

Limits, continuity and inverse functions¶

Problem	Source	Points
Continuity of $1/x^2$ from the $\varepsilon$ – $\delta$ definition	Mock exam, Q2(b)	14
Continuity and differentiability of $x^2\sin(A/x)$	Week 3 test, 2025-26	7
The inverse of $x+x^3$	Week 5 test	7

Derivatives, extrema and convexity¶

Problem	Source	Points
Global extrema of $x+1/x$ on a closed interval	Week 3 test	7
Stationary points, convexity and an alternating series	Week 6 test	7
A function with three local extrema	Mock exam, Q3(b)	14

Limits of type 0/0 and series¶

Problem	Source	Points
A limit via asymptotic equivalence	Mock exam, Q4(b)	14
Convergence of three series and $\displaystyle \sum\sin n$	Mock exam, Q1(b)–(c)	14

Integrals¶

Problem	Source	Points
Area between two curves and $\int_0^1\ln x\,\mathrm{d}x$	Mock exam, Q5(b)–(c)	14
FTC with a variable limit and a scaled improper integral	Week 7 test	7

Back to the Math I: Calculus course page.

Symbols: delta, epsilon, integral, sum