MATH 100 practice
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Topic-based questions, mock exams and step-by-step solutions for first-year differential calculus. Free, no account needed.
- Questions
- 451
- Topics
- 23
- Mock papers
- 44
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LimitsIntroductory2 min
Evaluate the limit.
What do you want to practise?
The topics of MATH 100, in the order the course teaches them, grouped by the test they come before. 451 questions across them, from introductory to capstone, each with hints and a worked solution.
Before the midterm
Practise a mixed set for the midtermPower functions and first sketches
Which term dominates for small and large x, sketching two-term polynomials, and the familiar functions.
Functions, graphs and inverses
Domain, composition, transformations, inverses and the trigonometry the course assumes.
Limits
Limit laws, the indeterminate form 0/0, one-sided and infinite limits, limits at infinity and asymptotes.
Continuity
The definition of continuity, classifying discontinuities, the constants that make a piecewise function continuous, and the Intermediate Value Theorem.
The definition of the derivative
Secants and tangents, average and instantaneous velocity, the difference quotient and differentiability.
Differentiation rules
Power, sum, product and quotient rules, tangent lines and higher-order derivatives.
Exponential and trigonometric functions
Derivatives of eˣ, aˣ, sine, cosine and tangent, and the limits behind them.
The chain rule and logarithms
Compositions, nested rules, the natural logarithm and logarithmic differentiation.
Implicit differentiation and inverse trigonometric functions
Differentiating relations, tangent lines to curves, and the derivatives of inverse and inverse trigonometric functions.
Related rates
Translating a moving situation into an equation, then differentiating.
Before Quiz 2
Practise a mixed set for Quiz 2L’Hôpital’s rule
Indeterminate forms 0/0, ∞/∞, 0·∞, ∞−∞, 1∞, 00 and ∞0, and when the rule does not apply.
Curve sketching
Increase, concavity, inflection points, asymptotes and full sketches.
Before the final
Optimization
Building an objective function and justifying the extreme value.
Linear approximation
Tangent-line estimates, whether they run high or low, and approximating a value without a calculator.
Taylor polynomials
Quadratic and higher approximations, the Maclaurin polynomials of the standard functions and, for flavour A, bounds on the error.
Newton’s method
Following the tangent line towards a root: an iteration or two by hand, and the starting points where it fails.
Flavours A and B only
Exponential growth and decay
The differential equations y′ = ky and y′ = a − by: half-lives, doubling times, Newton’s law of cooling and inflow and outflow.
Flavours A and B only
Slope fields and Euler’s method
Checking that a function solves a differential equation, reading a slope field, and stepping a solution forward with Euler’s method.
Flavours A and B only
Logistic growth and phase lines
Steady states and their stability from a state-space diagram (phase line), the logistic equation, and harvesting.
Flavour B only
Functions of two variables and partial derivatives
Evaluating and reading functions of two variables, level curves, partial derivatives as marginal rates, and second partials.
Flavour C only
Optimization in two variables and Lagrange multipliers
Critical points and the second derivative test, the extremes of a function on a closed region, and Lagrange multipliers.
Flavour C only
Not in the course
The Mean Value Theorem
Rolle’s theorem, the MVT, counting roots and proving inequalities.
Antiderivatives
Reversing differentiation, the constant of integration and initial conditions.
How it works
Choose a topic
Topics follow the course in the order it is taught, grouped by the test they come before.
Solve questions
Hints stay hidden until you ask for one. Answers are read the way you write them, fractions and exact values included.
Review mistakes
Every solution names the mistake the question is built to catch. A question you miss comes back the next day, then after three days and a week; the count beside Progress says how many are due.
Mock exams
Papers for the midterm, Quiz 2 and the final, with a running clock, question flags and marks by topic. Every paper can also be sat untimed.
Midterm
Everything up to the midterm, set out like a written test: five short answers, then three longer problems in parts, on limits and continuity, on differentiation, and on related rates. 10 papers, each 14 questions, 28 marks and 60 minutes, timed or untimed.
Midterm skills check
Twenty short answers across everything up to the midterm, from functions and limits to inverse trigonometric functions and related rates: a quick check of every skill. 10 papers, each 20 questions, 40 marks and 70 minutes, timed or untimed.
Quiz 2
L’Hôpital’s rule and curve sketching: five short answers, then a sketching problem in three parts. 8 papers, each 8 questions, 16 marks and 30 minutes, timed or untimed.