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Further proof, calculus and differential equations: lesson checks flashcards

This deck has 8 check-yourself questions and 7 spot-the-error cards. The misconception each lesson corrects, and its check-yourself question with the worked answer. Rate each answer, and the card comes back sooner or later depending on how you rated it.

Further Maths · all 584 maths cards

How the five kinds differ

Standard results ask you to recall a formula or a value. Definitions ask you to state a mathematical term precisely. Recall prompts ask for a rule, condition or method. Check yourself cards are a short question with its answer. Spot the error cards ask you to correct a statement that is wrong.

Every card in this deck

Choose any card to try it in the player above. Each card links to the lesson that teaches it.

• Prove by induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n.Check
• The inductive step does all the work; the base case is a formality that can be skipped.Spot the error
• Prove by induction that 9ⁿ + 11 is divisible by 4 for all positive integers n.Check
• For a region between two curves, subtract the curves first and square after: V = π∫(y_outer − y_inner)² dx.Spot the error
• The region under y = 2x from x = 0 to 3 is rotated about the x-axis. Find the volume, and name the solid.Check
• An integral with an infinite limit must give an infinite answer.Spot the error
• Find the mean value of f(x) = 1/x² on the interval [1, 2].Check
• Since arcsin is an inverse trigonometric function, its derivative must involve sines and cosines.Spot the error
• Evaluate ∫ 1/(1 + x²) dx from 0 to 1, exactly.Check
• If the variables in a first order equation do not separate, the equation cannot be solved.Spot the error
• Write down the integrating factor for dy/dx + 2xy = x, and the equation it produces.Check
• The auxiliary equation method applies to any second order differential equation.Spot the error
• Find the general solution of y'' + 4y' + 3y = 6.Check
• Adding any amount of damping to an oscillator removes the oscillation entirely.Spot the error
• Classify the damping in y'' + 6y' + 9y = 0 and state the long-term behaviour of any solution.Check

The lessons these cards come from

• Calculus with inverse trigonometric functions
• First order equations and integrating factors
• Induction: divisibility and matrices
• Mean values and improper integrals
• Modelling with differential equations
• Proof by induction: sums and series
• Second order equations
• Volumes of revolution

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