Scholarship Calculus 93202
Complex numbers, differentiation, integration and differential equations, with strategy prompts, worked steps and s/o marking.
350 problems · 10 free to try now
Free problems
- Complex numbers: de Moivre's theorem and rootsBridge
- Complex numbers: loci on the Argand diagramBridge
- Complex roots of real polynomialsBridge
- De Moivre's theorem: multiple-angle identities and cubic equationsBridge
- Complex numbers: powers of cis θ and integrationBridge
- Roots of unity and exact trigonometric valuesBridge
- Cubic equations: roots in arithmetic progressionBridge
- Exponential equations reducible to quadratics; parameter analysisBridge
- Tangency and number of solutions of transcendental equationsBridge
- Inverse functions and numerical solution (Newton's method)Bridge
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Integration techniques · 39
- Integration by parts: reduction formulae and limitsScholarship
- Hyperbolic functions and arc length (definitions supplied)Scholarship
- Polar coordinates: intersections and area (definitions supplied)Scholarship
- Arc length of parametric curves with sign changesScholarship
- Definite integrals by symmetry (the reflection property)Scholarship
- Volumes of revolution about both axes (formulae supplied)Scholarship
- Integration by parts: improper integrals and factorialsScholarship
- Integration by parts twice (cyclic)Scholarship
- Partial fractions, logarithmic and improper integralsScholarship
- Integrals of trigonometric powersScholarship
- Probability density functionsScholarship
- Arc length of Cartesian curves (formula supplied)Scholarship
- Areas using horizontal stripsScholarship
- Numerical integration in context: river flowScholarship
- Volumes of revolution with trig and exponentialsScholarship
- Completing the square: inverse tangent integralsScholarship
- Reduction formula for powers of cosineScholarship
- Work done pumping waterScholarship
- Average value in a temperature modelScholarship
- Integrals with circle geometry and odd partsScholarship
- Integration by parts with logarithmsScholarship
- Integrating inverse tangentScholarship
- Log and inverse-tan substitutionsScholarship
- Area under ln x and its chordScholarship
- Exponential lifetimes (probability density supplied)Scholarship
- Area between intersecting trig curvesScholarship
- Integration by recognising a derivativeScholarship
- Volume of liquid in a bowlScholarship
- Trapezium rule error and convexityScholarship
- Centroids of regions (formulae supplied)Scholarship
- Bounding integrals with inequalitiesScholarship
- Accumulated change from a rateScholarship
- Reduction formula for a rational integral and a series for ln 2Scholarship
- Area of a loopScholarship
- Parametric curves: second derivative, self-crossing and loop areaScholarship
- Parametric arc length: the logarithmic spiralScholarship
- The astroid: tangents, arc length and areaScholarship
- The deltoid: cusps, area and arc lengthScholarship
- Rolling coins and the cardioidScholarship
Differentiation techniques · 28
- Higher-order derivatives: pattern, proof and evaluationScholarship
- Parametric curves: stationary points and second derivativeScholarship
- Cycloid: parametric derivatives and a geometric propertyScholarship
- Implicit second derivativesScholarship
- Logarithmic differentiationScholarship
- Tangents from an external point and enclosed areaScholarship
- Common tangents to two parabolasScholarship
- nth derivatives and patternsScholarship
- Parametric curves: tangents and self-crossingScholarship
- The Mean Value Theorem (statement supplied)Scholarship
- Inflection points with a parameterScholarship
- Newton's method: convergence and failureScholarship
- Implicit tangents with a direction conditionScholarship
- Chain and quotient rulesScholarship
- Area of a parametric loop (formula supplied)Scholarship
- Rational functions with oblique asymptotesScholarship
- Linear approximation and its errorScholarship
- Verifying differential relationsScholarship
- Cusps and non-differentiable pointsScholarship
- Tangents, normals and the closest point to the originScholarship
- Parametric projectile: a conversion kickScholarship
- Parametric second derivatives and concavityScholarship
- The semicubical parabola: tangents meeting the curve againScholarship
- Normals to a parametric hyperbolaScholarship
- A rational parametrisation of the circleScholarship
- The flange of a train wheel: a prolate cycloidScholarship
- Parametric curves with logarithms: stationary and inflection pointsScholarship
- Normals to a parabola through a given pointScholarship
Limits, FTC and continuity · 25
- Average value of a function and limits with a parameterScholarship
- Integral equations via the Fundamental Theorem of CalculusScholarship
- Improper integrals: infinite limits and unbounded integrandsScholarship
- The Fundamental Theorem with variable limitsScholarship
- Standard limitsScholarship
- Continuity and differentiability at a pointScholarship
- Integral functions and their limitsScholarship
- Limits of exponential expressionsScholarship
- Jump discontinuities from exponentialsScholarship
- Squeeze theoremScholarship
- Derivative of sine from first principlesScholarship
- Limits recognised as derivativesScholarship
- Limits of integral sequencesScholarship
- Asymptotes of surd expressionsScholarship
- Removing discontinuities with trig limitsScholarship
- Differentiability of x|x|Scholarship
- Recovering functions from integral equationsScholarship
- Functions defined by integrals: the Fresnel integralScholarship
- Integral equations with a constant integralScholarship
- A Volterra integral equation leading to coshScholarship
- Filling a rainwater tank: accumulation from a rateScholarship
- Running averages and the FTCScholarship
- Limits of integrals by squeezingScholarship
- Recovering functions from integral identitiesScholarship
- An integral equation with an exponential sourceScholarship
Differential equations · 14
- Orthogonal trajectoriesScholarship
- Homogeneous differential equations: substitution y = vxScholarship
- Logistic growth (partial fractions supplied)Scholarship
- Linear DEs with an integrating factor (method supplied)Scholarship
- Simple harmonic motion: verifying solutionsScholarship
- Drug infusion to steady stateScholarship
- Separable DEs with trigonometric termsScholarship
- Falling with air resistanceScholarship
- Forensic time of death (Newton's law of cooling)Scholarship
- Separable DEs with inverse tangentsScholarship
- Harvesting and equilibriaScholarship
- Mixing with a changing volume (integrating factor)Scholarship
- Orthogonal families of circlesScholarship
- Second-order DEs by repeated integrationScholarship
Complex numbers · 51
- Complex roots and exact trigonometric sumsScholarship
- Euler's form: complex exponentials and logarithmsScholarship
- Regular polygons in the Argand diagramScholarship
- Argand regions: area and extreme modulusScholarship
- Quadratics with complex coefficientsScholarship
- Summing trigonometric series with complex exponentialsScholarship
- Roots of complex numbers and linked equationsScholarship
- Argument loci: arcs of circlesScholarship
- Conic loci in the complex planeScholarship
- Equations involving conjugatesScholarship
- Rotations: squares and equilateral trianglesScholarship
- Polynomials with a given complex rootScholarship
- Areas of regions defined by moduliScholarship
- Exact trig values via complex multiplicationScholarship
- Möbius-type transformations and circlesScholarship
- De Moivre: powers of sine, trig equations and limitsScholarship
- Half-angle factorisation of 1 + cis 2θScholarship
- Cubics with one real root: calculus and complex rootsScholarship
- Cube roots of unity: identities and factorisationScholarship
- Complex impedance in AC circuitsScholarship
- Fourth roots and real quadratic factorsScholarship
- Perpendicular bisector loci and half-disc regionsScholarship
- Loci defined by reciprocalsScholarship
- Square roots of complex numbers and quadraticsScholarship
- Extreme distances on circlesScholarship
- When are powers real or imaginary?Scholarship
- Forming quadratics from complex rootsScholarship
- Arguments of products and inverse-tan identitiesScholarship
- Equations mixing z and |z|Scholarship
- Products over roots of unityScholarship
- Sector regions and their image under w = 1/zScholarship
- Mapping lines and circles under w = z²Scholarship
- Complex-number proofs in geometry: Varignon's theoremScholarship
- Polynomials with complex coefficients: no conjugate pairsScholarship
- Binomial sums with complex exponentialsScholarship
- Adding waves with phasorsScholarship
- Euler's formula: compound angles, chords and the circumferenceScholarship
- The complex logarithm and solving e^z = wScholarship
- Complex powers: i to the power iScholarship
- Cosine of a complex numberScholarship
- Rotations in computer graphicsScholarship
- Damped oscillations with complex exponentialsScholarship
- Summing trigonometric series with a geometric seriesScholarship
- The exponential map on the Argand diagramScholarship
- Exponential equations in the complex planeScholarship
- An integral evaluated with complex exponentialsScholarship
- Fractal iterations: the Mandelbrot ruleScholarship
- Regions and their images under squaring and inversionScholarship
- Summing cosines: the Dirichlet kernelScholarship
- Noise-cancelling headphones: adding waves with phasorsScholarship
- Telescoping sums of inverse tangents using argumentsScholarship
Conics and coordinate geometry · 28
- Ellipse tangent properties: proofs with fociScholarship
- Tangency conditions and the director circle of an ellipseScholarship
- Rectangular hyperbola: tangent and normal propertiesScholarship
- Loci from focal chords of a parabolaScholarship
- Circles: equations, tangents and tangent lengthsScholarship
- Areas of ellipses and elliptical segmentsScholarship
- Hyperbola focal distancesScholarship
- Closest point on a parabolaScholarship
- Classifying conics from general equationsScholarship
- Reflective property of the ellipseScholarship
- Parabolic dishes: locating the focusScholarship
- Elliptical whispering galleriesScholarship
- Polar equations of conics (formula supplied)Scholarship
- Tangents from an external point to a circleScholarship
- Intersecting conics: angles and tangencyScholarship
- Locus of midpoints of chordsScholarship
- Focal chord length of a parabolaScholarship
- Ellipse conditions and eccentricityScholarship
- Hyperbolas from asymptotes and a pointScholarship
- Intersecting circles: common chord and angleScholarship
- Rectangles inscribed in ellipsesScholarship
- Focal property of parabola tangentsScholarship
- Hyperbolas from focus–directrix lociScholarship
- Planetary and comet orbitsScholarship
- Chords of rectangular hyperbolasScholarship
- Chord lengths of lines through ellipsesScholarship
- Circles tangent to both axes and a lineScholarship
- Parabolic segment area (Archimedes)Scholarship
Optimisation and rates · 28
- Optimisation in context: carrying a pole around a cornerScholarship
- Optimisation with parameters, scenarios and constraintsScholarship
- Related rates on an ellipse: coordinates, distance and angleScholarship
- Modelling with scenarios: optimal boat speedScholarship
- Open box optimisation with an invalid rootScholarship
- Norman window: maximum area for fixed perimeterScholarship
- Related rates: moving shadowsScholarship
- Related rates: inflating balloonsScholarship
- Fencing with different costsScholarship
- Least-time paths: the lifeguard problemScholarship
- Profit optimisation with a capacity constraintScholarship
- Radar tracking: angle and distance ratesScholarship
- Largest cylinder inside a sphereScholarship
- Drug concentration: peak, fastest decline, effective windowScholarship
- Related rates: filling a triangular troughScholarship
- Poster design with marginsScholarship
- Sliding ladder ratesScholarship
- Minimising average costScholarship
- Rectangles inscribed in trianglesScholarship
- Trip cost with a speed limitScholarship
- Sand piles: rates in conesScholarship
- Distance from a point to an exponential curveScholarship
- Projectile range on a slope (formula supplied)Scholarship
- Fencing along a riverScholarship
- Pulling a boat to a dock: speed and accelerationScholarship
- Largest cone from a circular sheetScholarship
- Parametric speed on an elliptical racetrackScholarship
- Closest approach of two dronesScholarship
Functions and inverse functions · 28
- Inverse trigonometric functions: derivatives, identities and integralsScholarship
- Absolute value functions: piecewise derivatives and weighted mediansScholarship
- Functional equations: symmetry, periodicity and sumsScholarship
- Derivatives of inverse functions at a pointScholarship
- Inverses of rational functions and 2-cyclesScholarship
- Even and odd functions: decomposition and integralsScholarship
- Integrals involving the floor function (definition supplied)Scholarship
- Functional equations: solving by substitutionScholarship
- Minimum-value functions and integralsScholarship
- Simplifying inverse-trig compositions by differentiationScholarship
- Domains of composite functionsScholarship
- Inverses on restricted domainsScholarship
- Functional equations with shiftsScholarship
- Differences of absolute valuesScholarship
- Inverse of a cubic: gradients, areas and intersectionsScholarship
- Self-inverse rational functionsScholarship
- Compositions of trig and inverse trig functionsScholarship
- Inverting a thermistor's temperature lawScholarship
- The inverse of x e^xScholarship
- Inverting a tax rule: from net to gross incomeScholarship
- The area between a function and its inverseScholarship
- The maximum of two functionsScholarship
- An even function built from |x|Scholarship
- A function defined as a maximum over a parameterScholarship
- An odd periodic extensionScholarship
- Comparing mobile data plansScholarship
- An additive functional equationScholarship
- Exploring x|x − 2|Scholarship
Trigonometry · 19
- Trigonometric equations: general solutions and systemsScholarship
- Triple-angle tangent identity and exact productsScholarship
- Range of trigonometric rational expressionsScholarship
- Sum-to-product identity and restricted solutionsScholarship
- Exact values of sin 18° and cos 36°Scholarship
- Tides modelled with the R-formScholarship
- t-substitution for linear trig equationsScholarship
- Trigonometric inequalitiesScholarship
- Product identity and exact products of sinesScholarship
- Sums of inverse tangentsScholarship
- Heights by angles of elevationScholarship
- General solutions using co-functionsScholarship
- Half-angle identity proofsScholarship
- Ranges of quadratic trig expressionsScholarship
- Counting solutions with a parameterScholarship
- Pairing complementary anglesScholarship
- A triangle identityScholarship
- Ferris wheel modellingScholarship
- Equations with inverse tangentsScholarship
Equations, logs and inequalities · 27
- Logarithmic systems with change of baseScholarship
- Quadratics with a parameter: conditions on rootsScholarship
- Inequalities with surds, absolute values and logarithmsScholarship
- Symmetric non-linear systemsScholarship
- Exponential equations and uniqueness argumentsScholarship
- Logarithmic equations with change of baseScholarship
- Remainder theorem and polynomial divisionScholarship
- Symmetric functions of rootsScholarship
- Absolute value equations and solution countsScholarship
- Mixing alloys: dependent systems and costsScholarship
- Radical equations and extraneous rootsScholarship
- Factor theorem with a parameterScholarship
- Logarithmic inequalities with domains and changing signsScholarship
- Parametric systems of planesScholarship
- Comparing growth modelsScholarship
- Biquadratic inequalities, trig substitution and counting rootsScholarship
- Rational and modulus inequalities with a parameterScholarship
- Tangent lines to circles through a pointScholarship
- Integer solutions by factorisationScholarship
- Systems with logarithms and powersScholarship
- Designing a rectangular gardenScholarship
- A trigonometric system and adding unit vectorsScholarship
- Exponential systems by substitutionScholarship
- A circle meets a family of hyperbolasScholarship
- Comparing bacterial culturesScholarship
- A symmetric system in three unknownsScholarship
- When does log(x + y) = log x + log y?Scholarship
Sequences and series · 5
- Recurrence relations: convergence and invariantsScholarship
- Geometric series: bouncing ballsScholarship
- Telescoping and differentiated seriesScholarship
- Loan repayments via geometric seriesScholarship
- Convergent recursive surdsScholarship
Geometry proofs · 5
- Maximising a viewing angle: calculus and geometryScholarship
- Triangle centres with coordinatesScholarship
- Largest triangle in a circleScholarship
- Median lengths via vectorsScholarship
- Shortest paths by reflectionScholarship
Combinatorics · 3
- Distributions with constraints: stars and bars, inclusion–exclusionScholarship
- Committees with conditionsScholarship
- Arrangements with repeated lettersScholarship