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Calculus

Exercises in C++

Calculus: 0% done

Module 1 · Rates of change: the question calculus answers

Calculus is the mathematics of change

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1 / 237 · Rates of change: the question calculus answers

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Calculus is the mathematics of change
, 2 to practiseThe average rate of change of a function
An average rate is the slope of a secant line
The C++ in this course
, 2 to practiseSpeed from a distance graph
, 1 to practiseSpeed right now is a puzzle
Shrink the interval yourself
, 1 to practiseThe area under a speed graph is the distance
, 1 to practiseTwo questions, slopes and areas
, 1 to practiseWhere calculus came from
, 4 to practisePractice problems
Recap

A limit is where a function is heading
, 1 to practiseGuessing a limit with a table
, 1 to practiseLimits from the left and from the right
, 1 to practiseThree ways a limit can fail to exist
, 1 to practiseLimit laws let you split the work
, 2 to practiseWhen substitution gives zero over zero
, 1 to practiseThe squeeze theorem
, 1 to practiseThe most important trig limit
, 1 to practiseLimits at infinity and horizontal asymptotes
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

Continuous means no breaks
, 2 to practiseThe kinds of discontinuity
, 1 to practiseMost formulas are continuous wherever they make sense
, 2 to practiseChoosing a value to join two pieces
The intermediate value theorem
, 1 to practiseThe theorem needs continuity
, 1 to practiseBisection: halve the interval until the root is trapped
, 2 to practiseBisection as code
, 1 to practiseThe extreme value theorem
, 4 to practisePractice problems
Recap

The derivative is the limit of secant slopes
, 2 to practiseWorking out a derivative from the definition
, 2 to practiseThe equation of the tangent line
, 1 to practiseThe derivative is a function too
, 1 to practiseWays to write the derivative
, 1 to practiseDifferentiable means smooth, not only unbroken
, 1 to practiseReading the derivative from a graph
How a computer estimates a derivative
, 1 to practiseFinite differences as code
, 4 to practisePractice problems
Recap

, 1 to practiseThe power rule
, 2 to practiseSums and constant multiples
, 1 to practiseDifferentiating a polynomial in code
The number e makes the exponential its own derivative
, 1 to practiseThe derivative of ln x
The derivatives of sine and cosine
, 1 to practiseThe product rule
, 2 to practiseThe quotient rule
, 1 to practiseHigher derivatives
, 1 to practisePosition, velocity and acceleration
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseFunctions inside functions
The chain rule: rates multiply
, 2 to practiseThe chain rule in practice
, 1 to practiseThe mistake everyone makes once
, 1 to practiseThe chain rule as code
, 1 to practiseImplicit differentiation
, 1 to practiseImplicit differentiation, step by step
, 1 to practiseDerivatives of inverse functions
, 1 to practiseLogarithmic differentiation
, 1 to practiseDerivatives of the inverse trig functions
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

The sign of the derivative says rising or falling
, 1 to practiseCritical points are where the slope is zero or undefined
, 2 to practiseThe first derivative test
The second derivative says which way it bends
, 1 to practiseThe second derivative test
, 2 to practiseThe mean value theorem
, 1 to practiseSketching a curve from its derivatives
, 1 to practiseThe biggest value on a closed interval
, 1 to practiseOptimisation: finding the best design
, 1 to practiseFinding a maximum by computer
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseNear a point, a curve is almost its tangent
, 1 to practiseApproximations worth knowing by heart
, 1 to practiseA linearisation as code
, 1 to practiseDifferentials: small changes in, small changes out
, 1 to practiseHow big is the error?
, 1 to practiseRelated rates
, 1 to practiseThe sliding ladder
Newton’s method rides tangents to a root
, 1 to practiseNewton’s method doubles the digits each step
, 1 to practiseWhen Newton’s method fails
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

The area problem
Approximating with rectangles
, 2 to practiseSigma notation for long sums
, 2 to practiseRiemann sums in general
The definite integral is the limit of the sums
, 1 to practiseArea below the axis counts as negative
, 1 to practiseProperties of the integral
, 2 to practiseDisplacement and distance from velocity
Which Riemann sum is most accurate
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

An area that grows as you move the end
, 1 to practisePart 1: the area’s rate of growth is the height
, 2 to practisePart 1 with the chain rule
, 1 to practiseAntiderivatives, and the constant C
, 1 to practiseA table of antiderivatives
Part 2: evaluate an integral with any antiderivative
, 1 to practiseWorking with indefinite integrals
, 3 to practiseEvaluating a definite integral without slips
, 1 to practiseThe net change theorem
, 2 to practiseBoth parts of the theorem, in code
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseEvery derivative rule runs backwards
, 2 to practiseSubstitution undoes the chain rule
, 2 to practiseWith a definite integral, change the limits too
, 2 to practiseIntegration by parts undoes the product rule
, 1 to practiseTrig identities turn powers into things you can integrate
, 1 to practisePartial fractions split a fraction into easy pieces
, 2 to practiseTry it: cover-up and checking by differentiating
, 2 to practiseA strategy for choosing a method
, 1 to practiseSome integrals have no formula at all
, 4 to practisePractice problems
Recap

Slice it thin, then add the slices
, 2 to practiseThe area between two curves is top minus bottom
, 1 to practiseWhen the curves cross, split the integral
, 2 to practiseVolumes by slicing: disks
, 1 to practiseWashers have a hole in the middle
, 1 to practiseShells: slicing parallel to the axis
, 2 to practiseThe average value of a function
, 1 to practiseThe length of a curve
, 1 to practiseWork done by a varying force
, 1 to practiseA preview: probability as area
, 3 to practisePractice problems
, 3 to practiseMore practice problems
Recap

, 1 to practiseComputers integrate by sampling
, 2 to practiseThe trapezoid rule joins the samples with straight lines
, 3 to practiseThe midpoint rule samples the middle of each piece
, 3 to practiseSimpson's rule fits parabolas
, 1 to practiseError and order: what halving the step buys
Watch the errors shrink
, 1 to practiseAdaptive methods spend samples where they are needed
, 1 to practiseMonte Carlo: integrate by random sampling
, 4 to practisePractice problems
Recap

, 1 to practiseSome limits cannot be read off
, 3 to practiseL'Hôpital's rule compares the speeds
, 2 to practiseRewrite the other forms as a fraction
, 1 to practiseLogs are slow, powers are steady, exponentials win
, 2 to practiseAn integral over an infinite interval is a limit
, 1 to practiseThe p-integrals: the yardstick for convergence
, 2 to practiseIntegrands that blow up are improper too
, 2 to practiseCompare with something you know
, 1 to practiseGabriel's horn: finite volume, infinite surface
, 3 to practisePractice problems
Recap

, 1 to practiseA sequence is a list that goes on forever
, 2 to practiseA series is the limit of its partial sums
, 2 to practiseGeometric series: the one you can always sum
, 1 to practiseIf the terms do not shrink to zero, the series diverges
, 2 to practiseThe integral test, and the p-series
, 1 to practiseComparison tests: measure against a known series
, 1 to practiseRatio and root tests: is it eventually geometric?
, 2 to practiseAlternating series converge if the terms shrink
, 2 to practiseAbsolute and conditional convergence
, 4 to practisePractice problems
Recap

, 1 to practiseA power series is a polynomial that never ends
, 1 to practiseEvery power series has a radius of convergence
, 2 to practiseTaylor series: match every derivative
, 1 to practiseFive series worth knowing by heart
, 3 to practiseBuild new series from old ones
, 2 to practiseTaylor's remainder: how much to trust a polynomial
, 1 to practiseHow a calculator computes sin
, 2 to practiseEuler's formula: e to an imaginary power
, 1 to practiseWhen a Taylor series lets you down
, 3 to practisePractice problems
Recap

, 1 to practiseAn equation whose unknown is a function
, 1 to practiseA slope field shows every solution at once
, 2 to practiseSeparable equations: split the variables and integrate
, 2 to practiseExponential growth and decay
, 1 to practiseLogistic growth levels off
, 2 to practiseNewton's law of cooling
, 1 to practiseEuler's method: follow the slope a step at a time
, 1 to practiseSecond order: a mass on a spring
, 1 to practiseSystems: several quantities changing together
, 4 to practisePractice problems
Recap

, 1 to practiseA function of two variables is a surface
, 1 to practiseLevel curves: the contour map of a surface
, 2 to practiseA partial derivative changes one variable at a time
, 1 to practiseThe gradient points straight uphill
, 2 to practiseDirectional derivatives: the slope in any direction
, 2 to practiseThe tangent plane is the best flat approximation
, 1 to practiseCritical points: minima, maxima and saddles
, 1 to practiseLagrange multipliers: the best point on a constraint
, 1 to practiseGradient descent: walk downhill
, 3 to practisePractice problems
Recap

, 2 to practiseA double integral adds up over a region
, 2 to practiseOver a curved region, the inner limits are functions
, 1 to practiseTriple integrals add up over a solid
, 2 to practisePolar coordinates: dA becomes r dr dθ
, 1 to practiseThe Jacobian: how a change of variables stretches area
, 1 to practiseA vector field puts an arrow at every point
, 1 to practiseLine integrals add up a field along a path
, 1 to practiseDivergence measures spreading, curl measures spinning
, 1 to practiseGreen's theorem: the boundary knows what is inside
, 1 to practiseStokes' and the divergence theorem: one idea in every dimension
, 4 to practisePractice problems
Recap

, 2 to practiseWhat a limit really means
, 2 to practiseThe order of the quantifiers is the whole point
, 1 to practiseThe real numbers have no gaps
, 2 to practiseA proof that is also an algorithm
, 1 to practiseUniform continuity: one δ for the whole interval
, 1 to practiseThe chain of theorems behind the fundamental theorem
, 1 to practiseBeyond Riemann: measure and the Lebesgue integral
, 1 to practiseCalculus of variations: the best function
, 2 to practiseAutomatic differentiation: the chain rule as code
, 4 to practiseProblems that mix everything
, 3 to practisePractice problems that mix everything
, 2 to practiseMore practice problems that mix everything
Recap, and where to go next