piatra . institutepathwaysFinding the next step…
Connect limits, derivatives, integrals and differential equations through worked problems and visual models.
No subject-specific background required; work through the opening foundation.
A derivation and numerical model of a changing quantity, with an error estimate.
Gilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingCalculus: functions. Model a quantity and specify its domain.
Gilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingCalculus: the limit of a function. Distinguish the limit from the function value.
Calculus: limit laws. State the hypotheses behind each calculation.
Calculus: defining the derivative. Derive a difference quotient.
Position is s(t) = t² + 3 metres, with t in seconds. What is the instantaneous velocity at t = 3, and why does the constant 3 disappear?
Differentiate the position before substituting the time.
s′(t) = 2t, so s′(3) = 6 metres per second. Adding a fixed 3 metres changes the starting position but does not change how position varies with time; its derivative is zero.
What to look for
Common mistake: Substituting t = 3 into position and calling the result a velocity.
Start this pathway to keep your answers in My learning.
Gilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingCalculus: differentiation rules. Explain each rule through a simple example.
Calculus: the chain rule. Trace inner and outer functions.
Calculus: approximating areas. Compare left, right and midpoint sums.
Calculus: the definite integral. Interpret signed area and accumulation.
Velocity is v(t) = 2t metres per second from t = 0 to 3. Compare three right-endpoint rectangles of width 1 second with the exact displacement.
Use velocities at t = 1, 2 and 3 for the rectangles, then integrate.
The rectangles give 1 × (2 + 4 + 6) = 12 metres. The integral gives ∫₀³ 2t dt = [t²]₀³ = 9 metres. The right-endpoint estimate is larger because velocity increases throughout each interval.
What to look for
Common mistake: Adding velocities without multiplying by the time width, or treating an approximation as exact.
Start this pathway to keep your answers in My learning.
Gilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingGilbert Strang · Edwin Herman · reference
Named section and worked examples
Check reading access
Open the readingCalculus: precise definition of a limit. Write an epsilon–delta argument.
Calculus: the mean value theorem. Explain why continuity and differentiability matter.
Calculus: fundamental theorem. Connect accumulation with local change.
The function f(x) = |x| is continuous at 0. Does it have a derivative there? Use the left and right difference quotients.
Examine |h| / h for positive and negative h.
No. At x = 0 the difference quotient is |h| / h. It is 1 for h > 0 and −1 for h < 0, so the one-sided limits disagree. Continuity means the function value has no jump; it does not guarantee a unique tangent slope.
What to look for
Common mistake: Assuming that every continuous function is differentiable.
Start this pathway to keep your answers in My learning.
Editorial perspectives based on selected works, rather than author-endorsed reading lists.
An editorial comparison of the core argument, alternative explanations and practical evidence.
A curated synthesis cannot be neutral or exhaustive. Follow the primary sources when an interpretation matters.
An editorial route that starts with working examples.
The order supports a particular study method; it does not establish which account is true.
An editorial route that starts with competing explanations.
The order supports a particular study method; it does not establish which account is true.
Background, different viewpoints, and further reading.
Model a quantity and specify its domain.
Named section and worked examples
Compare growth rates and shapes.
Named section and worked examples
Check when an inverse is defined.
Named section and worked examples
Translate between multiplicative growth and logarithms.
Named section and worked examples
Connect a secant slope with a local rate.
Named section and worked examples
Distinguish the limit from the function value.
Named section and worked examples
State the hypotheses behind each calculation.
Named section and worked examples
Find a discontinuity and classify it.
Named section and worked examples
Write an epsilon–delta argument.
Named section and worked examples
Derive a difference quotient.
Named section and worked examples
Explain each rule through a simple example.
Named section and worked examples
Trace inner and outer functions.
Named section and worked examples
Differentiate a relation rather than a solved function.
Named section and worked examples
Keep units consistent in a changing geometry.
Named section and worked examples
Separate local and global conclusions.
Named section and worked examples
Explain why continuity and differentiability matter.
Named section and worked examples
Translate constraints into a feasible domain.
Named section and worked examples
Compare left, right and midpoint sums.
Named section and worked examples
Interpret signed area and accumulation.
Named section and worked examples
Connect accumulation with local change.
Named section and worked examples
Use derivatives and gradients to explain how a training update changes the model.
Use continuous-change models for the optional advanced dynamics readings; algebra is enough to begin.
Use derivatives and change models if you pursue the optional reaction–diffusion modelling.
Use marginal change for the later mathematical selections; the historical arguments can be read directly.
Use calculus for advanced monetary models after learning the stock-and-flow vocabulary.
Derivatives prepare part of the mathematical vocabulary; variational reasoning and differential equations need further background.
Compare geometric models with explanations of continuous change.