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Calculus Ideas
YGK These Ideas From Calculus
| Question | Answer |
|---|---|
| Value a function approaches as input approaches a point | Limit |
| Formal definition of a limit using Greek letters | Epsilon-delta |
| Mathematician who developed the epsilon-delta definition | Augustin-Louis Cauchy |
| Ability to draw a graph without lifting the pen | Continuity |
| Requirement for continuity at a point | Limit exists and equals function value |
| Operation giving a function's rate of change | Derivative (differentiation) |
| Derivative’s value at a specific point | Slope of the tangent line |
| Lagrange’s notation for a derivative | f′(x) |
| Leibniz notation for a derivative | df/dx |
| Function property required for a derivative to exist | Differentiability |
| Stronger property: differentiability or continuity? | Differentiability |
| A function that can be differentiated infinitely many times | Smooth |
| Example of a continuous but non-differentiable function at x=0 | Absolute value |
| Operation giving the signed area under a curve | Definite integration |
| Integral symbol origin | Long S (∫) |
| Approximating area under a curve using rectangles | Riemann sums |
| Concept relating integration and differentiation as opposites | Fundamental Theorem of Calculus |
| Function whose derivative is the original function | Antiderivative |
| Rule for differentiating composite functions like f(g(x)) | Chain rule |
| Integration rule derived from the chain rule | Substitution |
| Rule for differentiating the product of two functions | Product rule |
| Integration rule derived from the product rule | Integration by parts |
| Rule for differentiating one function divided by another | Quotient rule |
| Infinite sum of monomials used to approximate a function | Taylor series |
| Taylor series centered at x = 0 | Maclaurin series |
| Theorem giving a bound on Taylor series truncation error | Taylor’s theorem |
| Equations relating a function to its derivatives | Differential equations |
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