Study Guide & Educational Overview
### Calculus Rules & Theorems:
* **Derivatives**: The Power Rule (d/dx[x^n] = n*x^(n-1)), Product Rule (u'v + uv'), Quotient Rule ((u'v - uv')/v^2), and Chain Rule (f'(g(x))*g'(x)).
* **Special Functions**: d/dx[e^x] = e^x, d/dx[ln(x)] = 1/x, d/dx[sin(x)] = cos(x), and d/dx[cos(x)] = -sin(x).
* **The Fundamental Theorem of Calculus**: Connects differentiation and integration; the definite integral of a rate of change equals the net change in the anti-derivative: ∫_a^b f(x) dx = F(b) - F(a).
* **Optimization & Curve Sketching**: Critical points occur where f'(x) = 0 or does not exist; inflection points mark changes in concavity where f''(x) changes sign.
* **Derivatives**: The Power Rule (d/dx[x^n] = n*x^(n-1)), Product Rule (u'v + uv'), Quotient Rule ((u'v - uv')/v^2), and Chain Rule (f'(g(x))*g'(x)).
* **Special Functions**: d/dx[e^x] = e^x, d/dx[ln(x)] = 1/x, d/dx[sin(x)] = cos(x), and d/dx[cos(x)] = -sin(x).
* **The Fundamental Theorem of Calculus**: Connects differentiation and integration; the definite integral of a rate of change equals the net change in the anti-derivative: ∫_a^b f(x) dx = F(b) - F(a).
* **Optimization & Curve Sketching**: Critical points occur where f'(x) = 0 or does not exist; inflection points mark changes in concavity where f''(x) changes sign.
Frequently Asked Questions
L'Hôpital's Rule applies when evaluating limits that yield indeterminate forms of 0/0 or ±∞/±∞, differentiating numerator and denominator separately.
The definite integral represents the net signed area bounded between the function curve and the horizontal x-axis over an interval [a, b].
