| Theorem | Required Conditions | Guaranteed Result | Formula / Definition |
|---|---|---|---|
| IVT (Intermediate Value) | f is continuous on [a, b] | Takes all values between f(a) and f(b) | ∃ c ∈ (a, b) such that f(c) = k |
| MVT (Mean Value) | Continuous on [a, b], Differentiable on (a, b) | Instantaneous rate equals average rate | f'(c) = [f(b) - f(a)] / (b - a) |
| EVT (Extreme Value) | f is continuous on closed interval [a, b] | Attains absolute maximum and minimum | Test critical points & endpoints |
| FTC Part 1 & 2 | f is continuous on [a, b] | Derivative of accumulator & Net Change | d/dx ∫[a to x] f(t)dt = f(x) |
| Rule Name | Standard Differentiation | Chain Rule Version |
|---|---|---|
| Product Rule | (fg)' = f'g + fg' | Keep first × d(second) + keep second × d(first) |
| Quotient Rule | (f/g)' = (f'g - fg') / g² | "Low d-High minus High d-Low over Low squared" |
| Trigonometry | d/dx sin(x) = cos(x), d/dx tan(x) = sec²(x) | d/dx cos(x) = -sin(x), d/dx sec(x) = sec(x)tan(x) |
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Key Conceptual Scaffolding & Diagnostic Intuition
Tested Competency: Derivative theorems, Riemann sums & Taylor series. In modern competitive exams, questions are engineered to test whether you understand the underlying structural relationship—not just whether you memorized a formula.
Forgetting the chain rule on composite functions or mixing up Average Rate of Change with Instantaneous Rate of Change.
Always write out your hypotheses and state conditions explicitly before calculating test statistics. Graders award up to 40% of partial credit for proper setup, regardless of arithmetic precision.
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