Calculus Midterm Preparation Checklist
Theory – Key Definitions
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Riemann sums: Σ f(x_i*) ∆x — approximation of area under a curve using rectangles.
Definite integral: ∫_a^b f(x) dx — net signed area under the curve from a to b.
Antiderivative: F'(x) = f(x).
Indefinite integral: ∫ f(x) dx = F(x) + C.
Relationship via FTC: ∫_a^b f(x) dx = F(b) − F(a).
Key Theorems from Calculus I
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Definition of derivative: f'(x) = lim_{h→0} (f(x+h) − f(x)) / h.
Mean Value Theorem: f'(c) = (f(b) − f(a)) / (b − a).
Intermediate Value Theorem: continuous functions take all values between f(a) and f(b).
Squeeze Theorem: if g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L then lim f = L.
Know standard derivative formulas (power, trig, exponential, logarithmic).
Proofs to Know
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Antiderivatives are unique up to a constant.
FTC Part 1 implies FTC Part 2.
Techniques of Integration
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u■substitution (reverse chain rule).
Integration by parts: ∫u dv = uv − ∫v du.
Trig integrals using identities.
Trig substitution for radicals (x = a sinθ, a tanθ, a secθ).
Important Tricks
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Integral of sec(x): ∫ sec x dx = ln|sec x + tan x| + C.
Long division before integrating rational functions.
Completing the square for quadratic denominators.
Boomerang integrals like ∫ e^x sin x dx requiring integration by parts twice.