l(x) ≤ f(x) ≤ u(x) lim l(x) = L and lim u(x) = L then lim f(x) = L

Formal Definition of a Limit

For every ε > 0, we can find a δ > 0 such that if 0 < | x - a | < δ, then | f(x) - L | < ε

Intermediate Value Theorem

f is continuous on the interval [a,b] and f(a) ≠ f(b), for any number 'k' between f(a) and f(b), there is at least one c∈(a,b) such that f(c) = k

Extreme Value Theorem

f is continuous on the interval [a,b] then there exists values 'm' and 'M' such that f(m) ≤ f(x) ≤ f(M) for every x∈[a,b]

Differentiability => Continuity

if f is differentiable at x=a, then f is continuous at x=a

if f is continuous at x=c, differentiable at x=c, and has a local max. or min. at x=c, then f has a horizontal tangent at x=c

a,b∈ℝ with a<b. if f is continuous on [a,b], is differentiable on (a,b), and satisfies f(a) = f(b), then there exists at least one c∈(a,b) such that f'(c)=0

a,b∈ℝ with a<b. if f is continuous on [a,b], and differentiable on (a,b), then there exists at least one value c∈(a,b) such that f'(c)=f'(b)-f'(a)/b-a

quotient if the remainder approaches 0 as x → ±∞

Fundamental Theorem of Calculus I

f(x) is continuous on [a,b], and A(x) = ₐ ∫ˣ f(t)dt. then A(x) is differentiable and A'(x) = f(x) for all x∈[a,b]

Fundamental Theorem of Calculus II

f(x) is continuous on [a,b], and F(x) is any antiderivative of f. Then ₐ ∫ᵇ f(x)dx = F(b) - F(a)

Change of Variables Theorem

Let u be a differentiable function whose range lies within the domain of a continuous function f. Then ∫f'(u(x)) (du/dx) dx = ∫f'(u(x)) du

Mean Value Theorem for Integrals

f is continuous on [a,b], then there exists at least one c∈(a,b) such that f(c)(b-a) = ₐ ∫ᵇ f(x)dx

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MATH1200

Calculus I

Term	Definition
sin^2(θ) + cos^2(θ) =	1
1 + cot^2(θ) =	csc^2(θ)
1 + tan^2(θ) =	sec^2(θ)
The Squeeze Theorem	l(x) ≤ f(x) ≤ u(x) lim l(x) = L and lim u(x) = L then lim f(x) = L
Formal Definition of a Limit	For every ε > 0, we can find a δ > 0 such that if 0 < \| x - a \| < δ, then \| f(x) - L \| < ε
Intermediate Value Theorem	f is continuous on the interval [a,b] and f(a) ≠ f(b), for any number 'k' between f(a) and f(b), there is at least one c∈(a,b) such that f(c) = k
Extreme Value Theorem	f is continuous on the interval [a,b] then there exists values 'm' and 'M' such that f(m) ≤ f(x) ≤ f(M) for every x∈[a,b]
Differentiability => Continuity	if f is differentiable at x=a, then f is continuous at x=a
Fermat's Theorem	if f is continuous at x=c, differentiable at x=c, and has a local max. or min. at x=c, then f has a horizontal tangent at x=c
Rolle's Theorem	a,b∈ℝ with a<b. if f is continuous on [a,b], is differentiable on (a,b), and satisfies f(a) = f(b), then there exists at least one c∈(a,b) such that f'(c)=0
Mean Value Theorem	a,b∈ℝ with a<b. if f is continuous on [a,b], and differentiable on (a,b), then there exists at least one value c∈(a,b) such that f'(c)=f'(b)-f'(a)/b-a
Slant Asymptote	quotient if the remainder approaches 0 as x → ±∞
Fundamental Theorem of Calculus I	f(x) is continuous on [a,b], and A(x) = ₐ ∫ˣ f(t)dt. then A(x) is differentiable and A'(x) = f(x) for all x∈[a,b]
Fundamental Theorem of Calculus II	f(x) is continuous on [a,b], and F(x) is any antiderivative of f. Then ₐ ∫ᵇ f(x)dx = F(b) - F(a)
Change of Variables Theorem	Let u be a differentiable function whose range lies within the domain of a continuous function f. Then ∫f'(u(x)) (du/dx) dx = ∫f'(u(x)) du
Mean Value Theorem for Integrals	f is continuous on [a,b], then there exists at least one c∈(a,b) such that f(c)(b-a) = ₐ ∫ᵇ f(x)dx

Created by: user-2005776

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