Calc 1 Basic Definitions and Theorems

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Limit

The limit of f(x) as x approaches a is L

Continuity

f is continuous at x=c iff lim_(x→c)f(x)=f(c)

Epsilon-delta

lim_(x→a)f(x)=L iff given E<0, then there is a d<0 such that |d(x)-L|<E whenever 0<|x-a|<d

Intermediate Value Theorem (IVT)

Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b) where f(a)≠f(b). Then there exists a number c in (a,b) such that f(c)=N

Derivative of a function

lim_(h→0)[f(x+h)-f(x)]/h

Average velocity

displacement/time - [f(a+h)-f(a)]/h

Extreme Value Theorem (EVT)

If f is continuous on [a,b] then f has an absolute max and an absolute min on [a,b]

Fermat’s Theorem

If f has a local min at x=c, and f’(c) exists the f’(c)=0 exists

Rolle’s Theorem

Let f be continuous on [a,b] and differentiable on (a,b). Then if f(a)=f(b), there is a number C ∈(a,b) such that f’(c)=0

Mean Value Theorem (MVT)

Let f be continuous on [a,b] and differentiable on (a,b). Then there is a number c ∈(a,b), such that f’(c)=[f(b)-f(a)]/[b-a]

FTC 1

Suppose f is continuous on [a,b]. If g(x)=∫_a^x f(t)dt, then g’(x)=f(x)

FTC 2

∫_a^b f(x)dx=F(b)-F(a), where F is any antiderivative of f, that is F’=f

FTC 1 → FTC 2

Assume d/dx∫_a^x f(t)dt=f(x). Then ∫_a^x f(t)dt=F(x)+C. Let x=a: ∫_a^a f(t)dt=0=F(a)+C. Isolate C: C=-F(a). Let x=b: ∫_a^b f(t)dt=F(b)-F(a). And so ∫_a^b f(x)dx=F(b)-F(a)

FTC 2 → FTC 1

Assume ∫_a^b f(x)dx=F(b)-F(a). Then d/dx∫_a^x f(t)dt=d/dx[F(x)-F(a)] → (d/dx)F(x)-0=f(x)