# Theorems BC Sem 1

## 12 terms

### Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and f(a)<N<f(b) with f(a)≠f(b) then there is a c S.T. a≤c≤b with f(c) = N

### Extreme Value Theorem

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

### Mean Value Theorem

If f is a function S.T.
1. f is cont on [a,b]
2. f is differentiable on (a,b)
then there is a c∈(a,b) S.T. f'(c) = (f(b)-f(a))/(b-a)

### Mean Value Theorem for Integrals

If f is cont on [a,b] then there exists a c∈[a,b] S.T. f(c) = favg = 1/(b-a) ∫(a,b) f(x)dx

### Fermat's Theorem

If f has local extrema at x=c, and f'(c) exists then f'(c)=0

### Rolle's Theorem

If f is a function S.T.
1. f is cont. on [a,b]
2. f is differentaible on (a,b)
3. f(a)=f(b)
then there is a c∈(a,b) S.T. f'(c)=0

### Fundamental Theorem of Calculus Part 1

If g(x) = ∫(a,x) f(t)dt where g(x) is continuos on [a,b] and differentiable on (a,b) then g'(x) = f(x)

### Fundamental Theorem of Calculus Part 2

If f is continuous on [a,b] then
∫(a,b) f(x)dx = F(b) - F(a) where F'(x)=f(x)

### L'Hôpital's Rule

if f and g are differentiable on an open interval I that contains a except at a and lim(x->a) f(x)/g(x)= 0/±∞ then lim(x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x)

### Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a, and lim (x->a) f(x) = lim (x->a) h(x) = L then lim (x->a) g(x) = L

### First Derivative Test

Given f is a continuous function
If f'(c)=0 or DNE then:
1. If f'(x) switches from + to - at x=c, f(c) is a local max
2. If f'(x) switches from - to + at x=c, f(c) is a local min
3. If f'(x) goes + to + or - to - or else it's a saddle point

### Second Derivative Test

If f''(x) is continuous near x=c
f''(x) exists ->f' diff -> f' cont -> f diff -> f cont
and f'(c) = 0 then
1. If f''(c)>0 then f(c) is local min
2. If f''(c)<0 then f(c) is local max
3. If f''(c) = 0 then the test fails and use 1st derivative test