Final Exam (other theorems)

Suppose f:E→R is uniformly continuous and E is bounded, then

f(E) is bounded

Let f:E→R be continuous with E compact, then

f(E) compact

Suppose f:E→R is continuous with E compact, then there are

x_1, x_2 in E such that for all x in E, f(x_1) <= f(x) <= f(x_2)

Suppose f:E→R is continuous and 1-1 with E compact, then

f^-1:f(E)→E is continuous

If f:[a,b]→R is continuous. Then, there are

c and d such that f([a,b]) = [c,d]

Suppose f:A→R is continuous and 1-1 with A connected. Then,

f is monotone

Suppose f:D→R, x_o in D, and x_o is an accumulation point of D. Then f is differentiable at x_o if and only if

for every sequence {x_n}n in J of points of D\{x_o} converging to x_o, the sequence {f(x_n)-f(x_o)/x_n-x_o} n in J converges

Let f:D→R be differentiable at x_o, then

f is continuous at x_o

Chain Rule

Let f: D→R and g:E→R with f(D) c E. If f is differentiable at x_o and g is differentiable at f(x_o), then gof is differentiable at x_o and (gof)’(x_o) = g’(f(x_o))f’(x_o)

Suppose f:[a,b]→R and suppose f has either a relative max or relative min at x_o in (a,b). If f is differentiable at x_o, then

f’(x_o) = 0

Suppose that f and g are continuous on [a,b] and differentiable on (a,b) and that f’(x) = g’(x) for all x in (a,b). Then,

there is a real number k such that f(x) = g(x) + k for all x in [a,b]

Suppose f is differentiable on [a,b] and r is a real number such that f’(a) < r < f’(b) or f’(b) < r < f’(a). Then,

there is a c in (a,b) such that f’( c) = r

(L’Hospital’s Rule) Suppose f and g are continuous on [a,b] and differentiable on (a,b). If x_o in [a,b],

g’(x) /= 0 for all x in (a,b), f(x_o) = g(x_o) = 0, f’/g’ has a limit at x_o, then f/g has a limit at x_o and lim x→x_o f/g(x) = lim x→x_o f’/g’(x)

Suppose f:[a,b]→R is continuous and differentiable with f’(x) /= 0 for all x in [a,b]. Then,

f is 1-1, f^-1 is continuous and differentiable on f([a,b]), and (f^-1)’(f(x)) = 1/f’(x)