theorems

Question 1

Continuity

Answer 1

Let f be defined at x=c and in an open interval containing x=c

Question 2

Conditions for continutity

Answer 2

1. The limit of f(x) as x approaches c is equal to f(c)
2. f(c) needs to be defined

Question 3

Intermediate Value Theorem

Answer 3

Let f(x) be continuous on [a,b], let g be a number between f(a) and f(b). Then, there is at least one a less than or equal to x and x is less than or equal to b such that f(x)=g

Question 4

Squeeze Theorem

Answer 4

if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them

Question 5

Existance of Zeroes Corollary

Answer 5

Let f(x) be continuous on [a,b]. Suppose f(a)f(b) ≤ 0, then there exists at least one a≤x≤b such that f(x) = 0

Question 6

Product Rule Theorem

Answer 6

Let f and g be differentiable functions. Then (fg)’ = f’g+fg’

Question 7

Quotient Rule Theorem

Answer 7

Let f and g be differentiable functions. (f/g)’ = (f’g-fg’)/g²

Question 8

Chain Rule Theorem

Answer 8

Assume h(x) and g(x) are differentiable fuctions. Then d/dx(h(g(x))= h’(g(x))g’(x)

Question 9

General Power Rule Theorem

Answer 9

Assume g(x) is differentiable and a natural number. The, d/dx(g(x)ⁿ) = ng(x)^n-1 dy/dx

Question 10

Extreme Value Theorem (EVT)

Answer 10

Assume f(x) is continuous on [a,b]. Then, f attains both its max and min in [a,b].

Question 11

Local Extrema Definition

Answer 11

• Local min at x=c if f(c) is the minimum value of on some open interval containing c
• local max at x=c f(c) is the maximum value of on some open interval containing c

Question 12

Critical point definition

Answer 12

c in the domain is called a critical point if either f’(c)=0 or f’(c) does not exist

Question 13

Fermat’s Theorem for Local Extrema

Answer 13

Assume f(a) is a local max/min. Then x=a is a critical point of f

Question 14

Extreme Values on a Closed Interval Theorem

Answer 14

Assume f is continuous on [a,b] and f(c) is the min/max. Then x=c is either a critical point or endpoint.

Question 15

Rolle’s Theorem

Answer 15

Assume f is continuous on [a,b] and differentiable on (a,b) where a=b. Then there exists a point c such that a< c< b and f’(c)=0

Question 16

Mean Value Theorem

Answer 16

Assume f is continuous and differentiable on [a,b]. Then, there exists some point c that a< c< b and f’(c)= f(b)-f(a)/b-a

Question 17

Mean Value Theorem Corollary

Answer 17

If f (x)=0 for all x ∈ (a, b), then f is constant on the interval (a, b)