calc chapter 2

epsilon-delta definition of a limit

The limit of f(x) as x approaches c is L if and only if, for every epsilon greater than zero, there exists a delta greater than zero, such that if x is within delta units of c, but does not equal c, then f(x) is within epsilon units of the limit.

limit theorems

1. limit of a constant

2. limit of an identity function

3. limit of a sum

4. limit of a product

5. limit of a constant times a function

limit of a constant

lim x —> c f(x) = k where f(x) = k

limit of an identity function

lim x —> c f(x) = c where f(x) = x

limit of a sum

lim x —> c (f(x) + g(x)) = (lim x —> c f(x)) + (lim x —> c g(x))

limit of a product

lim x —> c (f(x) * g(x)) = (lim x —> c f(x)) * (lim x —> c g(x))

limit of a constant times a function

lim x —> c (k * f(x)) = k * lim x —> c (f(x))

definition of continuity of a point

f(x) is continuous at x = c if and only if f(c ) exists and the limit of f(x) as x approaches c exists and f(c ) equals the limit of f(x) as x approaches c.

when does a limit exist

if lim of f(x) as x approaches c from the left = lim f(x) as x approaches c from the right, then lim f(x) as x approaches c exists

rational algabraeic theorem

if the degree of the numerator equals the degree of the denominator, the limit exists and equals a specific value

intermediate value theorem (IVT)

If f(x) is continuous on the closed interval [a, b] and y is between f(a) and f(b), then there exists at least 1 value, x = c, on the open interval (a, b) such that y = f(c ).

extreme value theorem (EVT)

if f(x) is continuous on the closed interval [a, b], then there exists points x = c and x = d, where c and d are greater than or equal to a and less than or equal to b, such that f(c ) and f(d) are the absolute max and absolute min.

how to find derivative algebraically

(f(x) - f(c )) / (x-c) where (c, f(c )) is the point you’re finding the derivative of and f(x) is the function

how to find a limit algebraically

plug in, if that doesn’t work, simplify and plug in, and if its to infinity, use SLO LSN properties

how to solve for X in IVT

set function equal to y and solve for the x’s int he domain

what does it mean show that delta is positive for any epsilon

solve for delta in terms of epsilon

definition of derivative

f’(c ) = lim f(x) - f(c ) over x-c as x approaches c

define normal line

line that passes through point of tangency, P(c, f(c )), and is perpendicular to the tangent line