calc chapter 1

algebraically find a derivative

f(x) minus f(c ) divided by x - c

algebraically find a limit

1. plug in

2. simplify then plug in

3. if infinity, use SLO LSN properties

how to solve for an exact IVT value

set equal to y value and solve for the answer in the domain

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

solve for delta in terms of epsilon

how to solve for delta in terms of epsilon

just plug in knowns like L and f(x) and solve until you isolate x, keeping c in mind to get it into the form c minus delta < x < c plus delta

Epsilon-Delta definition of a limit

the limit of f(x) as x approaches c equals L if and only if, for every epsilon greater than 0, there exists a delta greater than 0, such that if x is within delta units of c, but x does not equal c, then f(x) is within epsilon units of L

make sure u can do piecewise limits

limit theorems

1. limit of a constant

2. limit of an identity

3. limit of a sum

4. limit of a product

5. limit of a constant times a function

intermediate value theorem

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

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

derivative definition

the limit of f(x) minus f( c ) divided by x - c as x approaches c