Calc Test 1

definitions and thereoms

Limits

1. If the left limit exists

2. If the right limit exists

3. Both y-values are equal

even if hole if both sides approach the same value, it exists

Continuity

1. Must be defined at entire interval [a,b]

2. limit must exist

3. point must be equal to limit

also passes vertical line test

Differentiability

1. the limit exists for every x value in the interval

2. non-differential at a sharp point (cusp)

Immediate value theorem

Summary: if a function is continuous over a domain [a,b] and f(a) =! f(b) then there is a value c for N. Basically any y value between f(a) and f(b) should have an “x” corresponding to it.

Limit definition for derivative

The limit definition for the derivative of a function f(x) at a point x=a is given by:

f'(a) = lim(h->0) [f(a+h) - f(a)] / h

where h represents a small change in x. This formula calculates the instantaneous rate of change of the function at the point x=a.

For these questions test each one