chapter 1 and 2 memory check

Numberical interpretation of derivative

Limit of the average rate of change over the interval from c to x as x approaches c

Geometrical Interpretation of Derivative

Slope of the tangent line

Meaning of derivative

Instantaneous Rate of Change

Meaning of Definite Integral

Product of (b - a) and f(x)

Geometrical Interpretation of Definite Integral

Area under the curve between a and b

Verbal Definition of Limit

L is the limit of f(x) as x approaches c if and only if for any positive number epsilon, no matter how small, there is a positive number delta such that if x is within delta units of c (but not equal to c), then f(x) is within epsilon units of L.

Limit of a Product of Functions

lim x-> c [f(x) * g(x)]= lim x->c f(x) • lim x->c g(x)
The limit of a product equals the product of the limits

Limit of a Sum of Functions

lim x-> c [.f(x) + g(x)] = lim x-> c f(x) + lim x-> c g(x) The limit of a sum equals the sum of the limits

Limit of a Quotient of Functions

lim(x→c) [f(x)/g(x)] = lim(x→c) f(x) / lim(x→c) g(x)
where lim(x→c) g(x) ≠ 0
The limit of a quotient equals the quotient of the limits

Limit of a Constant Times a Function

lim x->c [k * f(x)] = k• lim x->c f(x) The limit of a constant times a function equals the constant times the limit.

Limit of the Identity Function

lim(x→c) [x] = c
The limit of x as x approaches c is c

Limit of a Constant Function

If k is a constant, then lim x-> c k = k The limit of a constant is the constant.

Property of Equal Left and Right Limits

lim x->c f(x) exists if and only if lim x->c- f(x) = lim x->c+ f(x)

Definition of Continuity at a Point

its continuous at x = c if and only if
1. f(c) exists
2. lim f(x) exists
x->c
3. lim f(x) = f(c)
x->c

Horizontal Asymptote

If lim x-> Infinity f(x) = L or lim x-> -Infinity f(x) = L, then the line y = L is a horizontal asymptote.

Vertical Asymptote

If lim x-> c f(x) = ∞ or lim x-> c f(x) = -Infinity , then the line × = c is a vertical asymptote.

Intermediate Value Theorem

If f is continuous for all x in the closed interval [a, b], and y is a number between f(a) and f(b), then there is a number c in the open interval (a, b) for which f(c) = y.