Calc AB Memory Check: Limits

Calc limits

Derivative

Instantaneous rate of change

Numerical Interpretation of dervative

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

Definite integral

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

Geometrical interpretation of definite integral

Area under the curve between a and b

Limit of a Product of Functions

lim_x→c[f(x)* g(x)] = lim_x→cf(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→cf(x) + lim_x→cg(x)

The limit of a sum equals the sum of the limits

Limit of a Quotient of functions

lim_x→cf(x)/ lim_x→cg(x) = lim_x→cf(x)/ lim_x→cg(x)

,where lim 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 funtion

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→ck=k

The limit of a constant is a constant

Property of Equal Left and Right Limits

lim_x→cf(x) exists if and only if lim_x→c- f(x)=lim_x→c+f(x)

Definition of Continuity at a Point

f is continuous at x=c if and only if:

1. f( c ) exists

2. lim_x→c f(x) exists, and

3. lim_x→c f(x)= f( c )

Horizontal Asymptote

If lim_x→∞f(x) = L or lim_x→-∞ f(x) = L, then the line y=L is a horizontal asymptote

Vertical asymptote

If lim_x→cf(x)= ∞ or lim_x→c-f(x)= -∞, then the line x=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.

Definition of Derivative at a point (x=c form):

f’(C)= lim_x→c f(x) - f(C)/ x-c

Meaning: the instantaneous rate of change of f(x) with respect to x at x=c

Definition of Derivative as a Function(Δx or h form)

f’(x) =lim Δx_→0 Δy/Δx =limΔx→0 f(x+Δx)-f(x)/Δx

= lim h→0 f(x+h)-f(x)/h

Power Rule

If f(x)=x^n, where n is a constant, then f’(x)=nx^n-1

Derivatives Of a sum of Functions

If f(x)=g(x)+h(x), then f’(x)