2.1-2.6 Flashcards
Instantaneous rate of change other names
ex. with 3
units
limit (unofficial)
derivative
see what f(2.9) is then f(3.1) what number is in the middle is the IRC of 3
yunits/x units
Definite Integral
units
area under the curve
xunits times yunits
what number can you not divide by?
0
formal definition of limit
L= lim x-> c f(x) iff
for any number weirdsquiggly
E > 0
there is a number notop 8 > 0 such that x is within notop 8 units of c, but x cannot equal c then f(x) is within weirdsquiggly E units of L
a. what does E and 8 go with?
b. which 8 is better bigger of smaller?
a. y and x
b. smaller
LOOK AT PG 35 SOLVE EX 2
DO IT
The Limit Theorems
a. Product
b. Sum
c. Quotient
d. Constant Times
e. Identity
f. Constant
a. limit distributes over multiplication or the limit of a product = the product of the limits
b. limit distributes over addition, or the limit of a sum equals the sum of the limits
c. limit distributes over division, except for division by zero, or the limit of a quotient equals the quotient of the limits
d. the limit of a constant times a function equals the constant times the limit
e. lim(x->c)x = c
f. if f(x) = k and k is constant then lim(x->c)f(x) = k
Continuity at a point
function f is continuous at x = c iff
1. f(c) exists
2. lim(x->c) f(x) exists
3. lim(x->c) f(x) = f(c)
Continuity on an interval
function f is continuous on an interval of x-values iff its continuous at each value of x in that interval
At the endpoints of a closed interval, only one-sided limits need to equal the function value
One-sided limits:
a. - means?
b. + means?
c. L = lim(x->c) f(x) iff __________
a. left
b. right
c. lim(x->c-) f(x) = lim(x->c +) f(x) - left and right limits have to equal each other for a L = lim(x->c) f(x) to exist
a. If lim(x->oo) f(x) = L then ______, the same is true for______
b. If lim(x->c) |f(x)| = oo then the _________
a. y=L is a horizontal asymptote,
x -> -oo
b. line x = c is a vertical asymptote
a. IVT theorem definition
b. IVT is not what type of theorem and what does that mean?
a. If function f is continuous for all x in the closed interval [a, b], and y is a number between (NOT EQUAL TO) f(a) and f(b), then there is a number x = c in
(a, b) for which f(c) = y
b. IVT is not an iff theorem, which means the conclusion can be true even if IVT does not apply to a function
0/0 or oo/oo is _________ form
Intermediate
If a limit has sincostan ect. in it, it is DNE if approaching _____
oo or -oo
Fill in the blank phrases for IVT
it applies:
a. f is cont on [ , ] and y is between f( ) = and f( ) = therefore their is x =c on
( , )at f(c) = y
