C7-11

Question 1

what is an arithmetic sequence

Answer 1

when the difference between two successive terms in a sequence is always the same

Question 2

what is the formula for arithmetic sequence

Answer 2

Sn = n/2 [2a + (n-1) d]

where Sn, n is subscript
a= first term
d= common difference

Question 3

what is a geometric sequence

Answer 3

a sequence in which each term is a constant multiple of the preceding term

Question 4

what is the formula for geometric sequence

Answer 4

Sn= a. (1 - r^n) / (1 - r)

where a= first term
r= common ratio

Question 5

what is the definition which explains when a sequence converges to the limit 0

Answer 5

A sequence converges to a limit ℓ for every ε >0 there is a natural number N such that for all natural numbers n, if n is equal to or greater than N then |un - ℓ| < ε
we write
lim (as x tends to infinity) un = ℓ

Question 6

when adding limits

lim (un + vn)

Answer 6

lim (un) + lim (vn)

Question 7

lim (aun) where aER

Answer 7

a lim (un)

Question 8

lim (un.vn)

Answer 8

(lim (un)) (lim (vn))

Question 9

lim (un/vn)

Answer 9

lim un / lim vn

Question 10

binomial coefficient

Answer 10

(n choose r) n!/ r!(n-r)!

Question 11

(x + y)^n =?

Answer 11

= sum[r=0,n][(n choose r)x^(n-r)b^r]

Question 12

if f approaches the limit l near a if ∀ ε > 0 ∃ δ > 0 such that 0 < |x − a| < δ then what

Answer 12

f approaches the limit l near a if ∀ ε > 0 ∃ δ > 0 such that 0 < |x − a| < δ

then =⇒ |f(x)−l| < ε

Question 13

if f approaches both m and l near a then ?

Answer 13

then l=m

Question 14

lim[f(x) + g(x)]

x→a

Answer 14

= l + m

Question 15

lim αf(x) =?

as x tends to a

Answer 15

= αl

Question 16

lim[αf(x) + βg(x)] = ?

Answer 16

αl + βm

Question 17

lim f (x)g(x) = ?

Answer 17

= lm

Question 18

if m doesn’t equal 0

lim 1/g(x) = ?

Answer 18

=1/m

Question 19

lim f(x) /g(x) = ?

Answer 19

=l/m

Question 20

lim (x tends to 0) sin/x =?

Answer 20

1

Question 21

lim (x tends to 0) sin (mx) / nx =?

Answer 21

m/n

Question 22

when is a function f continuous at a

Answer 22

if lim (x tends to a) f(x) = f(a)

Question 23

if f and g are continuous at a then

Answer 23

f+g is continuous at a

f.g is continuous at a

if g(a) does not equal 0, then 1/g is continuous at a

Question 24

if g is continuous at a and f is continuous at g(a) then?

Answer 24

then fog is continuous at a

Question 25

when is a function f continuous in an open interval (a,b)?

Answer 25

if it is continuous at all points x ∈ (a, b)

Question 26

when is a function f continuous in a closed interval [a, b]?

Answer 26

if it is continuous in (a, b) and if lim f(x) = f(a) and lim f(x) = f(b)

Question 27

when is the function f differentiable at a?

Answer 27

if lim (f(a + h) − f(a))/h exists

Question 28

when f(x) = c (constant function)

Answer 28

f′(a) = lim (c − c)/h = lim 0 = 0

h tends to 0

Question 29

when f(x)=cx+d

Answer 29

f′(a) = lim [c(a + h) + d − (ca + d)] /h = lim ch/h = lim c = c
h tends to 0

Question 30

when f(x) = x^2

Answer 30

2a (more working)

Question 31

when f(x) = x^3

Answer 31

3a^2

Question 32

if f is differentiable at a

Answer 32

then f is continuous at a