New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places. 60. [T] A bank account earns 5% interest compounded monthly. Suppose that S 1000 is initially deposited into the account, but that $ 1 0 is withdrawn each month. a. Show that the amount in the account after n months is A n = ( 1 − .05 / 12 ) A n − 1 − 10 ; A 0 = 1000 b. How much money will be in the account after I year? c. Is the amount increasing or decreasing? d. Suppose that instead of $10. a fixed amount d dollars is withdrawn each month. Find a value of d such that the amount in the account after each month remains $1000. e. What happens if d is greater than this amount?
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places. 60. [T] A bank account earns 5% interest compounded monthly. Suppose that S 1000 is initially deposited into the account, but that $ 1 0 is withdrawn each month. a. Show that the amount in the account after n months is A n = ( 1 − .05 / 12 ) A n − 1 − 10 ; A 0 = 1000 b. How much money will be in the account after I year? c. Is the amount increasing or decreasing? d. Suppose that instead of $10. a fixed amount d dollars is withdrawn each month. Find a value of d such that the amount in the account after each month remains $1000. e. What happens if d is greater than this amount?
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence
z
n
+
1
=
x
n
−
f
(
x
n
)
f
'
(
x
n
)
. For the given choice of f and x0. write out the formula for
x
n
+
1
. If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places.
60. [T] A bank account earns 5% interest compounded monthly. Suppose that S 1000 is initially deposited into the account, but that $ 1 0 is withdrawn each month.
a. Show that the amount in the account after n months is
A
n
=
(
1
−
.05
/
12
)
A
n
−
1
−
10
;
A
0
=
1000
b. How much money will be in the account after I year?
c. Is the amount increasing or decreasing?
d. Suppose that instead of $10. a fixed amount d
dollars is withdrawn each month. Find a value of d such that the amount in the account after each month remains $1000.
Consider the function g(x) = x^2 + 3/16(a) This function has two fixed points, what are they?(b) Consider the fixed point iteration xk+1 = g(xk) for this g. For which of the points you found in(a) can you be sure that the iterations will converge to that fixed point? Justify your answer.(c) For the point(s) you found in (b), roughly how many iterations will be required to reduce theconvergence error by a factor of 10?
The size of an undisturbed fish population has been modeled by the formula
pn+1 = bpn / (a + pn)
where Pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 > 0.
a) If {pn} is convergent, then the only possible values for its limit are 0 andb −a. Justify.
b) Given that pn+1<(b/a)pn is true. justify.
c) Use part (b) to justify if a>b, then lim pn =0. where limit n is approach to infinity
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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