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Use the result of Exercise 79 to approximate the square roots in Exercises 80–83. In each case, start with x0=1and stop when xk+1xk<0.001.

82.4

Short Answer

Expert verified

The approximate value of the root of4is2

Step by step solution

01

Step 1. Given datax

The given term is 4and x0=1.

Here, we have to find the root of the functions.

02

Step 2. Finding the value of x1

Let us consider the functionf(x)=x2-4

We have the equation xk+1=xk-fxkf'xk.......Equation (1)

Therefore,

f(xk)=xk24f(xk)=2xk

Substituting the values in equation (1)

xk+1=xkxk242xk

Now to find the value of x1, substitute k=0in equation (2)

x0+1=x0x0242x0x1=x0x0242x0

Substitute x0=1

x1=(1)(1)242(1)=1142=132=1+32=42=2

Therefore,x1=2

03

Step 3. Finding the value of x2

Now to find the value of x2, substitute k=1in equation (2)

x2=x1x1242x1

Substitutex1=2

x2=(2)(2)242(2)=2444=204=2

Thereforex2=2

04

Step 4. Finding the value of x3

Now to find the value of x3, substitute localid="1649345412906" k=2in equation (2)

x3=x2x2242x2

Substitutex2=2

x3=(2)(2)242(2)=2444=204=2

Therefore,x3=2

05

Step 5. Finding the value of x4

Now to find the value of x4, substitute localid="1649345423154" k=3in equation (2)

x4=x3x3242x3

Substitutex3=2

x4=(2)(2)242(2)=2444=204=2

Therefore,x4=2

06

Step 6. Finding the root of the function  

Here,

|x4x3|=|22|=|0|

Since, role="math" localid="1649335839464" |x4x3|<0.001let us stop the iteration.

Therefore, the approximate value of the root of 4is 2

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Most popular questions from this chapter

Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that ifpksalmon spawn in Redfish Lake in a given year, then only 0.2pkfish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be pk+1=0.2(pk+h)..

(a) Show that the sustained number of fish returning approaches p=hk+10.2kas k→∞.

(b) Evaluate p.

(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

k=1k3k+100

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder,R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so thatRn10-6

k=11k2

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that localid="1649224052075" Rn10-6.

k=0e-k

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series k=1akin which ak0.

(b) A divergent p-series.

(c) A convergent p-series.

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