Homework 2 Solution Section 2.2 and 2.3.

Transcription

1 Homework Solution Section. and Write each of the following autonomous equations in the form x n+1 = ax n + b and identify the constants a and b. a) x n+1 = 4 x n )/3. x n+1 = 4 x n) 3 = 8 x n 3 = x n = 3 x n x n+1 = 3 x n a = 3, b = 8 3 b) 3x n+1 x n = 8. 3x n+1 x n = 8 3x n+1 = x n + 8 x n+1 = 3 x n c) x n+1 + x n )/ = 1. a = 3, b = 8 3 x n+1 + x n = 1 x n+1 + x n = x n+1 = x n + a = 1, b = d) x n+1 /x n = 0.9. x n+1 x n = 0.9 x n+1 = 0.9x n a = 0.9, b = Write x 1 = x 0, x = x 1 + 1, x 3 = 3x +, x 4 = 4x 3 + 3,. in the form x n+1 = a n x n + b n, and then compute and make a time-series plot of x 0, x 1,, x using x 0 = 1. If we use the general form x n+1 = a n x n + b n of a linear iterative model, then a 0 = 1, a 1 =, a = 3, a 3 = 4, a n = n + 1 1

2 and b 0 = 0, b 1 = 1, b =, b 3 = 3, b n = n. So x n+1 = n + 1)x n + n. Now x 1 = x 0 = 1, x = x = = 3, x 3 = 3x + = = 11, x 4 = 4x = = 47, x = x = = 39. The following graph is the time-series graph of x n...8. Write the following iteration x 1 = 1 x 0, x = 1 + x 1, x 3 = 1 x, x 4 = 1 + x 3,. in the form x n+1 = a n x n + b n, and then compute and make a time-series plot of x 0, x 1,, x using x 0 = 1. a 0 = 1, a 1 = 1, a = 1, a 3 = 1, a n = 1)n+1 b 0 = 1, b 1 = 1, b = 1, b 3 = 1, b n = 1 Therefore x n+1 = 1)n+1 x n + 1. So x 1 = 1 x = = 1,

3 x = 1 x = = 4, x 3 = 1 x + 1 = = 3 8, x 4 = 1 x = = 19 16, x = 1 x = = The time-series graph is the below Construct an iterative method of computing the partial sums S n and identify which N would make S N equal the entire sum for k k. Let a k = k 100 k. We would like to compute a k. Note that S n, the partial sum is the sum of first n terms. Thus S 0 = 0 and And S n+1 = S n + a n+1 = S n + k k = S 100. n + 1) n+1 = S n + n 1 n Construct an iterative method of computing the partial sums S n and identify which N would make S N equal the entire sum for ,

4 Note that this is a sum of squares, that is, If we define a k = k, then the sum we want to find is Let S 0 = 0 and a k = k. S n+1 = S n + a n+1 = S n + n + 1). The 100th term S 100 = k is what we want to find...4. Suppose someone borrows $10,000 at an annual interest rate of 6% compounded monthly and makes monthly payments on the loan. Construct an iterative model for the monthly unpaid balance if the monthly payments are: a) $1, $, $4, $8, $16,. The monthly interest rate i is 0.06/1 = Then from the Loan Payment Model, if we denote the remaining balance after n-th month as P n, P 1 = 1 + i)p 0 1 = 1.00P 0 1, P = 1 + i)p 1 = 1.00P 1, P 3 = 1 + i)p 4 = 1.00P 4, P 4 = 1 + i)p 3 8 = 1.00P 3 8, etc. If we denote the general iterative equation as P n+1 = 1.00P n d n, then d 0 = 1, d 1 =, d = 4 =, d 3 = 8 = 3,. Therefore d n = n and b) $100, $00, $100, $00, $100,. P n+1 = 1.00P n n, P 0 = By using the same idea, if we denote the general iteration as P n+1 = 1.00P n d n, then d 0 = 100, d 1 = 00, d 3 = 100, d 4 = 00,. We are able to describe d n as d n = ) n+1 0. Then P n+1 = 1.00P n ) n+1 0) = 1.00P n 10 1) n+1 0. Thus P n+1 = 1.00P n 10 1) n+1 0, P 0 = is the model describing the remaining balance. 4

5 .3.6. If P 0 = 1, construct an iterative equation for computing the partial products P n, identify which N would make P N equal the entire product, and compute P 1,, P for 1 1 ) 1 + ) 1 3 ) ) ) If we define a sequence then a formula for the n-th term is Then a 1 = 1 1, a = 1 + 4, a 3 = 1 3 8,, P n = a n = 1 + 1) n n n. n a n = n 1 + 1) n n n ), the product of the first n-term has the iteration P n+1 = a n+1 P n = 1 + 1) n+1 n + 1 ) n+1 P n, with P 0 = 1. Also P 0 is the product we want to compute. Finally,.3.8. For P = P 3 = P 4 = P = P 1 = 1 1 ) P 0 = 1, 1 + ) P 1 = ) P = ) P 3 = ) P 4 = x n+1 = n + n + 1 x n, = 3 4, 4 = 1 3, 3 = 7 18, 7 18 = the solution x n with x 0 = 1 is a product that telescopes to a relatively simple expression. Find that expression.

6 For a homogeneous equation x n+1 = a n x n, the closed formula is So n 1 x n = x 0 k=0 a k. n k n + 1 x n = = k 1 3 n = n + 1 n + 1 = = n + 1. n Find a formula for the exact solution x n for x n+1 = x n /3, x 0 =. Because x n+1 = 3 x n with x 0 =, x n = x 0 3) n = ) n Suppose that an investment of $1000 grows to $100 in years getting interest that is compounded monthly. a) What is the interest rate? Suppose that r is the annual interest rate. Then i = r/1, because it is compounded monthly. The total value follows the compound interest model P n+1 = 1 + i)p n, which is an autonomous homogeneous equation. From the condition given in the problem, after 4 months = years), P 0 = 1000 becomes 100. So Thus 100 = P 4 = P i) 4 = i) i) 4 = = i = 1.) 1 4 i = 1.) r = 1i = Therefore the annual interest rate is approximately 0.4%. b) How much interest in dollars) does the investment earn during the first year? The total interest in the first year can be computed by P 1 P 0. P 1 P 0 = P i) 1 P 0 = ) So the total interest is approximately $4.0. 6

7 .3.. Suppose that when a certain radioactive substance decays, its mass M n on day n satisfies M n+1 = am n. Assuming its mass is measured to be 0 grams initially and 100 grams on day 7: a) Find the formula for the mass M n. The iterative equation is autonomous homogeneous. So we know that M n = M 0 a n. Also from the assumption, M 0 = 0 and M 7 = 100. So 100 = M 7 = M 0 a 7 = 0a 7 a 7 = = a = ) 1 7. Therefore M n = 0 a n = 0 ) 1 ) n ) n 7 7 = 0. b) What is the approximate half-life of the substance, i.e., the time its takes to lose half its mass? We want to find n such that M n = M 0 /. M n = M ) n = ) n 7 1 = n 7 log n = 7 log ) 1 log ).9 ) = log So approximately it takes.3 days the answer days is also ok). ) Suppose that when a certain pendulum is set in motion, it swings in such a way that the greatest positive or negative) angle it makes on one side of the vertical is always 9% of the greatest negative or positive) angle it previously made on the other side of the vertical, which always occurs 3 seconds earlier. If the initial angle it makes with the vertical when let go is 0 : a) How many times will the pendulum cross the vertical before the magnitude of the angle it achieves is 1 or less? Suppose that A n is the angle from the vertical axis. Then from the assumption, A n+1 = 0.9A n. Note that in each step, the sign does change because the pendulum moves to the opposite side. Then A n = A 0 0.9) n = 0 0.9) n. We want to find the smallest n such that A n 1, because we are interested in the magnitude of the angle, not its direction. A n = 0 0.9) n = 00.9) n A n ) n n 1 0 7

8 ) 1 n log 0.9 log n log ) log Note that at the last step, the direction of the inequality does change because log 0.9 is a negative number.) Because n is an integer, the smallest n with the property is 9. b) Approximately how long will it take for this to happen? Although the swing distance decreases, a swing always takes 3 seconds. So the answer is 9 3 = 177 seconds. 8