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M ble Calculus 2016—03~03 (Thu) McCary 1varia O. A07/005/01 Q
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Course Due Date
Instructor mew @ e 3 n...“ O 20160224 09:38 O. AO7/005/02 Q 1. (10 points) In lecture, we discussed Newton’s method, which is a way to solve a nonlinear equation f = iteration. For a. function with signature R —> R, the underlying method is: fern) In+1 = 5511 _ f,($n) The primary idea is to start with an initial guess $0 and then iterate until (hopefully) the method converges. Your calculator (and computer) uses the following approximation to compute square roots. To compute x/E, your
machine uses Newton’s method with $60 = 1 to solve the equation f = $2 — b = 0. A little bit of algebra yields the
following, which is known as the divide and average formula. / ‘x (79/ 1 Lila =58 _m%—b__ 322+?) m l b
mn+1 — $11 _ n 2&1 H 2m“ 5 ($11 + A little bit of work can show that no “hope” is necessary in this case: the divide and average formula converges1 and
the rate at which it converges is incredible. Use the divide and average formula to ﬁll out the following tables. N ewton’s for V10 Newton’s for x/ﬁ l Grader Only if 5r“ O. AOT/OOS/OL Q 7 .0 points) Computers are machines which very quickly perform arithmetic: +, —, and *. Division is more complicated. One strategy computers take is when a, division 0/1; is requested, the computer ﬁrst computes 1/b and then multiplies
by a. To evaluate 1/1), many computers solve f I: 1/9: — b = 0 with Newton’s method. (a) Use Newton’s method to determine an update formula (analogous to the divide and average formula) which uses only +: —, and *. The name for this formula. is the Newton—Raphson division, and you can easily look it up to
check your answer. However, you should try to ﬁgure it out before looking it up! k, * (“2
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An A x. (b) Use the NewtonRaphson formula to ﬁll out the following tables. You’ll ﬁnd that the value of $0 is quite important
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3/, l, Grader Only J, 0 Ge 0 O) O. AOT/OOS/O4 Q 3. (10 points) Let ﬂx) = 233 + 3.1: +1. (5.) Show f(:c) is invertible by showing f is monotone. ig :a\,;:f_,i , 77‘ ‘ 3, _ ' , ' l ‘ r
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'.. r . w ‘ ; ‘ . (13) Even though f is invertible, that doesn’t mean it is easy to compute f ‘10)) for a speciﬁc 5: no easy analytic
formula exists! In general this must be solved numerically, and Newton’s method is one of the most common
techniques. Observe: f‘1(b)=w => b=f(sc) => f(z)—b=0 So setting 9(3) = ﬁns] — b, we can use New’mn‘s method on g to compute f‘1(b). In the same manner as the
previous two exercises, ﬁnd an update formula. (c) Use your update formula to ﬁll out the following tables. You can check to make sure you’re on the right track by
plugging your 2:9 into f. Newton’s for f'1(7] Newton‘s for f‘1(13.8) J. Grader Only l
. O® @@®@@®@®®®® . O. A07/005/05 . 0 points) Newton’s method also applies to functions with signature R” —> R”. In this case, to solve f = l3, you
start with an initial guess £0 and then iterate ifn+1 = in _ [D And in the same manner as the previous exercise, Newton’s method can be used to compute the inverse ﬁmction when {V5 2 Nu
the function is invertible. Let f 2: (I + 3'1”)?
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5 (a) Use Newton‘s method to compute f ‘1 (6 ). Your answer must be accurate to 4: places past the decimal. 5 (b) Use Newton’s method to compute f’1 (1 ). Your answer must be accurate to 4 places past the decimal. This can be done by hand, but by all means, use a symbolic calculator. To encourage you to use a. symbolic calculator:
you can use the free, web—based version of SAGE at the following URL: https://cloud.sagemath.com. Here is a
script which solves the ﬁrst part of this exercise: https://Cloud.sagemath.com/projectsl94263e783ae24ad388c55e6ce29465e5lfiles/Newton.sagews. t Grader Only J, 0 . \.../‘ \_, my a, Q. AO7/005/06 0 . GL0 f’r
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5. (10 points) For each function and point: determine if the inverse flmction theorem guarantees a local inverse. E.
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. O® @@®@@®©®@@® . .. AOT/OOS/OT . .0 points) Let E“ = R2 — {5} denote the punctured plane. Does f : E“ —> R2 have a local inverse at every point?
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