5

Current Attempt in ProgressUse Newton's Method, where needed, to approximate the X-coordinates of the intersections of the curves to nine decimal places_ and t...

Question

Current Attempt in ProgressUse Newton's Method, where needed, to approximate the X-coordinates of the intersections of the curves to nine decimal places_ and then use those approximations to approximate the area of the region that lies below the curvey = sinx and above the line y = 0.3x, wherex > 0.Round your answer to four decimal places_

Current Attempt in Progress Use Newton's Method, where needed, to approximate the X-coordinates of the intersections of the curves to nine decimal places_ and then use those approximations to approximate the area of the region that lies below the curvey = sinx and above the line y = 0.3x, wherex > 0. Round your answer to four decimal places_



Answers

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$ y = \cos x $ , $ y = x + 2 \sin^4 x $

Okay, so let's first find the second um degree polynomial, taylor polynomial of dysfunction. So we're gonna need to find the first two derivatives. So why prime is going to be equal to One half times the co sign affects to the negative 1/2 and then multiplied by negative sine of X. And um I just found that using the chain rule the second derivative is going to be um well its first actually um simplify this a little bit. So we're gonna get negative sine of X divided by two times the co sign X To the 1/2 power. So so let's now actually find the second derivative. So we're going to use the quotient rule which is the derivative of the first which is going to be negative co sign of X times the denominator. Just going to be two times co sign X to the 1/2 power minus the first which is negative sine of X. Multiplied by the derivative of the denominator, which is going to be 1/2 times two. So it's going to be co sign of X To the negative one half power, then multiplied by negative sine of X again. So I'm just since I'm running out of room, I'm going to change this to sine of X square and then this is all divided by two times co sign thanks to the one half power square. So this is equal to negative co sign vex times two co sign of X to the one off power minus sine squared X times co sign X To the negative 1/2 power. And this is all divided by four times co sign of X. So what we can do now is we're going to figure out what are derivatives are equal to at X is equal to zero as well as our original function. So f of zero is equal to Co sign of zero. It's the one have power which is just equal to one to the one half which is equal to one. Our first derivative of prime of zero is equal to negative sine of zero, divided by two times zero whatever, but sign of 00 So this one's gonna be equal to zero In the 2nd derivative Is negative coastline of zero. So negative one multiplied by two times co signed of 0 to 1 half power. So two times one I'm the squared of one which is just to and then we have minus sine of zero which is zero. So this is going to be minus zero and then we divide by four times a co sign of zero which is four times one. So we have negative two divided by four, which is equal to negative one half. So now that we have these three values, we can plug them into our equation to find The actual 2nd Taylor polynomial of our function. So F of zero is equal to one. F prime of zero is equal to zero. So we can kind of just skip that term and F double prime of zero is equal to negative one half. So we're gonna have minus one half, multiplied by x squared Divided by two factorial. So this is equal to 1 -1 X squared divided by four. So now we're gonna use this taylor polynomial to actually estimate the integral from From -1 to 1. So we're going to be looking at the integral from negative 1-1 of one minus X squared divided by four D. X. So we're just gonna use anti differentiation to figure out this integral. The anti derivative of one is just x. And the anti derivative of x squared is X to the third. So we're going to have Next to the 3rd divided by three divided before. And this is from negative 1 to 1. So this is equal to x minus X to the third, divided by 12 From -1 to 1. So when this is equal to negative one um we have well actually let's look at what it's equal to one first we have one minus 1/12. So we're gonna have 11 12th When X is equal to one And then when X is equal to negative one we have negative one minus negative one divided by 12. Just going to be equal to 11 12 minus Um -11/12. Which is equal to 11 12 plus 11 12 which is equal to 22 12th. So this is 20 to 12 which is equal to three, divide both by two. We get 11 6th. So this is our estimate for the area under the curve, from negative 1 to 1 of our original function, which was the square root of the cosine of X.

Okay. Well, you've been wise. You could do 11 of six over high on DDE. Why is a sign of X Okay, so let's draw out our graph. We have Hey, uh, supply line is one. Hi. So Oh, when that pirate for two and no, I pi okay. And then we have ah, why is equal to one minus X over pi? So we have an intercept of one. And our if X is equal, choose you. Okay? If why would you be over yet? One is equal to X over pie and that pie Z x. How? Except it's up the pies. We get something like this. All right. No, we're trying to find the stay here in a girl from a pie of the top craft, which is a sign of X. What? This one minus X over pi the ex. Okay, that's this equal to if you plug this into HQ actuator. The fine dad A is equal to 0.82 78585582 15 And you people got a into here. You get the area is you put two. So point it to four for 39 eggs 72

We're going to have in a girl or why is he could sign of X and wiseacre detention squared of X. And we're looking at some jail do higher. But I might This is and so I just fired too with you and attendants. Good. Excellent. Tended to zero with you, girl. And And you have something like this. You're looking at this area here. We're trying to find a c the sea right here. So we have the integral from zero to see of our top draft with the sign of X. When attention's squared of x x, you plug this or if you tried for see you figure out that C is equal tOdo 1661666 I found this from just applauding these two graphs online and finding their section Point. Okay, If this is C Dad, if you plug this in prissy and find their area, you find that it's equal to 0.9336 62

We want to find the area of the enclosed region. Of these two equations, the 1st 1 is y equals X sign of X squared, which is in red. And the second is why equals X to the four, which is in blue. And we also are interested in X greater than or equal to zero. So using a graphing calculator, we can see that there are two intersection points, Um, which are at the X equals zero and also X is approximately 0.896 So what we want to find is the area which is enclosed. So this small area right here, So as usual, we're going to determine this area by integrating eso area is equal to integral. We're going from our left most point, which is X equals zero to write most point, which was zero point 896 And we're going to do the top function minus the bottom function. So what we see here is that the top function is X sign of X squared, and the bottom function is X to the force. So we're just going to subtract those x sign of X squared minus X to the fore DX and after solving with this integral will get the approximate area. There is approximate because we we have an approximation for the top people. Intersection point. So that's what it's going to be the approximate area. So let's carry out the integration. The integral of X sign of X squared, uh, is negative half coasts of X squared. And if it's not clear how we do that, you could do a U substitution of U equals X squared. So D'You equals two x d. X. So that's how we get this first anti derivative and the second anti derivative is very clear. It's 1/5 x to the five, and this is again going from X equals 0 to 0.896 So after we plug in everything, we get the approximate answer of 0.37


Similar Solved Questions

5 answers
BI0121_A_FALL2O18_CHCourse Home{Ch 3 HWMultiple Choice - Chapter 3 Question 10Part AUntch ofthe tollowingdiseoleKater?organic molecults Kath hyarox;| groupscarbon skeletons bound hyaregenorginic hycrocanconsSubmitRequest AnskerProride Feedback
BI0121_A_FALL2O18_CH Course Home {Ch 3 HW Multiple Choice - Chapter 3 Question 10 Part A Untch ofthe tollowing diseole Kater? organic molecults Kath hyarox;| groups carbon skeletons bound hyaregen orginic hycrocancons Submit Request Ansker Proride Feedback...
5 answers
3=ii10o Etaluatec tbe duzba MElegral4il D = the tiangle funad by (-1,04 (019, (1,0pFuualtLlee?TtT4i= LESula &integat49 [ [~
3=ii10o Etaluatec tbe duzba MElegral 4il D = the tiangle funad by (-1,04 (019, (1,0p Fuualt Llee? TtT4i= LESula &integat49 [ [~...
5 answers
(2) Determine a suitable form for the particular solution Yp(z) if the method of undetermined coefficients is to be used. (DO NOT SOLVE THE EQUATION)(a) y" + 2y' + 2y 3e-1 + 2e-1 cos x + 4e-Ix2 sin % (b) y +3y 214 +r2e-&r + sin 3x
(2) Determine a suitable form for the particular solution Yp(z) if the method of undetermined coefficients is to be used. (DO NOT SOLVE THE EQUATION) (a) y" + 2y' + 2y 3e-1 + 2e-1 cos x + 4e-Ix2 sin % (b) y +3y 214 +r2e-&r + sin 3x...
5 answers
Queotiol 13Wbich of thesc phannaccuticals i5 not an 1somer?
queotiol 13 Wbich of thesc phannaccuticals i5 not an 1somer?...
5 answers
Diat # For te L an`and MI HLin "0lksom 1 1 nifnl 1 | 1 1 F 1 1 0 1 Vi 1 1 Aatehuch In 1 1 uAltunina 1 1
Diat # For te L an`and MI HLin "0lksom 1 1 nifnl 1 | 1 1 F 1 1 0 1 Vi 1 1 Aatehuch In 1 1 uAltunina 1 1...
5 answers
At $100^{circ} mathrm{C}$, the gaseous reaction $mathrm{A} ightarrow 2 mathrm{~B}+mathrm{C}$ is found to be of first order. Starting with pure $mathrm{A}$, if at the end of $10 mathrm{~min}$, the total pressure of the system is $176 mathrm{~mm}$ and after a long time it is $270 mathrm{~mm}$, the partial pressure of $mathrm{A}$ at the end of $10 mathrm{~min} 1$(a) $94 mathrm{~mm}$(b) $43 mathrm{~mm}$(c) $47 mathrm{~mm}$(d) $176 mathrm{~mm}$
At $100^{circ} mathrm{C}$, the gaseous reaction $mathrm{A} ightarrow 2 mathrm{~B}+mathrm{C}$ is found to be of first order. Starting with pure $mathrm{A}$, if at the end of $10 mathrm{~min}$, the total pressure of the system is $176 mathrm{~mm}$ and after a long time it is $270 mathrm{~mm}$, the pa...
5 answers
The hybridization of carbon atoms in $mathrm{C}-mathrm{C}$ single bond of $mathrm{HC} equiv mathrm{C}-mathrm{CH}=mathrm{CH}_{2}$ is(a) $mathrm{sp}^{3}-mathrm{sp}^{3}$(b) $mathrm{sp}^{2}-mathrm{sp}^{3}$(c) $mathrm{sp}-mathrm{sp}^{2}$(d) $mathrm{sp}^{3}-mathrm{sp}$
The hybridization of carbon atoms in $mathrm{C}-mathrm{C}$ single bond of $mathrm{HC} equiv mathrm{C}-mathrm{CH}=mathrm{CH}_{2}$ is (a) $mathrm{sp}^{3}-mathrm{sp}^{3}$ (b) $mathrm{sp}^{2}-mathrm{sp}^{3}$ (c) $mathrm{sp}-mathrm{sp}^{2}$ (d) $mathrm{sp}^{3}-mathrm{sp}$...
5 answers
If ffx) =a* , show that f(A + B) = f(A) f(B).Rewrite f(A + B) by substituting A + B for x in the given function:f(A + B)-0 Which law of exponents can be used t0 rewrite the expression above as product?OA.'-7-(J"(ab)s =a8.b8 a8 .at-a8+:(a8)' -a8:Rewrite the expression f(A + B) as product to show that f(A + B) = f(A) - f(B):f(A - B)-DSince each factor in the product above can be written as f(A) and f(B) , f(A + B) = f(A)-f(B):
If ffx) =a* , show that f(A + B) = f(A) f(B). Rewrite f(A + B) by substituting A + B for x in the given function: f(A + B)-0 Which law of exponents can be used t0 rewrite the expression above as product? OA. '-7-(J" (ab)s =a8.b8 a8 .at-a8+: (a8)' -a8: Rewrite the expression f(A + B) a...
5 answers
9. (12 Pts) Use the Divergence Theorem to evaluate ffs F n dS 3 1 where F = x3 ,y3 Z,xy2 >r ni is the outward unit normal, < and D is bounded by z = x2 + y2 and z = 4. Note that the surface $ is 6 o 8 closed and includes the circular cap at z = 4. <
9. (12 Pts) Use the Divergence Theorem to evaluate ffs F n dS 3 1 where F = x3 ,y3 Z,xy2 >r ni is the outward unit normal, < and D is bounded by z = x2 + y2 and z = 4. Note that the surface $ is 6 o 8 closed and includes the circular cap at z = 4. <...
5 answers
Find the sum4k + 1)2(k _ 4)StepRecall the definition of summation notation_Aa For the given problem, begins with and ends withTherefore,4(k + 1)2(k _ 4) is the sum ofterms in this series_
Find the sum 4k + 1)2(k _ 4) Step Recall the definition of summation notation_ Aa For the given problem, begins with and ends with Therefore, 4(k + 1)2(k _ 4) is the sum of terms in this series_...
5 answers
3) Consider the limacon given by the polar function r=1+ 2-cos(0). (i) Identify all tangents at the pole: (ii) Sketch the curve1 4) Find a power series for centered at 2 and give its interval ofconvergence_5) Consider the conic given by the polar function r = 5 sin (0)(i) Identify the conic and give its standard rectangular equation.(ii) Identify all important points and lines. (iii) Sketch the curve.
3) Consider the limacon given by the polar function r=1+ 2-cos(0). (i) Identify all tangents at the pole: (ii) Sketch the curve 1 4) Find a power series for centered at 2 and give its interval of convergence_ 5) Consider the conic given by the polar function r = 5 sin (0) (i) Identify the conic and ...
5 answers
Consider applying the method of separation of variables with u(€,t) = X(z) T(t) to the partial differential equation02 u Ox282 & +u. Ot2Select the option that gives the resulting pair of ordinary differential equations (where p is a non-zero separation constant):Select one: x" (2) = pX(c), i()+T(t) = uT(t) x" (c) = p, f()+1=p x" (c) +1 = pX(c), i(t)= = pT(t) x" (2) +1 = p, T()= k
Consider applying the method of separation of variables with u(€,t) = X(z) T(t) to the partial differential equation 02 u Ox2 82 & +u. Ot2 Select the option that gives the resulting pair of ordinary differential equations (where p is a non-zero separation constant): Select one: x" (2)...
1 answers
Evaluate the following integrals. $$\int \frac{d x}{\sqrt{x^{2}-49}}, x>7$$
Evaluate the following integrals. $$\int \frac{d x}{\sqrt{x^{2}-49}}, x>7$$...
5 answers
10. Given f(r) =r + 3x 18 find the following:Axis of symmetry: Write its equation. b) Vertex c) Range d) Maximum gr Minimum yalue Of f(x) c) y-intercept f) x-intercept(s); if any
10. Given f(r) =r + 3x 18 find the following: Axis of symmetry: Write its equation. b) Vertex c) Range d) Maximum gr Minimum yalue Of f(x) c) y-intercept f) x-intercept(s); if any...

-- 0.020286--