Answer:
180°
Step-by-step explanation:
cos(x) = - 1 has only 1 solution in the required interval , that is
x = [tex]cos^{-1}[/tex] (- 1) = 180°
I need help with this ( see image). Please show workings.
Answer:
32. (E)stopped at 2pm
33. (A)
Step-by-step explanation:
32. on the graph you go across the x axis and find where the straight line starts and look for the time
33. look at the y-axis and see what line and the number it is on
Find the median of the data in the bar chart below 4 11 5 8
Answer:
The median is 6.5 :)
Step-by-step explanation:
Brainliest please?
The median of the data in the bar chart given is 6.5 kilometers.
What is Median?Median of a data set is the element in the middle if the data are arranged in increasing or decreasing order.
Given is a bar chart which shows the distance that each family member run in a relay race.
The data is given as :
4, 11, 5, 8
In order to find the median, first arrange the data in an increasing order.
4, 5, 8, 11
Since there are even number (4) of data points, median is the average of the middle two elements.
Median = (5 + 8) / 2 = 6.5 km
Hence the median of the given data is 6.5 kilometers.
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find the measure of the exterior 1
Answer:
C
Step-by-step explanation:
The exterior angle of a triangle is equal to the sum of the 2 opposite interior angles, that is
∠ 1 = 60° + 25° = 85° → C
Answer:
C 85
Step-by-step explanation:
21) Michelle has just been informed that her cell phone plan will be increasing in price. Her new pricing
plan has a flat fee of $25.00 per month, plus a cost of $0.09 per minute of usage. Based on her budget,
Michelle has determined she can only afford to use a maximum of 209 minutes per month. Based on
this information, what is the highest amount Michelle will pay for her cell phone per month?
a) $18.81
b) $5.23
c) $43.81
d) $25.09
Answer:
c) $43.81
Step-by-step explanation:
= $25.00 + ($0.09 × 209)
= $25.00 + $18.81
= $43.81
Which is the proper interpretation of a 95% confidence interval of a proportion of U.S. adults who own a sports car with an upper limit of 27% and a lower limit of 22%? O l am 5% confident the true proportion of U.S. adults who own a sports car is between 22% and 27%. I am 95% confident the sample proportion of U.S. adults who own a sports car is between 22% and 27%. I am 95% confident the true proportion of U.S. adults who own a sports car is between 22% and 27%. o I am 5% confident the sample proportion of U.S. adults who own a sports car is between 22% and 27%.
There is a high level of confidence that the true proportion lies within the interval of 22% to 27%
The proper interpretation of a 95% confidence interval of a proportion of U.S. adults who own a sports car with an upper limit of 27% and a lower limit of 22% is:
I am 95% confident that the true proportion of U.S. adults who own a sports car is between 22% and 27%.
This interpretation accurately conveys the meaning of a confidence interval. It means that if we were to take multiple random samples and calculate 95% confidence intervals using the same methodology, approximately 95% of those intervals would contain the true proportion of U.S. adults who own a sports car. Therefore, there is a high level of confidence that the true proportion lies within the interval of 22% to 27%.
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Anyone please, thanks
Step-by-step explanation:
[tex](2y - 3) = 45 \\ 2y = 48 \\ y = 24[/tex]
Use a Maclaurin series in the table below to obtain the Maclaurin series for the given function f(x) = 2ex + e6x n=0 TABLE 1 | l = Sx" = 1 + x + x2 + x3 + . Important Maclaurin Series and Their Radii of Convergence R 1 no n! 2n+1 otsin x = n=0(-1)" (2n + 1).-x-31+51-71 R=0 0 2n+1 R 1 -0ntl k(k-1) (1+x)" = (k)x" = 1 +kx + k(k-1)(k-2) , x3 + , R=1
The radius of convergence for the Maclaurin series of f(x) is R = 1.
To obtain the Maclaurin series for the given function f(x) = 2ex + e6x, we can use the Maclaurin series expansions of the exponential function. The Maclaurin series for ex is given by e^x = Σ(x^n/n!) from n=0 to infinity, with a radius of convergence of R = ∞.
Using this, we can rewrite the given function as f(x) = 2e^x + e^6x. Substituting the Maclaurin series expansion of ex into the first term and the Maclaurin series expansion of e^6x into the second term, we get:
f(x) = 2(Σ(x^n/n!)) + (Σ((6x)^n/n!))
= 2Σ(x^n/n!) + Σ((6^n)(x^n)/n!)
Now we can combine the two series by collecting like terms and coefficients. The resulting Maclaurin series for f(x) is:
f(x) = Σ((2/n! + (6^n)/n!)(x^n))
The radius of convergence of this series will be determined by the smaller of the radii of convergence of the individual series used, which is R = 1 for the series of ex and R = 1 for the series of e^6x. Therefore, the radius of convergence for the Maclaurin series of f(x) is R = 1.
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Combine the like terms to create an equivalent expression for -5j+(−2j)+3
Answer:
-7j+3
Step-by-step explanation:
-5j+-2j+3
Combine -5j and -2j using like terms which gets you -7j.
-7j+3
So your answer is -7j+3.
The weight and land car crash talatyrtet per 100,000 populien Contact me ate of the recordo concert, og er 0 recentements and rhyre? Do the results so that importados cartes 480 CA Fate ПБ LED 167 300 TE 704 155 143 DA 0 OM H0 Oo ODHO 0 OA OR ое OD - - PE G D ale Thor காலபyea Thes the inco.th suficientevidence lo port the chambrere is shear cortion betwemon ports and whity rates for a good Do the west that point cu cutis OA There dominariyle Do The reported on were OC. The dog whip between the two OD These are tale car Listed below are annual data for various years. The data are weight (metric tons of imported lemons and car crash fotaitytes per 100,000 population Control, fed referent and find the value 0.05. Is there suficient evidence to conclude that there is a linear correlation between lemon ports and crash fatally rates? Do are set portemonnees Lemon Imports 232 264 357 534 Crash Fatality Rate 150 155 152 148 GUD What are the nut and tomative hypotheses? ОА коро Hepco OG Hypo Hp> OB Hypo Hp=0 OD HDO H00 Construct a scatterplot. Choose the correct graph below ОА AY 17- OD OG od 10 16 55 15 36 wo 30010000 The ne correlation coefficient The car correlation conti Rund to three decades a meded The best statt Round to the laces a ded) The valve Round to the meded) Because the than the suficance level 005, there Bicentence to supporto claim that there are correlation between Somon imports and crash tales for a significance of 0.05 Do the shape monsters? OA The tests estan increase in mpored mens causes in an increase in cartaaltyes O The results out that an increase in imported amons causes car fatality rates to remain the same Od. The root oest anycosteronship between the two variables OD The results suport that imported lemons une cartes
The analysis investigates the correlation between lemon imports and crash fatality rates using a significance level of 0.05. The results indicate a strong negative correlation, supporting the claim of a linear relationship between the variables.
Hypotheses:
The null hypothesis (H0) states that there is no linear correlation between lemon imports and crash fatality rates. The alternative hypothesis (Ha) suggests that there is a linear correlation between the two variables.
Scatterplot:
A scatterplot is created to visually examine the relationship between lemon imports and crash fatality rates. Each data point represents a different year, with lemon imports (in metric tons) on one axis and crash fatality rates (per 100,000 population) on the other. The scatterplot allows us to observe any potential patterns or trends in the data.
Correlation coefficient:
The correlation coefficient (r) is calculated to quantify the strength and direction of the linear relationship between lemon imports and crash fatality rates. In this case, the correlation coefficient is determined to be -0.979, indicating a strong negative correlation. This means that as lemon imports increase, the crash fatality rates tend to decrease.
Hypothesis test:
To test the significance of the correlation, a hypothesis test is conducted. The test aims to determine if the observed correlation is statistically significant or likely due to chance. The significance level chosen is 0.05, which represents a 5% chance of rejecting the null hypothesis when it is true.
Test statistic and critical value:
The calculated t-value, based on the correlation coefficient and sample size, is -12.4. Since the sample size is small (n = 4), a t-distribution is used instead of a standard normal distribution. The critical t-value at a significance level of 0.05 and 2 degrees of freedom is 4.303.
Conclusion:
By comparing the obtained test statistic (t-value) with the critical value, it is determined that the obtained t-value (-12.4) is significantly smaller than the critical t-value (4.303). This leads to rejecting the null hypothesis. Therefore, there is sufficient evidence to support the claim that there is a linear correlation between lemon imports and crash fatality rates.
In summary, the detailed analysis concludes that there is a strong negative correlation between lemon imports and crash fatality rates, with sufficient evidence to support this claim.
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Factor the expression: 2x^2 +21x+49
Answer:
[tex](2x + 7) ( x + 7)[/tex]
Step-by-step explanation:
Fred kicks a ball with a force of 20n to George, who is 5m away. How much work was done to the ball
Answer:
100 Nm
Step-by-step explanation:
Work is said to be done when a force moves a body through a distance. This may be expressed as
work done = Force * distance
where
work done is measured in joules or Newton-meter
Force is measured in newtons and distance in meters
Substituting the given values,
Workdone = 20 * 5
= 100 Nm
= 100J
a converging lens with a focal length of 6.70 cmcm forms an image of a 4.80 mmmm -tall real object that is to the left of the lens. the image is 1.50 cmcm tall and erect.
A converging lens with a focal length of 6.70 cm forms an erect image of a 4.80 mm tall real object positioned to the left of the lens. The resulting image is 1.50 cm tall.
A converging lens is a lens that bulges in the middle and causes light rays to converge. In this case, the lens has a focal length of 6.70 cm, which means that parallel rays of light incident on the lens will converge to a point 6.70 cm away from the lens. The object, positioned to the left of the lens, has a height of 4.80 mm. When the light rays from the object pass through the lens, they refract and intersect at a point to form the image. The image formed is erect, meaning it is in the same orientation as the object. The height of the image is 1.50 cm.
The magnification of the image can be calculated using the formula: magnification = height of image / height of object. In this case, the magnification is 1.50 cm / 4.80 mm. To convert the height of the object to centimeters, we divide 4.80 mm by 10, which gives us 0.48 cm. Therefore, the magnification is 1.50 cm / 0.48 cm, which equals approximately 3.125.
Since the image is erect and the magnification is greater than 1, we can determine that the image is larger than the object. The positive magnification indicates that the image is virtual, which means it cannot be projected onto a screen. The image is formed on the same side of the lens as the object, which is the left side in this case. The image distance can be calculated using the lens formula: 1/f = 1/v - 1/u, where f is the focal length, v is the image distance, and u is the object distance. Since the image is formed on the same side as the object, the object distance is negative (-u). By plugging in the values, we can solve for the image distance. However, additional information, such as the object distance, would be needed to calculate the exact position of the image.
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Can someone help me with this. Will Mark brainliest . Need answer and work/explanation. Thank you.
Answer:
Hope it helps...
Step-by-step explanation:
Write an equation in terms of x and y for the function that is described by the given characteristics. a cosine curve with a period of , an amplitude of 1, a left phase shift of , and a vertical translation down by 9/2 of a unit.
The equation for the described function can be written as:
y = cos(x - π) - 9/2
Let's break down the components of the equation:
The cosine function, cos(x), produces a periodic wave with an amplitude of 1.
The period of the cosine curve is determined by the coefficient in front of the angle, which is 1 in this case. A period of 1 corresponds to one complete cycle of the cosine curve.
The left phase shift of π shifts the entire curve to the right by π units.
The vertical translation down by 9/2 units shifts the entire curve downwards by 9/2 units.
Therefore, the equation y = -cos(x - π) - 9/2 represents a cosine curve with the given characteristics.
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factorise (2x²-5x-3)
Answer:
2x² - 5x - 3
= 2x² -6x + 1x -3
= 2x( x - 3) + 1 (x - 3)
= ( x - 3) (2x + 1)
1.0
Im in need for help..
Answer:
$3200
Step-by-step explanation:
i had the answer but wanted to make sure
when added up everything equals 273,900
-700
--------
273,200
his networth is 270,000 that means when you subtract it from the 273,200 you are left with 3,200
The data represent the age of world leaders on their day of inauguration. Find the five-number summary, and construct43 a boxplot for the data. Comment on the shape of the distribution 65 44 49 65 62 54 69 61 53 67 52 57 67 68 The five-number summary ?
The given data represents the ages of world leaders on their day of inauguration. The five-number summary consists of the minimum value, which is 44, the first quartile (Q1) at 53, the median at 61, the third quartile (Q3) at 67, and the maximum value at 69.
The five-number summary for the given data is as follows: Minimum: 44, First quartile (Q1): 53, Median (Q2): 61, Third quartile (Q3): 67, Maximum: 69.
To construct a boxplot, we can represent the five-number summary on a number line. The boxplot will have a box representing the interquartile range (from Q1 to Q3) with a line inside representing the median (Q2). Whiskers will extend from the box to the minimum and maximum values, and any outliers will be shown as individual data points.
The shape of the distribution can be inferred from the boxplot. If the box is symmetrically positioned and the whiskers are roughly equal in length, the distribution is likely to be approximately symmetric. If the box is skewed or the whiskers are unequal, the distribution may be skewed or have outliers.
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Use the predictor-corrector method to solve dy = x² + y²; ; y(1) = 0 dx for y(2) with h = 0.01.
y(1) equal to zero and x equal to 2, the differential equation dy/dx = x2 + y2 has an approximate solution of y(2) 0.0101005. y(2)_c = y(1) + (0.01/2)*[(12 + 02) + (22 + 0.012)]= 0.0101005.
In numerical analysis, the predictor-corrector method is an important tool for solving ordinary differential equations. It is a combination of two distinct strategies, the corrector and the indicator. The indicator technique uses a limited distinction conspire to predict the value of the dependent variable at the subsequent time step, whereas the corrector strategy uses this anticipated value to correct the indicator's error.
To deal with dy = x2 + y2 using the marker corrector method; When y(1) = 0 dx for y(2) and h = 0,01, we can begin by employing the Euler's technique as the indicator strategy and the adjusted Euler's technique as the corrector strategy. The following is the equation for Euler's method: Coming up next is the way we can decide the anticipated worth of y(2) utilizing this strategy: where f(x,y) = x2 + y2 and y(i+1) = y(i) + h*f(x(i),y(i)).
Utilizing the altered Euler's strategy, we can decide the adjusted worth of y(2) as follows: y(1) + h*f(x(1),y(1)) = 0 + 0.01*(12 + 02) = 0.01 y(i+1) = y(i) + (h/2)*(f(x(i),y(i)) + f(x(i+1,y(i+1)_p) The differential equation dy/dx = x2 + y2 has an approximate solution of y(2) 0.0101005 when y(1) is set to zero and x is set to 2. y(2)_c = y(1) + (0.01/2)*[(12 + 02) + (22 + 0.012)]= 0.0101005.
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Two distinct lines are said to be parallel if they have no point of intersection. true or false..
Two distinct lines are said to be parallel if they have no point of intersection. This statement is true.
A line is a straight figure that continues infinitely in both directions. It has no endpoints. The term "parallel lines" is used to describe two or more lines that do not intersect. When we draw parallel lines, they seem to be straight and never touch each other. Therefore, the two distinct lines are said to be parallel if they have no point of intersection and the statement is true.
A line is a boundlessly lengthy item with no width, profundity, or shape. Lines can be embedded in spaces of two, three, or higher dimensions, but they are one-dimensional objects. The word line may likewise allude to a line fragment in regular day to day existence, which has two focuses to mean its closures.
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I need help with this question
The ages of student volunteers at an event are: 13, 11, 12, 15, 16, 11, 12, 14, 15,
and 11. Find the median *
Answer:
12.5
Step-by-step explanation:
Jack bought 52 bags of pellets last year. This year he increased that amount by 40%. How many bags of pellets did he buy this year?
Answer:
He bought 73 bags of pellets this year.
Step-by-step explanation:
Since this year the amount increased by 40%, we can say that it is 100% + 40% = 140% = 1.4 of lasts years amount.
Last years amount was of 52 bags.
So, this year:
52*1.4 = 73
He bought 73 bags of pellets this year.
identify the vertical asymptote(s) of each function. check all of the boxes that apply. x = -8 x = -2 x = -1 x = 1 x = 2 x = 8
The correct boxes to check are x = 1 and x = 2.
To identify the vertical asymptotes of the function f(x) = (x - 8) / ([tex]x^{2}[/tex]- 3x + 2), we need to determine the values of x for which the denominator of the function becomes zero. These values will give us vertical asymptotes.
Breaking the denominator into the factors :
[tex]x^{2}[/tex] - 3x + 2 = (x - 1)(x - 2)
Setting the denominator equal to zero:
x - 1 = 0 --> x = 1
x - 2 = 0 --> x = 2
So, the vertical asymptotes of the function are:
x = 1 and x = 2.
Now, let's check for diffrent values of x:
x = -8 (This is not a vertical asymptote as it is not related to the function's denominator.)
x = -2 (This is not a vertical asymptote as it is not associated with the function's denominator.)
x = -1 (This is not a vertical asymptote as it is not related to the function's denominator.)
x = 1 (This is a vertical asymptote.)
x = 2 (This is a vertical asymptote.)
x = 8 (This is not a vertical asymptote as it is not related to the function's denominator.)
Therefore, the correct boxes to check are x = 1 and x = 2.
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The correct question is given below -
Identify the vertical asymptote(s) of each function. Check all of the boxes that apply. f(x)=x-8/[tex]x^{2}[/tex]-3x+2
x = -8
x = -2
x = -1
x = 1
x = 2
x = 8
Answer:
D and E for the first one. B and D for the second one
Step-by-step explanation: edg is stinky, i hope this helped a little
A recipe for cooking rice indicates that 14 cups of boiling water are needed for 8 cups of rice.
if henri has 28 cups of boiling water how much cups of rice should he add
(WILL GIVE BRAINLIST FOR BEST ANSWER!)
Hello!
This is a question about ratios and relating them to multiples of the ratio.
We are given that a recipe for cooking rice requires 14 cups of boiling water for every 8 cups of rice.
Then we are asked how many cups of rice are needing to be added if we have 28 cups of boiling water.
Since 28 is two times 14, and assuming that this ratio is proportional, we can multiply the cups of rice by 2 to find the equal ratio.
That means that 16 cups of rice should be added if we have 28 cups of boiling water.
In terms of ratios, we can set up the following proportion.
[tex]\frac{14}{8}=\frac{28}{x}[/tex]
Then here, we can cross multiply and solve for [tex]x[/tex].
[tex]14x=224[/tex]
[tex]x=16[/tex]
Hope this helps!
I will give a brainlist and more if someone can tutor me on how to do this so if you want to let me know, please!! anyone from ages 15-17
Solving Exponential and Logarithmic Equations: What are the potential solutions to the equation below? 2ln(x+3)= 0 X=-3 and X=-4 O x=-2 and x=-4 X= 2 and X=-3 O x=2 and X= 4
Answer:
x = -2.
Step-by-step explanation:
2 ln(x + 3) = 0
ln(x + 3) = 0
Now ln 1 = 0 so
x + 3 = 1
x = -2.
HELP ME FIND AREA I WILL MARK BRAINLEST
Answer:
472
Step-by-step explanation:
break it up into two rectangles
1) 12x18=216 in
2) 8x32=256
Hope this helped
Answer:
472 in²
Step-by-step explanation:
The figure is composed of 2 rectangles , then
area = area of lower rectangle + area of upper rectangle, that is
area = (8 × 32) + (12 × 18) = 256 + 216 = 472 in²
math question in pic above plz help !!!!!!!!:))))) plz i need it a lot :')brainlist to how even answers right first
Answer:
a
Step-by-step explanation:
a because they both have a right angle which is 90 degrees
and beacuse 90 - 42 = 48 degrees which means both triangles have a 48 degrees
5. (Joint Use of the Bisection and Newton's Method). (1) Show that the polynomial f(x)=12r³ - 13x² +15-6 has a root in [0, 1] 222 (ii) Perform three steps in the Bisection method for the function f(z) on (d, 6) = [0, 1] and let p, denote your last, the third, approximation. Present the results your calculations in a standard output table nas bn Pn f(an) (Pm) for the Bisection method (w/o the stopping criterion). In this and in the next subproblem all calculations are to be carried out in the FPA, (Answer: p=0.625; if your answer is incorrect, redo the subproblem.) (iii) Find the iteration function 9(z)=x-1(2) f'(x) for Newton's method (this time an analysis of convergence is not required). (iv) Use then Newton's method to find an approximation py of the root p of f(z) on (0.1] satisfying RE(PNPN-1) < 10-7 by taking Po = 0.625 as the initial approximation (so we start with Newton method at the last approximation found by the Bisection method). Present the results of your calculations in a standard output table for the method. (Your answers to the problem should consist of a demonstration of existence of a root, two output tables, and a conclusion regarding an approximation PN.)
(1) The given function f(x) = 12x³ - 13x² + 15x - 6 has a root in [0,1].
(ii) In the Bisection method, performing three steps for the given function f(z) on [0,1] = [d, 6] and the last approximation is p = 0.625. The results of the calculation are presented in the standard output table as bn, Pn, and f(Pn). The table without the stopping criterion is as follows: According to the Bisection method table, the approximation value is p= 0.625.
(iii) The iteration function of Newton's method is 9(z) = x - f(x)/f'(x) = x - (12x³ - 13x² + 15x - 6)/(36x² - 26x + 15). (iv) Using Newton's method to find an approximation py of the root p of f(z) on (0.1] satisfying RE(PNPN-1) < 10-7 by taking Po = 0.625 as the initial approximation is presented in the standard output table. Therefore, the root of the equation is approximately 0.6188.
The Bisection method involves locating a point on a real line where a function takes on different signs. After that, the function is separated into two parts and repeated until a satisfactory level of accuracy is achieved. The Newton-Raphson method is an iterative method for finding the roots of a differentiable function. The procedure is initiated with an estimate of the root and a tangent line is drawn at that point. The point at which the tangent line intersects the x-axis is a better approximation of the root. The procedure is repeated until the desired level of accuracy is achieved.
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In the binary dependent variable model, a predicted value of 0.6 means that A. the most likely value the dependent variable will take on is 60 percent. B. given the values for the explanatory variables, there is a 60 percent probability that the dependent variable will equal one. C. the model makes little sense, since the dependent variable can only be 0 or 1. D. given the values for the explanatory variables, there is a 40 percent probability that the dependent variable will equal one.
Answer:
B. Given the values for the explanatory variables, there is a 60 percent probability that the dependent variable will equal one.
Step-by-step explanation:
Given
Model: binary dependent variable
Predicted value: 0.6
Required
Interpret
The possible values of a binary dependent variable are 0 or 1.
So, a predicted value means that, the chance that the dependent variable will be 1 equals to the predicted value.
Since the predicted value is 0.6 (or 60%), this means that; there is a 60% chance that the dependent variable = 1.
Hence, option (b) is true.
Evaluate the indefinite integral as a power series.
a)
integrate x ^ 7 * ln(1 + x) dx
f(x) =C+ sum n=1 ^ infty [
What is the radius of convergence R?
R =
Express the function as the sum of a power series by first using partial fractions.
b)
f(x) = 13/(x ^ 2 - 5x - 36)
f(x)= sum n=0 ^ infty boxed - 1/9 * (x/9) ^ n - 1/4 * (- x) ^ n ]x
Find the interval of convergence(Enter your answer using interval notation.)
(-1,1)
R = 1 / L = ∞. Thus, the radius of convergence of the given power series is ∞. The interval of convergence is (-1,1) is the answer.
a)The indefinite integral of x7ln(1 + x) can be obtained by using the formula for integration by parts. For the same, we need to select the parts as u and dv, such that on differentiating u and integrating dv, the obtained integrals get easier to solve.
Let us select x7 as u and ln(1 + x)dx as dv.u = x7 => du/dx = 7x6 => du = 7x6dx, and v = ∫ ln(1 + x)dx.
Using u and v, we can express the integral as,x7ln(1 + x)dx= ∫ u dv= uv - ∫ v du= x7 ln(1 + x) - ∫ 7x6/ (1 + x) dx = x7ln(1 + x) - 7 ∫ x6/ (1 + x) dxThe indefinite integral of the term ∫ x6/ (1 + x) dx can be obtained by the substitution method, let t = 1 + x, then x = t - 1, and dx = dt.∫ x6/ (1 + x) dx= ∫ (t - 1)6/t dt= ∫ (t6 - 6t5 + 15t4 - 20t3 + 15t2 - 6t + 1)/ t dt= ∫ t6/t dt - 6 ∫ t5/t dt + 15 ∫ t4/t dt - 20 ∫ t3/t dt + 15 ∫ t2/t dt - 6 ∫ t/t dt + ∫ 1/t dt= ∫ t5 dt - 6 ∫ t4 dt + 15 ∫ t3 dt - 20 ∫ t2 dt + 15 ∫ t dt - 6 ln|t| + ln|t| + C= t6/6 - 6t5/5 + 15t4/4 - 20t3/3 + 15t2/2 - 6 ln|t| + C.
Substituting the value of t, we get the indefinite integral of the original expression as,x7 ln(1 + x)dx= x7ln(1 + x) - 7 [x6/6 - 6x5/5 + 15x4/4 - 20x3/3 + 15x2/2 - 6 ln|1 + x|] + C= x7ln(1 + x) - x6 - 42x5/5 - 280x4/4 - 1125x3/3 - 1875x2/2 - 3150x - 735 ln|1 + x| + C.
Now, we need to obtain the power series for f(x) = x7ln(1 + x).
The formula to obtain the power series for f(x) = (1 / 1 - x)2 is as follows,f(x) = Σn=0 ∞ (n + 1)xn.
The integral x7ln(1 + x) can be written as Σn=1 ∞ (-1)n-1 xn / n.
Therefore, the power series for x7ln(1 + x) can be written as,f(x) = ∑n=1 ∞ (-1)n-1 xn / n= -x + x2/2 - x3/3 + x4/4 - x5/5 + x6/6 - x7/7 + ...= C + ∑n=1 ∞ (-1)n-1 xn / n, Where C is a constant, we can evaluate the value of C by substituting x = 0 in the power series. f(0) = 0, therefore, the constant C = 0.
Now, we need to obtain the radius of convergence of the obtained power series using the formula for the radius of convergence, R = 1 / lim supn→∞ |an|where an is the nth term in the power series.
In this case, |an| = |(-1)n-1 / n| = 1 / n.Let L = lim supn→∞ |an| = limn→∞ |an| = 0Therefore, R = 1 / L = ∞Therefore, the radius of convergence is ∞.
b)To obtain the power series for the given function f(x) = 13 / (x2 - 5x - 36), we need to first perform the partial fraction decomposition of the given function. The partial fraction decomposition of the given function is given as follows,f(x) = 13 / (x2 - 5x - 36)= 13 / [(x - 9)(x + 4)] = A / (x - 9) + B / (x + 4) where A and B are constants.
To obtain the values of A and B, we can equate the numerators on both sides and solve for A and B.13 = A(x + 4) + B(x - 9)At x = 9, we get 13 = 13B, B = 1.At x = -4, we get 13 = -4A, A = -13/4.
Therefore, the partial fraction decomposition of the given function is,f(x) = 13 / (x2 - 5x - 36)= -13/4 * 1 / (x + 4) + 1 / (x - 9)
Now, we can write the power series for the above partial fractions. The power series for 1 / (1 - x) is given by,f(x) = Σn=0 ∞ xn, |x| < 1
The power series for 1 / (x + 4) is given by,f(x) = -1/4 * Σn=0 ∞ (-x / 4)n, |x / 4| < 1
The power series for 1 / (x - 9) is given by,f(x) = Σn=0 ∞ (x / 9)n, |x / 9| < 1
Substituting the above power series in the original function, we get the power series for the given function as,f(x) = -1/4 * Σn=0 ∞ (-x / 4)n + Σn=0 ∞ (13 / 4) (x / 9)n= Σn=0 ∞ [-1/9 * (x / 4)n - 1/4 * (-x) n](x / 9)
Therefore, the power series for the given function is,f(x) = Σn=0 ∞ [-1/9 * (x / 4)n - 1/4 * (-x) n](x / 9)
The given function f(x) has the power series representation, f(x)= ∑n=0 ∞ [-1/9 * (x/4)n - 1/4 * (-x)n] * (x/9)n where c= 0.
Now, the radius of convergence of a power series is given by the formula,R = 1 / lim supn→∞ |an| where an is the nth term in the power series.
In this case, the nth term in the power series is given by |an| = |(-1)n-1 / 9 * 4n-1 | + |(-1)n / 4 * n|Let L = lim supn→∞ |an|. L = limn→∞ |(-1)n-1 / 9 * 4n-1 | + |(-1)n / 4 * n|L = 0 + 0 = 0.
Therefore, R = 1 / L = ∞Thus, the radius of convergence of the given power series is ∞.The interval of convergence is (-1,1)
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