The solution graphs for the differential equations y' = 1 + x2 + y2, y' = xe?x2 ? y2, y' = 1/(1 + ex2 + y2), and y' = sin(xy) cos(xy) are labeled I, II, III, and IV respectively. The slope of the graphs when x is less than 0 and greater than 0, as well as the horizontal tangent line when y is equal to 0, are used to determine the correct graph for each equation.
Match the differential equations with the solution graphs labeled I-IV. Give reasons for your choices.
I 9-1-013a.gif II 9-1-013b.gif
III 9-1-013c.gif IV 9-1-013d.gif
(a)
y' = 1 + x2 + y2
y' = 1 + x2 + y2 ? (? < ? > = )1 and y' ? ? (1 ? 0 -1 -? )as x ? ?.
The only curve satisfying these conditions is labeled ? (I II III IV) .
(b)
y' = xe?x2 ? y2
y' = xe?x2 ? y2 ?( > ? ? = <) 0 if x > 0 and y' ? (? = > < ?) 0 if x < 0.
The only curve with ---Select---( negative positive horizontal vertical) tangent slopes with
x < 0
and ---Select--- positive negative horizontal vertical tangent slopes when
x > 0
is labeled ? (I II III IV) .
(c) y' =
1
1 + ex2 + y2
y' =
1
1 + ex2 + y2
? ? = ? > < 0 and y' ? ? (-? ? 1 0 -1 )as x ? ?. The only curve satisfying these conditions is labeled ?( I II III IV) .
(d)
y' = sin(xy) cos(xy)
y' = sin(xy) cos(xy) ? (> ? < ? )= 0 if y = 0,
which is the solution graph labeled ?( I II III IV) .
a) I
b) IV
c) II
d) III
a) The differential equation is y' = 1 + x2 + y2, which has a positive slope when x is less than 0 and a negative slope when x is greater than 0. The only graph with these conditions is labeled I.
b) The differential equation is y' = xe?x2 ? y2, which has negative tangent slopes when x is less than 0 and positive tangent slopes when x is greater than 0. The only graph with these conditions is labeled IV.
c) The differential equation is y' = 1/(1 + ex2 + y2), which has a negative slope when x is less than 0 and a positive slope when x is greater than 0. The only graph with these conditions is labeled II.
d) The differential equation is y' = sin(xy) cos(xy), which has a horizontal tangent line when y is equal to 0. The only graph with this condition is labeled III.
The solution graphs for the differential equations y' = 1 + x2 + y2, y' = xe?x2 ? y2, y' = 1/(1 + ex2 + y2), and y' = sin(xy) cos(xy) are labeled I, II, III, and IV respectively. The slope of the graphs when x is less than 0 and greater than 0, as well as the horizontal tangent line when y is equal to 0, are used to determine the correct graph for each equation.
Learn more about equation here
https://brainly.com/question/29657992
#SPJ4
explain precisely why we cannot apply the mean value theorem to function f(x) on the provided interval:
The Mean Value Theorem states that there exists a point c in the interval (a,b) such that f'(c) equals the function's average rate of change throughout [a,b] if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b).
Describe the mean value theorem.The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then at some point c on the interval (a, b), f'(c) must equal the function's average rate of change over [a, b].
There is at least one point on a planar arc between two endpoints where the tangent to the arc is parallel to the secant between its ends.
To learn more mean value theorem refer to:
brainly.com/question/19052862
#SPJ4
The complete question is: Explain why the Mean Value Theorem does not apply to the function on the interval [0,6].
What is 3(4 + x) + 2 x 3 = ?
50 POINTS!!! Simplify each expression by combining like terms. NO SPACES in your answer.
−5.2t+8.2−7.8t
What is the answer in simpelest form?
Answer: -13t+8.2
Step-by-step explanation:
When combining like terms, only the same terms can be combined. In this case, it is -5.2t and -7.8t are the same. If we add them together, we got -13, unfortunately, 8.2 is different, so our simplest form is
-13t+8.2.
Four students used the same meter stick to measure an edge of the same wooden cube four times. They obtained the following results
Which student'$ measurements represent the greatest precision? A Student I B Student II C. Student III D Student IV
the standard deviation of 2nd student data is least so the measurement of 2nd student is more precise.
In this problem we have four sample of four students which measure by same meter stick. Average of all student's data is 12.4 which is same. we have to check whose measurement is precision. according to least standard deviation is least which data is more precision.
the standard deviation=[tex]\sqrt{\frac{\sum(xn-x')^2}{n-1} }[/tex] ,where x' is average of data .
by the above formula.
We have standard deviation for first student is 0.40.
2nd student is 0.23, 3rd student is 0.52 and 4th student is 0.36.
after examining the standard deviation of all data 2nd student measurement is more precision.
The complete question is,
Four students used the same meter stick to measure an edge of the same wooden cube four times. They obtained the following results:
Student I II III IV
11.90 cm 11.95 cm 11.90 cm 11.95 cm
12.25 cm 12.35 cm 12.15 cm 12.30 cm
12.70 cm 12.60 cm 12.45 cm 12.55 cm
12.75 cm 12.70 cm 13.10 cm 12.80 cm
Average 12.40 cm 12.40 cm 12.40 cm 12.40 cm
Which student’s measurements represent the
greatest precision?
A. Student I
B. Student II
C. Student III
D. Student IV
learn more about standard deviation.
https://brainly.com/question/23907081
#SPJ4
What are the domain and range of the following function: (3, 5), (-2, 4), (6, 1), (8, 7)?
ron ate 2/5 of a pie, carolyn ate 3/10 of the same pie. how much of the pie is left
Answer: 3/10
Hope this helps :)
Step-by-step explanation:
2/5= 4/10
4/10 + 3/10= 7/10
3/10 of the pie is left
PLEASE SOMEONE EXPLAIN HOW TO DO THIS QUESTION SO CONFUSED AND DUE IN AND HOUR
The minimum number of students is 3.
Average of grouped data:The mean is a value that is used to determine the central tendency of the data and is the average or computed central value of a set of values.
The method of determining the mean of a set of data that are grouped together in several categories is known as the mean of grouped data.
A frequency table must be set across the data's frequencies in order to calculate the mean of grouped data, which makes the calculation simple.
[ Here we need to create a frequency table to solve the problem ]
Here we have
Score 10 20 30 40
No of students 6 7 8 ?
To calculate the average of the score make the table as follows
Let 'x' be the number of students that the class average over 23
Score (x) No of students(f) fx
10 6 60
20 7 140
30 8 240
40 x 40x
∑f = 21 + x ∑fx = 440 + 40x
Average of score = ∑fx/ ∑f = [ 440 + 40x ]/21+x
As we assumed the average score is 23
=> [440 + 40x ]/21+x = 23
=> [440 + 40x ] = [ 21+x ] (23)
=> 440 + 40x = 483 + 23x
=> 40x - 23x = 483 - 440
=> 17x = 43
=> x = 43/17 = 2.53 (approx 3 )
Therefore,
The minimum number of students is 3.
Learn more about Average at
https://brainly.com/question/16612044
#SPJ1
HELP!!!
Ivan must choose a number between 55 and 101 that is a multiple of 4, 10, and 20. Write all the numbers that he could choose. If there is more than one number, separate them with commas.
Answer:
60, 80, 100
Step-by-step explanation:
What you need to do in this case is find animver between 55 and 101 that has a first number as an even number and the second number as zero
Answer:
60,80,100
Step-by-step explanation:
I don't even know
the sine of the angle at a is equal to side bc divided by side ab. the sine of angle a is therefore equal to:
sin(a) = bc/ab. is the angle of a is therefore the sine of angle at a is equal to side bc divided by side ab.
this is explained as below as :
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. In this case, the angle at "a" is opposite side "bc" and the hypotenuse is side "ab". Therefore, the sine of angle "a" is equal to the ratio of side "bc" divided by side "ab", or sin(a) = bc/ab.
The sine of an angle in a triangle is a trigonometric function that is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The hypotenuse is the side of a right triangle that is opposite the right angle.
In this particular case, the statement "the sine of the angle at a is equal to side bc divided by side ab" is referring to a right triangle where angle "a" is one of the angles, side "bc" is the side opposite angle "a" and side "ab" is the hypotenuse.
This is known as the sine ratio and it is used in trigonometry to calculate the sine of an angle in a right triangle. It is important to note that the sine ratio is only valid for right triangles, and the measure of the angle must be in radians.
It's also worth mentioning that this relationship holds true for any angle in a triangle, not only in right triangles, but in non-right triangles sine, cosine and tangent ratios are defined differently.
So, the sine of angle "a" is calculated as the ratio of side "bc" to side "ab" (bc/ab).
To know more about sine click here:
brainly.com/question/30162646
#SPJ4
Two times the interior angle of a regular polygon is equal to seven times is exterior angle. Find the interior angle of the polygon and the number of sides in it.
The interior angle of a regular polygon with n sides is (n-2) times 180 degrees divided by n.
The exterior angle of a regular polygon with n sides is 360 degrees divided by n.
A polygonal shape is what?
A polygon is a two-dimensional, closed form that is flat or planar and is limited by straight sides. Its sides are not curled. A polygon's edges are another name for its sides. The vertices (or corners) of a polygon are the places where two sides converge.
Given that 2 * interior angle = 7 * exterior angle, we can set up the following equation:
2 * ( (n-2) * 180 / n ) = 7 * ( 360 / n )
Solving for n, we get:
n = (1440 / (2*(7-180)) ) = 9
So the polygon has 9 sides. To find the interior angle, we substitute n back into the formula:
(n-2) * 180 / n = (9-2) * 180 / 9 = 160
So the interior angle of the polygon is 160 degrees.
learn more about polygonal shape
https://brainly.com/question/1592456
#SPJ4
there are four balls in a bag, but it is not known of what colours they are ; one ball is drawn and found to be white : find the chance that all the balls are white.
So the chance of drawing a white ball in the first draw is 1/4 and the chance that all the balls are white is the same.
What is the straightforward meaning of probability?
A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
If one ball is drawn and found to be white, and it is not known of what colors the other balls are, the probability that all the balls are white is 1/4 or 25%. This is because there is only one outcome where all the balls are white (all four balls are white) out of the four total possible outcomes (all four balls are white, three are white and one is not, two are white and two are not, one is white and three are not, and none are white). So the chance of drawing a white ball in the first draw is 1/4 and the chance that all the balls are white is the same.
Learn more about probability
https://brainly.com/question/13604758
#SPJ4
The numeric components of the electric field in a region of space can be calculated using the equations E; fzy and Ey (0, Y) and (X,Y) ? Which of the following expressions vields the potentia difference between the points (0,y) and (x,Y)a. -6yb. 6yc. -3x^2yd. 3x^2ye. 3x^2y-6
The potential difference between the points (0, y) and (x, y) is given by the equation V = 3x^2y - 6yx.
e. 3x^2y-6y
The potential difference between two points (x, y) and (0, y) is given by the equation V = -E⋅x. The electric field in the region is given by the equations E; fzy and Ey (0, Y) and (X,Y). Using these equations, the electric field in the region can be calculated as E = 3x^2y - 6y. Therefore, the potential difference between the points (0, y) and (x, y) is given by V = -(3x^2y - 6y)⋅x = 3x^2y - 6yx.
The potential difference between the points (0, y) and (x, y) is given by the equation V = 3x^2y - 6yx.
Learn more about point here
https://brainly.com/question/28224145
#SPJ4
Ishi bought a $6.95 canvas and 8 tubes of paint. She spent a total of $24.95 on the canvas and paints. Determine the cost of each tube of paint.
Answer:
1 tube of paint is $2.25.
Step-by-step explanation:
We can set this as an equation first. Since we know that the canvas is $6.95 and the total is $24.95, those are the values that we will work with to determine the tube price. We can set the tube price as 8x, x being the tube price, and 8 being the number of tubes Ishi plans to buy. Now, we can set an equation:
6.95 + 8x = 24.95
8x = 18
x = 18/8 (simplified is 9/4, or 2.25)
show that the least squares residuals e are heteroskedastic and correlated, even if the disturbance vector e is homoskedastic and uncorrelated.
The least squares residuals e are heteroskedastic and correlated, even if the disturbance vector e is homoskedastic and uncorrelated.
The least squares residuals e are heteroskedastic and correlated even if the disturbance vector e is homoskedastic and uncorrelated because the variance of the observed errors, Var(e) = Var(X'e) = X'Var(e)X. As X'e is not a vector of independent random variables, the variance-covariance matrix of the errors is not diagonal and therefore the errors are correlated and heteroskedastic. This can be calculated by the formula Var(X'e) = X'Var(e)X. This formula shows that the variances of the least squares residuals e depend on the parameters of the model, and that the errors are therefore correlated and heteroskedastic.
Learn more about least squares here:
https://brainly.com/question/30044885
#SPJ4
In a randomly generated list of numbers from 0-6, what is the chance that each number will concur
In a randomly generated list of numbers from 0 to 6, each number has an equal chance of occurring. Since there are 7 numbers total (0, 1, 2, 3, 4, 5, and 6), the chance of each number occurring is 1/7 or about 14.3%.
A probability simple definition is what?
A probability is a measure of the likelihood of a particular event occurring. It is a value between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur. A probability of 0.5, for example, indicates that the event has an equal chance of occurring or not occurring. A probability can be expressed as a fraction, decimal, or percentage.
Learn more about probability
https://brainly.com/question/13604758
#SPJ1
Find approximate values of the solution of the given initial value problem at T=0.1, 0.2, 0.3, and 0.4 using Euler method with h=0.1y'= 0.5-t+2y ; y(o)=1
A starting value or starting point is what is meant by the term "initial value" (plural "initial value").
What is meant by initial value?The definition of an initial value is accurate: It is a starting value or starting point. When the input value is 0, the initial value of a function is its output. For instance, the starting value would be the amount of money you had on day 0 if your function was to track how much money you made over a specific period of time. A function's initial value is the value it has when x is zero. This corresponds to the y-intercept. The graph reveals that our line crosses the y-axis at the coordinates 0 and 8. When x=0 is entered into an exponential function, the result is the function's initial value. The initial value of the exponential function f(x) = abx f (x) = a b x is a.Given,
[tex]$y_{n+1}=y_n+h\left(x_n-y_n\right)^2$[/tex]
Since
[tex]$x_0=0, y\left(x_0\right)=0.5$[/tex]
[tex]\frac{h}{x} & =0.1 \\ x & =0.3\end{aligned}$[/tex]
To learn more about initial value, refer to:
https://brainly.com/question/10155554
#SPJ4
three students reported the length of a pencil as 12 cm 12,0 cm and 12,00 cm do all three readings have the same information
yes three readings have the same information for three students reported the length of a pencil as 12 cm 12,0 cm and 12,00 cm
The precision was good while the accuracy was poor. Precision is defined
as how close measurements are with each other.
We were told the data from the measurement was between 12cm-
12cm which shows how close the measurements are hence having a good precision.
The accuracy is defined as how close a measurement is to the theoretical
or actual value. In this case, the actual value is 12cm which is far from the
values gotten from measurements hence signifying the accuracy was poor.
yes three readings have the same information for three students reported the length of a pencil as 12 cm 12,0 cm and 12,00 cm
Read more of length on
brainly.com/question/5792909
#SPJ4
X=4y 4y+X=24 what the answer
x= 12, y= 3 answer for the equation
x= 4y equation 1
4y +x= 24 equation 2
Substituting the value of x in the equation 2
4y +4y = 24
8y= 24
y= 24/8
y= 3
Now putting the derived value of y in equation 1
x=4y
x= 4*3
x= 12
y= 3
To know more about the topic please click here
https://brainly.com/question/26094713
The function f has the property that as x gets closer and closer to 3, the values of f(x) get closer and closer to 5. Which of the following statements must be true?
A) f(3) = 5
B) f(5) = 3
C) lim f(x) = 5
x-->3
D) lim f(x) = 3
x-->5
The function f has the property that as x gets closer and closer to 3, the values of f(x) get closer and closer to 5.
The function f has the property that as x gets closer and closer to 3, the values of f(x) get closer and closer to 5.
A) f(3) = 5
C) lim f(x) = 5
x-->3
The function f has the property that as x gets closer and closer to 3, the values of f(x) get closer and closer to 5. This means that the limit of f(x) as x approaches 3 is 5, so C) lim f(x) = 5 x-->3 must be true. Additionally, since the values of f(x) approach 5 as x approaches 3, it follows that f(3) must also equal 5, so A) f(3) = 5 must be true as well. D) lim f(x) = 3 x-->5 is not true since the limit of f(x) as x approaches 5 is not 3. Similarly, B) f(5) = 3 is not true since f(5) does not necessarily equal 3.
The statements A) f(3) = 5 and C) lim f(x) = 5 x-->3 must be true for the given function.
Learn more about function here
https://brainly.com/question/29633660
#SPJ4
The angle of elevation to a nearby tree from a point on the ground is measured to be
53. How tall is the tree if the point on the ground is 89 feet from the tree? Round
your answer to the nearest hundredth of a foot if necessary.
The height of the tree is approximately 51.4 feet, to the nearest hundredth of a foot.
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. Trigonometry is based on the study of the ratios of the sides of a right triangle to its angles, these ratios are called trigonometric functions and are represented by sine, cosine and tangent.
To solve the problem, we can use the tangent function. The tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle.
Given that the angle of elevation to the tree is 53 degrees, we can use the tangent function to find the height of the tree.
Let h be the height of the tree and x be the distance from the point on the ground to the tree.
tan 53 = h/x
h = x * tan 53
h = 89 * tan 53
= 89*(√3/3) ≈ 89*0.577 ≈ 51.4
Hence, the height of the tree is approximately 51.4 feet, to the nearest hundredth of a foot.
To learn more about trigonometry, visit:
https://brainly.com/question/22525788
#SPJ1
find two nonzero vectors perpendicular to 2 in r3 that are not scalar multiples of each other, 2 looks like. 2
v1 and v2 be two nonzero vectors perpendicular to 2 in R3.
We can calculate the dot product of each vector with 2 to determine if they are perpendicular.
v1•2 = (x1, y1, z1)•(2, 0, 0) = 2x1
v2•2 = (x2, y2, z2)•(2, 0, 0) = 2x2
If 2x1 = 0 and 2x2 = 0, then v1 and v2 are perpendicular to 2. To make sure they are not scalar multiples of each other, we can check that the coordinates of each vector are not all the same. Let's say v1 = (a, b, c) and v2 = (d, e, f). If a≠d, b≠e, and c≠f, then v1 and v2 are not scalar multiples of each other.
For example, let v1 = (1, 2, 3) and v2 = (4, 5, 6). Then v1•2 = 2 and v2•2 = 8, which both equal 0. Therefore, v1 and v2 are perpendicular to 2. Furthermore, since 1≠4, 2≠5, and 3≠6, v1 and v2 are not scalar multiples of each other.
Learn more about perpendicular here:
https://brainly.com/question/29268451
#SPJ4
Squaroot of p^10 where p>0
Answer:
p^5
Step-by-step explanation:
p^10/2=p^5
Which of the following is an appropriate statement about Wilt’s free-throw shooting, based on this dotplot?
a) If Wilt were still only a 56% shooter, the probability that he would make at least 34 of his shots is about 0,03.
b) If Wilt were still only a 56% shooter, the probability that he would make at least 34 of his shots is about 0,03.
c) If Wilt is now shooting better than 56%, the probability that he would make at least 34 of his shots is about 0,03.
d) If Wilt is now shooting better than 56%, the probability that he would make at least 34 of his shots is about 0,03.
If Wilt were still only a 56% shooter, the probability that he would make at least 34 of his shots is about 0,03.The correct option is a
To perform this simulation, we can use a computer program to generate a random number between 0 and 1 for each shot.
If the random number is less than or equal to 0.56, then the shot is counted as a make. We can then repeat this process 50 times and count the number of shots that were made.
We can repeat this process many times, and count the proportion of times that 34 or more makes are achieved. This proportion is the estimated probability that a 56% free-throw shooter would make 34 or more in a sample of 50 shots.
Hence a) If Wilt were still only a 56% shooter, the probability that he would make at least 34 of his shots is about 0,03.
For more questions like Probability click the link below:
brainly.com/question/8942260
#SPJ4
How is a thermometer different than a number line
Answer:
A thermometer is different than a number line because a thermometer goes from the bottom up with a minimum count. As a number line goes both ways with a pointer indicating where the point stands on either side.
Step-by-step explanation:
Simplify. Rewrite the expression in the form x^nx n x, start superscript, n, end superscript. \dfrac{x^{5}}{x^2}= x 2 x 5
The rewritten expression in the form of xⁿ is x^3
In math, the expression refers the combination of variables, constants and exponents.
Given,
Here we have the expression x^5 / x^2
Now, we have to rewrite this expression into the form of xⁿ.
As per the law of exponents, we have rewrite the given expression as,
=> x^5*x^-2
Now, as per the law of exponent of multiplication, we can write it as,
=> x^5-2
Now, it can be further simplified as,
=> x^3
expression can be with one variable two variable or any number similar concept for constant also that is it can be any number
To know more about Expression here.
brainly.com/question/14083225
#SPJ4
Cold cabin? The Fathom screen shot below shows the results of taking 500 SRSs of 10 temperature readings from a population distribution that is and recording the sample minimum each time. (a) Describe the approximate sampling distribution. (b) Suppose that the minimum of an actual sample is . What would you conclude about the thermostat manufacturer's claim? Explain.
The sample distribution is approximately normal with a mean of 10 and a standard deviation of 1.
The distribution is slightly skewed to the left, with a slightly higher probability of lower readings than higher readings.
(b) If the sample minimum is lower than 9, this would indicate that the thermostat manufacturer's claim is false. This is because the manufacturer claims that the temperature will not go below 10 degrees, but the sample minimum of 9 suggests that it does. The probability of observing a sample minimum of 9 or lower is 0.25, which indicates that the probability of the temperature dropping below 10 degrees is relatively high.
Learn more about probabilityhere:
https://brainly.com/question/30034780
#SPJ4
Find the perimeter of the following trapezoid:
Answer:
17 feet
Step-by-step explanation:
perimeter is the measurement of the border of a shape
2.5 + 6 + 8 + 2.5 = 17
John Corporation produces widgets. The fixed expenses are $45,000,
and the variable expenses are $9 per widget. Express the total expense E
in terms of q, where q is the quantity of widgets produced. (Hint: E = V + F).
The total cost E expressed as a function of q, wherein q is the number of widgets manufactured= E = 9q + 45,000.
What is fixed expenses?Changes in sales or production volume have no effect on costs or expenses that are constant. They cover costs like rent, insurance, membership dues and subscriptions, equipment leases, loan payments, depreciation, management salaries, and advertising.
What exactly are variable expenses?Variable costs are expenses that alter as the volume of a product or service that a company produces fluctuates. The total of marginal costs over all units produced is what is referred to as variable costs. They may also be regarded as ordinary expenses. The two parts of total cost are fixed costs as well as variable costs.
According to the given information:The fixed expenses = $45,000,
The variable expenses = $9
E = V + F
E = 9q + 45,000
To know more about Variable costs visit:
https://brainly.com/question/27853679
#SPJ1
Use the Remainder Estimation Theorem and the method of Example 1 to prove that the Taylor series for sinx about x=π/4 converges to sinx for all x.
We know that,
f(x) = ∑[tex]\frac{f^{k} (x_{0} ) }{k!} (x-x_{0} )^{k}[/tex]
We know that,
f(x) = ∑[tex]\frac{f^{k} (x_{0} ) }{k!} (x-x_{0} )^{k}[/tex]
holds at a point x if and only if
[tex]\lim_{n \to \infty} R_n(x) = 0[/tex]
Let f(x) = sin x. The Taylor series for f(x) is
f(x) = sin x = ∑ [tex](-1)^{k}[/tex] [tex]\frac{x^{2k+1} }{(2k+1)!}[/tex]
According to the above theorem, we must show that [tex]R_{n}[/tex] ⇒0 or all x as n ⇒ infinity. For all x we have
[tex]f^{n+1} (x) =[/tex] ± sin x
or, [tex]f^{n+1} (x)[/tex] = ± cosy
and in all cases we have [tex]| f^{n+1} (x) |\leq 1.[/tex]
We need to conclude that
[tex]0\leq |R_{n}(x) | \frac{|x|^{n+1} }{(n+1)!}[/tex]
Furthermore, it follows that
[tex]\lim_{n \to \infty} \frac{|x|^{n+1} }{(n+1)!} = 0[/tex]
Using this result, it follows that [tex]|R_{n}|[/tex] ⇒0 and hence that [tex]R_{n}[/tex] ⇒0 is n⇒infinity.
Since this, is true for all x, we have proved that the Taylor series for sin x converges to sin x for all x.
Holds at a point x if
[tex]\lim_{n \to \infty} R_n(x) = 0[/tex]
Let f(x) = sin x. The Taylor series for f(x) is
f(x) = sin x = ∑ [tex](-1)^{k}[/tex] [tex]\frac{x^{2k+1} }{(2k+1)!}[/tex]
According to the above theorem, we must show that [tex]R_{n}[/tex] ⇒0 or all x as n ⇒ infinity. For all x we have
[tex]f^{n+1} (x) =[/tex] ± sin x
or, [tex]f^{n+1} (x)[/tex] = ± cosy
and in all cases we have [tex]| f^{n+1} (x) |\leq 1.[/tex]
We need to conclude that
[tex]0\leq |R_{n}(x) | \frac{|x|^{n+1} }{(n+1)!}[/tex]
Furthermore, it follows that
[tex]\lim_{n \to \infty} \frac{|x|^{n+1} }{(n+1)!} = 0[/tex]
Using this result, it follows that [tex]|R_{n}|[/tex] ⇒0 and hence that [tex]R_{n}[/tex] ⇒0 is n⇒infinity.
Since this, is true for all x, we have proved that the Taylor series for sin x converges to sin x for all x.
Read more about the Taylor series:
https://brainly.com/question/23334489
#SPJ4
if y^2-2xy+3x^3=5, then at the point (-1,2) dy/dx =
Answer:
5
Step-by-step explanation:
To find the derivative of a function at a given point, we need to use the concept of limits and the definition of a derivative. In this case, we are trying to find dy/dx at the point (-1,2) for the given function y^2-2xy+3x^3=5
Since the function is equal to 5, we can assume that y^2-2xy+3x^3-5 =0, then we can find the partial derivative with respect to x and y.
∂/∂x (y^2-2xy+3x^3-5) = -2y + 9x^2
∂/∂y (y^2-2xy+3x^3-5) = 2y - 2x
Now we can substitute the point (-1,2) into the equation,
dy/dx = -2(2) + 9(-1)^2 = -4 + 9 = 5
Therefore, dy/dx at the point (-1,2) is equal to 5
It's important to note that this result is valid only for this specific point, it does not mean that it's the slope of the function in general.
