The limit along the line y = x is different from the limits along the x-axis and y-axis, the limit as (x, y) approaches (0, 0) does not exist for the given expression.
To find the limit as (x, y) approaches (0, 0) of the expression y^2 sin(x^2) / (x^4 y^4), we can analyze the limit along different paths and see if they converge to the same value. If they do not, the limit does not exist.
Let's consider the limit along the x-axis first, where y = 0:
lim(x->0) [0^2 sin(x^2) / (x^4 * 0^4)] = 0.
Next, let's consider the limit along the y-axis, where x = 0:
lim(y->0) [y^2 sin(0^2) / (0^4 * y^4)] = 0.
Now, let's examine the limit along the line y = x:
lim(x->0) [x^2 sin(x^2) / (x^4 * x^4)] = lim(x->0) [sin(x^2) / x^6].
By applying L'Hôpital's rule repeatedly, we can find the limit:
lim(x->0) [sin(x^2) / x^6] = lim(x->0) [2x cos(x^2) / 6x^5] = lim(x->0) [2 cos(x^2) / 6x^4] = lim(x->0) [cos(x^2) / 3x^4] = lim(x->0) [(-2x sin(x^2)) / (12x^3)] = lim(x->0) [(-2 sin(x^2)) / (12x^2)] = lim(x->0) [(-4x cos(x^2)) / (24x)] = lim(x->0) [(-4 cos(x^2)) / 24] = (-4/24) = -1/6.
Since the limit along the line y = x is different from the limits along the x-axis and y-axis, the limit as (x, y) approaches (0, 0) does not exist for the given expression.
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find y' and y''. y = ln x 8 x2 y' = y'' =
The derivative of y = ln(x) is y' = 1/x. Taking the second derivative, we have y'' = -1/x^2.
To find the derivative of y = ln(x), we can use the basic differentiation rule for logarithmic functions. The derivative of ln(x) with respect to x is 1/x. Therefore, the first derivative of y = ln(x) is y' = 1/x.
To find the second derivative, we need to differentiate y' = 1/x with respect to x. Applying the differentiation rule for 1/x, we obtain y'' = -1/x^2.
The second derivative y'' = -1/x^2 indicates the rate of change of the slope of the original function y = ln(x). It tells us how quickly the slope of the function is changing at each point.
Since the derivative of y' is negative, it means that the slope of y' is decreasing as x increases. In other words, as x gets larger, the rate of change of the slope becomes smaller and smaller.
In summary, the derivative of y = ln(x) is y' = 1/x, and the second derivative is y'' = -1/x^2.
These derivatives help us understand the behavior of the logarithmic function and provide information about its rate of change and concavity.
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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
2
�
+
�
=
2x+y=
3
3
−
2
�
−
�
=
−2x−y=
−
6
−6
The given system of equations has no solutions.
To determine the number of solutions for the given system of equations, let's analyze the equations:
Equation 1: 2x + y = 3
Equation 2: -2x - y = -6
We can solve this system of equations using the method of elimination or substitution.
Method 1: Elimination
If we add both equations, we get:
(2x + y) + (-2x - y) = 3 + (-6)
2x + y - 2x - y = -3
0 = -3
Since 0 does not equal -3, we have a contradiction. The left side of the equation simplifies to 0, but the right side is -3. This means that the system of equations is inconsistent and has no solutions. The lines represented by the equations are parallel and will never intersect.
Therefore, the given system of equations has no solutions.
Alternatively, we can also visualize this geometrically. The first equation represents a line, and the second equation represents another line. Since the lines are parallel, they will never intersect, indicating that there are no solutions.
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im confused on how to solve it. I just need clarification
The requried, Cole purchased 11 songs and 15 movies.
Let x be the number of songs that Cole purchased, and let y be the number of movies that he purchased.
We can write a system of equations based on the information given:
1.25x + 2.75y = 55 (the total amount spent is $55.00)
x + y = 26 (the total number of songs and movies purchased is 26)
To solve this system of equations, we can use substitution or elimination. Here, we will use substitution.
First, we can rearrange the second equation to solve for one of the variables:
x + y = 26
x = 26 - y
Then, we can substitute this expression for x into the first equation:
1.25x + 2.75y = 55
1.25(26 - y) + 2.75y = 55
32.5 - 1.25y + 2.75y = 55
1.5y = 22.5
y = 15
Finally, we can substitute this value of y back into the equation we found for x:
x = 26 - y
x = 26 - 15
x = 11
Therefore, Cole purchased 11 songs and 15 movies.
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i need help with this
Answer:
y < 3/4x -2
Step-by-step explanation:
You want the inequality expression that corresponds to the given graph.
SlopeThe boundary line rises 3 squares for each 4 to the right. Its slope is ...
m = rise/run = 3/4
Y-interceptThe boundary line crosses the y-axis at y = -2. Its y-intercept is ...
b = -2
Boundary line equationThe slope-intercept form of the equation of the boundary line is ...
y = mx +b
y = 3/4x -2
ShadingThe shading is below the dashed line, so the line is not part of the solution set. Only y-values less than those on the line are in the solution set.
The inequality that describes the graph is ...
y < 3/4x -2
Test the series for convergence or divergence using the Alternating Series Test. Σ(-1) 7n – 5 8n + 5 n = 1 Identify b n' Evaluate the following limit. lim bn n-00 Since limb n n00 ? O and bn + 1 ? v bn for all n, ---Select---
The given series is Σ(-1)^n (7n – 5)/(8n + 5) for n = 1 to infinity.
To apply the Alternating Series Test, we need to check if the series satisfies the following two conditions:
1) The terms of the series alternate in sign.
2) The absolute value of the terms decreases as n increases.
1) The given series alternates in sign because of the (-1)^n factor.
2) To check if the absolute value of the terms decreases as n increases, we can find the ratio of consecutive terms:
b_n = (7n – 5)/(8n + 5)
b_n+1 = (7(n+1) – 5)/(8(n+1) + 5)
So, b_n+1/b_n = [(7n+12)/(8n+13)] * [(8n+5)/(7n-5)]
= (56n^2 + 43n + 60)/(56n^2 - 41n - 65)
We can observe that the numerator is always greater than the denominator for n >= 1. Therefore, b_n+1/b_n < 1 for all n >= 1, which means that the absolute value of the terms decreases as n increases.
Since the series satisfies both conditions of the Alternating Series Test, we can conclude that the series converges.
To evaluate lim bn as n approaches infinity, we can use the fact that bn is a rational function of n. By dividing both numerator and denominator by n, we can write:
b_n = (7 - 5/n)/(8 + 5/n)
As n approaches infinity, both the numerator and denominator approach constants (7 and 8, respectively). Therefore, lim bn = 7/8.
So, the series Σ(-1)^n (7n – 5)/(8n + 5) converges to a limit of 7/8.
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What’s the answer to this question?
Answer: The answer is D. 1/4
Harper leans a 24-foot ladder against a wall so that it forms an angle of 69 degrees with the ground. What’s the horizontal distance between the bade of the ladder and the wall? Round your answer to the nearest tenth of a foot if necessary.
The horizontal distance between the base of the ladder and the wall is 8.6 feet.
What is distance?distance is the length between two points.
To calculate the horizontal distance between the ladder and the wall, we use the formula below.
Formula:
[tex]\sf cos \ \theta=\dfrac{adjacent(y)}{hypotenuse(h)}[/tex]
Make y the subject of the equation
[tex]\sf h = \dfrac{y}{ cos\ \theta}[/tex]........ Equation 2
Where:
[tex]\theta[/tex] = Angle between the wall and the laddery = horizontal distance between the base of the ladder and the wallh = Length of the ladder lean against the wallFrom the question,
Given:
y = 24 foot[tex]\theta[/tex] = 69°Substitute these values into equation 2
[tex]\sf y = \dfrac{24}{(cos \ 69^\circ)}[/tex]
[tex]\sf y = \dfrac{24}{0.358}[/tex]
[tex]\sf y =\bold{8.6 \ feet}[/tex]
Hence, The horizontal distance between the base of the ladder and the wall is 8.6 feet.
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(-7/8+-5/12)+3/16 perform the indicate operations
Answer:
Step-by-step explanation:
, (-7/8 - 5/12) + 3/16 = -53/48.
answer fast please and explain how you got it!!
Answer:
-35.375
Step-by-step explanation:
(-1.5+9.5)=8
5/8 =0.625
7+11=18
0.4*18=36/5
=7.2
7.2/-0.2=
-283/8=
-35.375
Translate the sentence into an equation.
Six times the sum of a number and 4 equals 3.
Use the variable y for the unknown number.
Answer:
6(y+4) = 3
Step-by-step explanation:
in rolling 2 fair dice what is the probability of a sum greater than 3 but not exceeding 6
We can list all possible outcomes in a table and count the number of outcomes that meet the criteria. There are 9 such outcomes, which gives a probability of 9/36, or 1/4.
To calculate the probability of rolling a sum greater than 3 but not exceeding 6 with 2 fair dice, we first need to determine the total number of possible outcomes when rolling the dice. Since each die has 6 possible outcomes, there are a total of 6 x 6 = 36 possible outcomes when rolling 2 fair dice.
| Die 1 | Die 2 | Sum |
|-------|-------|-----|
| 1 | 3 | 4 |
| 1 | 4 | 5 |
| 1 | 5 | 6 |
| 2 | 2 | 4 |
| 2 | 3 | 5 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 4 | 1 | 5 |
| 5 | 1 | 6 |
There are a total of 9 outcomes where the sum is greater than 3 but not exceeding 6. Therefore, the probability of rolling a sum greater than 3 but not exceeding 6 with 2 fair dice is 9/36, which simplifies to 1/4.
The probability of rolling a sum greater than 3 but not exceeding 6 with 2 fair dice can be calculated by dividing the number of outcomes where the sum is greater than 3 but not exceeding 6 by the total number of possible outcomes.
There are a total of 36 possible outcomes when rolling 2 fair dice, as each die has 6 possible outcomes. To determine the number of outcomes where the sum is greater than 3 but not exceeding 6, we can list all possible outcomes in a table and count the number of outcomes that meet the criteria. There are 9 such outcomes, which gives a probability of 9/36, or 1/4.
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The probability of the sum of two dice being greater than 3 but not exceeding 6 is 1/3 as there are 12 successful outcomes out of a total 36.
Explanation:This question is about the probability in rolling two dice.
When rolling two fair dice, there will be total 36 possible outcomes (6 possible outcomes for one die times 6 for the second die). The outcomes where the sum is greater than 3 but not exceeding 6 are: {1,3}, {1,4}, {1,5}, {2,2}, {2,3}, {2,4}, {3,1}, {3,2}, {3,3}, {4,1}, {4,2}, {5,1}. There total 12 such outcomes.Therefore, the probability of this event is 12/36 or 1/3.
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Assume that the Federal Reserve increases the money supply This will cause i. Interest rates to decrease ii. Consumption and investment to decrease iii. Inflation to fall 1. l and ll only 2. II and III only 3. I. II. and III 4. I only
The correct answer is 1. I and II only. Interest rates to decrease. when the Federal Reserve increases the money supply, it can lead to a decrease in interest rates and an increase in consumption and investment.
When the Federal Reserve increases the money supply, it injects more money into the economy. This can lead to a decrease in interest rates, as there is more money available for borrowing and lending. This is because an increase in the money supply can lead to a decrease in the demand for money, which in turn causes the interest rates to fall.
A decrease in interest rates can lead to an increase in consumption and investment. Lower interest rates make it cheaper for consumers to borrow money to buy goods and services, and for businesses to borrow money to invest in new projects. As a result, an increase in the money supply can lead to an increase in consumption and investment, as businesses and consumers have more money available to spend.
However, an increase in the money supply can also lead to inflation. This is because more money is chasing the same amount of goods and services, leading to an increase in prices. Inflation can erode the purchasing power of money and lead to a decrease in the standard of living.
In conclusion, when the Federal Reserve increases the money supply, it can lead to a decrease in interest rates and an increase in consumption and investment. However, it can also lead to inflation, which can have negative effects on the economy. Therefore, the correct answer is 1. I and II only.
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Write the y-intercept of the function: f(x)=2x^2-2x+1
The y-intercept of the function f(x) is 1.
The y-intercept of a function is the point where the graph of the function intersects the y-axis. It represents the value of the function when x=0. To find the y-intercept of a function, we can substitute x=0 into the function and evaluate it.
In the case of the function [tex]f(x) = 2x^2 - 2x + 1[/tex], when x=0, we have:
[tex]f(0) = 2(0)^2 - 2(0) + 1 = 1[/tex]
Therefore, the y-intercept of the function f(x) is 1. This means that the graph of the function intersects the y-axis at the point (0, 1).
Knowing the y-intercept is important when graphing the function, as it provides a reference point for drawing the graph. Additionally, the y-intercept can provide information about the behavior of the function as x approaches infinity or negative infinity.
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g students arriving to take an exam are handed one of two versions, a or b, of the test randomly when they enter the room. the tests were handed out in the following order: aabababaaabaabbbabba at 5% significance would you say this sequence is non-random?
We cannot conclude that the sequence is non-random at 5% significance.
How to determine if the the sequence is non-randomTo determine whether the sequence is non-random, we need to perform a chi-square goodness-of-fit test.
calculating the expected frequency for each version (a and b). Since there are 15 tests, we expect each version to be handed out 7.5 times (50% of 15).
Calculating the observed frequency for each version. Version a was handed out 9 times, and version b was handed out 6 times.
Using the chi-square formula, we can calculate the chi-square statistic:
χ² = ∑((Observed - Expected)²/Expected)
χ² = ((9 - 7.5)²/7.5) + ((6 - 7.5)²/7.5)
χ² = 0.5
We calculate the critical value at 5% significance using a chi-square distribution table with one degree of freedom (because there are two categories - a and b - and we already know the total number of tests).
We cannot reject the null hypothesis that the sequence is random because the calculated chi-square statistic of 0.5 is less than the critical value of 3.84.
Therefore, we cannot conclude that the sequence is non-random at 5% significance.
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in a two-player game in which each player has four available strategies, how many outcomes can there be?
The total number of possible outcomes in this two-player game is 12 Option C
Game theory is a branch of mathematics that studies decision-making in situations where multiple players are involved.
In this particular game, one player has four available strategies,
1 x 4 = 4
While the other player has three available strategies
1 x 3 = 3,
This problem can be solved by using Unitary Method. To determine the total number of possible outcomes, we need to consider all the possible combinations of strategies that each player can use. We can do this by multiplying the number of strategies available to each player.
4 x 3 = 12
Using the multiplication rule, the total number of outcomes is equal to the product of the number of strategies available to each player. Therefore, the number of outcomes in this game is 12.
This means that there are twelve possible outcomes when the two players choose their strategies independently. Each outcome represents a different combination of strategies chosen by both players.
The total number of possible outcomes in this two-player game is 12, Option C which is equal to the product of the number of strategies available to each player.
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Complete Question
In a two-player game in which one player has four available strategies and the other player has three available strategies, how many outcomes can there be?
A. 8
B. 10
C. 12
D. 16
E. 64
{ASAP}
Triangle XYZ is similar to triangle JKL.
Determine the length of side LJ.
4.59
5.13
12.48
13.12
(Use Image added)
Answer:
LJ = 13.12
Step-by-step explanation:
given the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{LJ}{ZX}[/tex] = [tex]\frac{JK}{XY}[/tex] ( substitute values )
[tex]\frac{LJ}{8.2}[/tex] = [tex]\frac{13.92}{8.7}[/tex] ( cross- multiply )
8.7 × LJ = 8.2 × 13.92 = 114.144 ( divide both sides by 8.7 )
LJ = [tex]\frac{114.144}{8.7}[/tex] = 13.12
find the general solution of the differential equation 9y'' + 48y' + 64y=0Use C1, C2, ... for the constants of integration.
The general solution of the given differential equation is y = C1e^(-8/3x) + C2xe^(-8/3x), where C1 and C2 are constants of integration.
To find the general solution of the differential equation 9y'' + 48y' + 64y = 0, we can assume a solution of the form y = e^(rx), where r is a constant to be determined.
Taking the first and second derivatives of y with respect to x, we have:
y' = re^(rx)
y'' = r^2e^(rx)
Substituting these derivatives into the differential equation, we get:
9(r^2e^(rx)) + 48(re^(rx)) + 64(e^(rx)) = 0
Factoring out e^(rx) and dividing through by e^(rx), we obtain the characteristic equation:
9r^2 + 48r + 64 = 0
To solve this quadratic equation, we can apply the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 9, b = 48, and c = 64. Substituting these values into the quadratic formula, we have:
r = (-48 ± √(48^2 - 4964)) / (2*9)
r = (-48 ± √(2304 - 2304)) / 18
r = -48 / 18
r = -8/3
Since we have repeated roots (r1 = r2 = -8/3), the general solution of the differential equation is:
y = C1e^(r1x) + C2xe^(r2x)
Substituting the values of r1 and r2, we have:
y = C1e^(-8/3x) + C2xe^(-8/3x)
Therefore, the general solution of the given differential equation is y = C1e^(-8/3x) + C2xe^(-8/3x), where C1 and C2 are constants of integration.
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the monthly rents for five apartments advertised in a newspaper were $650, $650, $800, $1900, and $820. find the mean, median, and mode of the rents.
The mean monthly rent for these five apartments is $764, the median monthly rent is $800, and the mode monthly rent is $650.
To find the mean of the rents, we add all the rents and divide by the total number of rents:
Mean = (650 + 650 + 800 + 1900 + 820) / 5 = $764
The median is the middle value when the rents are arranged in numerical order. In this case, the rents arranged in numerical order are:
650, 650, 800, 820, 1900
The middle value is the third rent, which is $800.
The mode is the value that appears most frequently in the data set. In this case, the mode is $650 because it appears twice, which is more than any other value.
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Consider the quadrilateral below below. Which statement below correctly describes whether or not the quadrilateral is a parallelogram based upon the measurements given?
The statement that correctly describes whether or not the quadrilateral is a parallelogram based upon the measurements given is this: A. The quadrilateral is a parallelogram because opposite angles are congruent.
What makes a parallelogram?A parallelogram is a four-sided representation that has two pairs of equal sides and two pairs of equal angles. The easy way to identify parallelograms is by the congruency they feature.
So, we qualify the quadrilateral as a parallelogram because the parallel angles are congruent.
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Differential Equation
Consider the system of differential equations
dxdt=?5ydydt=?5x.
Convert this system to a second order differential equation in y by differentiating the second equation with respect to t and substituting for xfrom the first equation.
Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0)=4 and y(0)=1, what are x and y?
The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.
Differentiating the second equation with respect to t, we get: d^2y/dt^2 = -5 dx/dt, Substituting dx/dt from the first equation, we get: d^2y/dt^2 = -5(-5y) = 25y.
This is a second order differential equation in y. The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.
To find x as a function of t, we can substitute y(t) into the first equation and solve for x: dx/dt = -5y = -5(c1 cos(5t) + c2 sin(5t)) , Integrating both sides with respect to t, we get: x(t) = -c1 sin(5t) + c2 cos(5t) + k
where k is a constant of integration. Using the initial conditions x(0) = 4 and y(0) = 1, we can solve for the constants c1, c2, and k: x(0) = -c1 sin(0) + c2 cos(0) + k = c2 + k = 4, y(0) = c1 cos(0) + c2 sin(0) = c1 = 1
Substituting c1 = 1 and c2 + k = 4 into the equation for x, we get:
x(t) = -sin(5t) + 4
So the solution to the system of differential equations with initial conditions x(0) = 4 and y(0) = 1 is x(t) = -sin(5t) + 4 and y(t) = cos(5t).
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Help me please......
Based on the given diagram 1 to 5, each picture represent a number, diagram 5 is 661.
How to solve algebra?Based on the diagram;
Diagram 1;
90 = 30 + 30 + 30
Each picture in diagram 1 represents 30
Diagram 2:
1 × 1 × 0 = 0
Diagram 3:
30 ÷ 1 = 30
Diagram 4:
22 × 1 - 1 = 21
Hence,
Diagram 5:
1 + 30 × 22 + 0
Using PEMDAS
P = parenthesis
E = Exponents
M = Multiplication
D = Division
A = Addition
S = Subtraction
1 + 30 × 22 + 0
= 1 + 660 + 0
= 661
Ultimately, diagram 5 equals 661.
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Solve for x. Assume that lines appear tangent are tangent.
5 is the missing value of x.
From the intersecting of two chords theorem,
Given angle= 14x-1
Arcs are 13x+8, 65°
From the theorem,
14x-1 = (13x+8 + 65)/2
14x-1 = (13x+73)/2
28x-2=13x+73
15x=75
x= 5
Therefore, the value of x will be 5 for the given figure.
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what is the coefficient of x 40 in the expansion of (x 3 x 4 x 5 x 6 x 7 · · ·) 4 ?
The coefficient of x^40 in the given expression is 220.
The coefficient of x^40 in the expansion of the given expression can be found using the concept of generating functions and combinatorics.
We can write the given expression as:
(x^3 + x^4 + x^5 + x^6 + x^7 + ...) ^ 4
= (x^3/(1-x) - x^8/(1-x)) ^ 4 [using the formula for infinite geometric series]
Now, we can expand this expression using the binomial theorem. The term x^40 will appear in the expansion of the product only if we choose the terms x^3, x^4, x^5, x^6, x^7, and x^8 in such a way that their sum is equal to 40.
Let the number of times we choose x^3 be a, the number of times we choose x^4 be b, and so on up to x^8 which we choose c times. Then, we have the following equation:
3a + 4b + 5c + 6d + 7e + 8f = 40
We need to find the number of non-negative integer solutions to this equation, which can be found using the concept of stars and bars. We can represent the equation using stars and bars as follows:
***|****|*****|****|***|**
The six bars divide the 40 stars into 7 groups. The number of stars in each group represents the number of times we choose a particular term in the product. Hence, the number of solutions to the equation is equal to the number of ways of arranging the 40 stars and 6 bars, which is (40 + 6) choose 6 = 46C6.
Therefore, the coefficient of x^40 in the given expression is the same as the coefficient of x^7 in the expression (x^3/(1-x) - x^8/(1-x))^4, which can be found by extracting the coefficient of x^7 from the expanded form of the expression. Using this method, we can find that the coefficient of x^7 is 220.
Hence, the coefficient of x^40 in the given expression is 220.
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a rock is thrown from (2,1) to (12,18). how far did the rock travel
Answer:
18.136 units of distance
Step-by-step explanation:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) = (2, 1) and (x2, y2) = (12, 18)
d = sqrt((12 - 2)^2 + (18 - 1)^2)
= sqrt(10^2 + 17^2)
= sqrt(329)
≈ 18.136
Therefore, the rock traveled approximately 18.136 units of distance.
evaluate the integral. (use c for the constant of integration.) 6 tan2(x) dx
The value of the integral ∫ 6 tan^2(x) dx is 6tan(x) - 6x + c.
We can start by using the identity tan^2(x) = sec^2(x) - 1 to rewrite the integral as:
6 tan^2(x) dx = 6(sec^2(x) - 1) dx
Now we can integrate each term separately:
∫ 6(sec^2(x) - 1) dx = 6∫sec^2(x) dx - 6∫dx
The antiderivative of sec^2(x) is tan(x), so:
6∫sec^2(x) dx - 6∫dx = 6tan(x) - 6x + c
where c is the constant of integration.
Therefore, the value of the integral ∫ 6 tan^2(x) dx is 6tan(x) - 6x + c.
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(LCM) of 24 and 64
The LCM of 24 and 64 is 192.
#1 )) . which inequality best represents the range of the graphed exponential function ?
a . y<0
b . y<-1
c . x<0
d . x<-1
#2 )) . which function is best represented by this graph ?
a . f(x)= ^2-1
b . f(x)= ^2+1
c . f(x)= -x^2+x-1
d . f(x)= -x^2+1
(( PLEASE HELP , I HAVE MORE QUESTIONS TO POST FEEL FREE TO HELP )) .
The range of the graphed exponential function is b . y < -1.
The function which is best represented by the graph is f(x) = -x² + 1.
1) Given an exponential function.
We have to find the range of the function.
The range of the function is the set of all the y values for the x values where the function is defined.
From the graph, it is clear that for any x values, the y values are all either -1 or numbers less than -1.
So the range is y < -1.
2) Given a graph of a parabola opens downwards.
So the function will be quadratic. That is, the highest degree of the variable will be 2.
For a function of the form, (parent function), y = -x², the parabola passes through the point (0, 0), which will be the vertex and the parabola is opened downwards.
Here vertex is (0, 1).
That is the parabola is shifted up to 1 unit.
A function f(x) after the translation to k units up becomes f(x) + d.
So here since the original function is shifted up 1 units, it becomes,
f(x) = -x² + 1
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Roland works in a local factory
Here is the completed piecewise function that models Roland's pay:
[tex]\[f(x) = \begin{cases} 95x & \text{if } x \leq 100 \\1.25(x-100) + 95(100) & \text{if } 101 \leq x \leq 300 \\1.55(x-300) + 95(100) + 1.25(300-100) & \text{if } x > 300\end{cases}\][/tex]
This piecewise function represents Roland's pay based on the different pay rates for the respective ranges of units produced.
To create a piecewise function to model Roland's pay, we need to consider the different ranges of units produced and the corresponding pay rates.
Let's complete the missing portions of each expression:
[tex]\[f(x) = \begin{cases} 95x & \text{if } x \leq 100 \\1.25(x-100) + 95(100) & \text{if } 101 \leq x \leq 300 \\1.55(x-300) + 95(100) + 1.25(300-100) & \text{if } x > 300\end{cases}\][/tex]
In the piecewise function:
- For [tex]\(x \leq 100\)[/tex], Roland receives 95 cents for each unit, so the expression is [tex]\(f(x) = 95x\).[/tex]
- For [tex]\(101 \leq x \leq 300\),[/tex] Roland receives $1.25 for each unit between 101 and 300. The base pay for the first 100 units (at 95 cents each) is added, resulting in the expression [tex]\(f(x) = 1.25(x-100) + 95(100)\).[/tex]
- For [tex]\(x > 300\)[/tex], Roland receives $1.55 for each unit over 300. Both the base pay for the first 100 units and the additional pay for units between 101 and 300 are added, leading to the expression [tex]\(f(x) = 1.55(x-300) + 95(100) + 1.25(300-100)\).[/tex]
This piecewise function models Roland's pay based on the different pay rates for the different ranges of units produced.
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8. give a recursive definition of the sequence {an}, n = 1, 2, 3,…if a) an = 4n − 2. b) an = 1 (−1)n. c) an = n(n 1). d) an = n2.
a) Recursive definition: a1 = 2, an = an-1 + 4 for n > 1.
b) Recursive definition: a1 = 1, an = (-1)^(n+1) for n > 1.
c) Recursive definition: a1 = 0, a2 = 2, an = (n - 1) * n + a(n-2) for n > 2.
d) Recursive definition: a1 = 1, an = a(n-1) + (2n - 1) for n > 1.
In mathematics, a sequence is an ordered list of numbers or other elements. A recursive definition of a sequence is one that defines each term of the sequence in terms of one or more previous terms. To find the nth term of the sequence, we need to know the previous terms up to n-1.
a) For the sequence {an} given by an = 4n - 2, the first few terms are 2, 6, 10, 14, 18, ... To define this sequence recursively, we can say that a1 = 2 and for n > 1, an = an-1 + 4.
b) For the sequence {an} given by an = 1^(-1)n, the first few terms are 1, -1, 1, -1, 1, ... To define this sequence recursively, we can say that a1 = 1 and for n > 1, an = -an-1.
c) For the sequence {an} given by an = n(n-1), the first few terms are 0, 2, 6, 12, 20, ... To define this sequence recursively, we can say that a1 = 0 and for n > 1, an = (n-1)an-1.
d) For the sequence {an} given by an = n^2, the first few terms are 1, 4, 9, 16, 25, ... To define this sequence recursively, we can say that a1 = 1 and for n > 1, an = an-1 + 2n - 1.
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The figure is a trapezoid. Find the value of the
variables.
a) x = 85, y = 75
b) x = 75, y = 85
c) x =95, y = 105
d) x = 105, y = 95
Step-by-step explanation:
the kind of the math
sug u. e i hmm f j. ok
