Answer: The slope is -1/2, the y-intercept is 4, and the equation is y=-1/2x+4
Step-by-step explanation:
evaluate f(x) = 3/6x+6 when x = -1
a) undefined
b) 0
c) 3
d) 3/36
Answer:
a) Undefined.
Step-by-step explanation:
\(f(x) = \dfrac 3{6x+6} = \dfrac{3}{6(x+1)} = \dfrac{1}{2(x+1)}\\\\\\f(-1) =\dfrac{1}{2(-1+1)}= \dfrac{1}{0}\)
Boran Stockbrokers, Inc., selects four stocks for the purpose of developing its own index of stock market behavior. Prices per share for a year 1 base period, January year 3, and March year 3 follow. Base-year quantities are set on the basis of historical volumes for the four stocks. Price per Share (s) Year 1 Stock Industry Quantity Year 1 January March Year 3 Year 3 BaseY 29.50 20.75 22.50 65.00 40.0031.00 18.00 A Oil B Computer C Steel D Real Estate 100 150 75 50 49.00 47.50 29.50 4.75 6.50 Compute the price relatives for the four stocks making up the Boran index. Round your answers to one decimal place.) Price Relative Stock March Use the weighted average of price relatives to compute the January year 3 and March year 3 Boran indexes. (Round your answers to one decimal place.)
As per the concept of average, the price relatives for the four stocks making up the Boran index are as follows:
Stock A: January Year 3 - 73.88, March Year 3 - 67.16
Stock B: January Year 3 - 75.38, March Year 3 - 73.08
Stock C: January Year 3 - 82.50, March Year 3 - 73.75
Stock D: January Year 3 - 32.50, March Year 3 - 18.75
To calculate the price relatives for each stock, we need to compare the prices of each stock in different periods to the base-year price. The base-year price is the price per share in the year 1 base period. The formula for calculating the price relative is:
Price Relative = (Price in Current Period / Price in Base Year) * 100
Now let's calculate the price relatives for each stock based on the given data:
Stock A:
Price Relative for January Year 3 = (24.75 / 33.50) * 100 ≈ 73.88
Price Relative for March Year 3 = (22.50 / 33.50) * 100 ≈ 67.16
Stock B:
Price Relative for January Year 3 = (49.00 / 65.00) * 100 ≈ 75.38
Price Relative for March Year 3 = (47.50 / 65.00) * 100 ≈ 73.08
Stock C:
Price Relative for January Year 3 = (33.00 / 40.00) * 100 ≈ 82.50
Price Relative for March Year 3 = (29.50 / 40.00) * 100 ≈ 73.75
Stock D:
Price Relative for January Year 3 = (6.50 / 20.00) * 100 ≈ 32.50
Price Relative for March Year 3 = (3.75 / 20.00) * 100 ≈ 18.75
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Tracy ran 5 blocks north and then 7 blocks east. How far away is she from her starting point.
There are 10 people waiting on standby at an airport to get on the next flight when 2 seats open up. One of the seats is in first class, and the other is in coach. What error is made in the work shown below to calculate the number of ways to choose the passengers to fill the seats
Step-by-step explanation:
As I do not have the work shown to calculate the number of ways to choose the passengers, I cannot identify the specific error made. However, I can provide some insight on how to approach this type of problem correctly.
When choosing 2 people from a group of 10, we can use combinations, which are denoted as "n choose k" and represented by the formula:
n choose k = n! / (k! * (n-k)!)
where n is the total number of items in the group and k is the number of items to be chosen.
In this case, we want to choose one passenger for first class and one passenger for coach. Therefore, we can separate the group of 10 into two subgroups: one subgroup with 1 person (for first class) and another subgroup with 9 people (for coach).
Using combinations, we can calculate the number of ways to choose 1 person from the subgroup of 1 and 1 person from the subgroup of 9:
(1 choose 1) * (9 choose 1) = 1 * 9 = 9
Therefore, there are 9 ways to choose the passengers to fill the seats.
Two adjacent angles are on a straight line. One angle is (5x−28)° and the other angle is 8x°. What is the degree value of the second angle?(1 point)
Responses
128 degrees
120 degrees
52 degrees
16 degrees
Answer:
128°
Step-by-step explanation:
Adjacent angles on a straight line must add up to 180°
Using the angle measures given,
(5x−28)° + 8x° = 180°
5x - 28 + 8x = 180
13x - 28 = 180
13x = 180 + 28
13x = 208
x = 208/13 = 16
Therefore second angle = 8x° = 8(16)° = 128°
the residual plots from two different sets of bivariate data are graphed below. explain, using evidence from graph a and graph b which graph indicates
According to the evidence from graph a and graph b, graph a indicates that the model for the data was a good fit because it is a random.
Hence, the correct option is B.
Based on graph a, the residual plot shows a random scattering of points around the horizontal line at y=0. This indicates that the residuals (the differences between the observed and predicted values) are distributed randomly and do not show any clear pattern. This suggests that the data points are evenly distributed around the regression line, indicating a good fit for a linear regression model.
On the other hand, graph b shows a residual plot with a clear pattern or trend. The residuals are not randomly scattered, but instead show a curved pattern or a fan-like shape. This indicates that the model used to fit the data may not be appropriate, as it is not capturing the underlying relationship between the variables accurately.
Hence, the correct option is B.
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-- The given question is incomplete, the complete question is
"The residual plots from two different sets of bivariate data are graphed below. Explain, using evidence from graph a and graph b which graph indicates that the model for the data was a good fit and what is the reason?
A. Graph A because it is symmetrical
B. Graph A because it is random
C. Graph B because there is a pattern
D. Graph B because it is symmetrical" --
Put the following equation of a line into slope intercept form, simplifying all fractions 6x-2y=10
Answer:
y = 3x - 5
Step-by-step explanation:
1. Since we know the equation of a line in slope-intercept form is y = mx + b, we have to solve for the variable, y.
2. (Solving)
Step 1: Add -6x to both sides
\(6x - 2y + (-6x) = 10 + (-6x)\) \(-2y = -6x + 10\)Step 2: Divide both sides by -2.
\((-2y)/(-2) = (-6x+10)/(-2)\) \(y = 3x - 5\)Therefore, the slope-intercept form of 6x - 2y = 10 is equal to y = 3x - 5.
The slope - intercept form of the equation is, y = 3x - 5.
What is slope - intercept from of the equation?
y=mx+b, where m is the slope and b is the y-intercept, is the slope intercept form. The graph of the linear equation can be drawn on the x-y coordinate plane using this form of the equation.
The given equation is, 6x - 2y = 10
Since we are aware that the slope-intercept form equation for a line is y = mx + b, we must find the value of the variable y.
Consider, the equation 6x - 2y = 10
Subtract 6x from both sides,
6x - 2y - 6x = 10 - 6x
-2y = 10 - 6x
Divide both sides by 2,
\(\frac{-2y}{2} = \frac{10 - 6x}{2} \\-y = 5 - 3x\)
Multiply both sides by -1.
y = -5 + 3x
y = 3x - 5
Therefore, the equation of a line into slope - intercept form is, y = 3x - 5.
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- 4u + 7 = - 10u + 37
Solve for u
Answer:
5
Step-by-step explanation:
The sum of two numbers is twenty-four. Four less than three times the
smaller is twelve less than twice the larger. Find the
two numbers.
can you please give me the equation for this tyyyy
With the help of two variable equations, the two numbers are 8 and 16.
How to solve the system of equations?There are three ways to solve equations-
1. Substitution method
2. Elimination method
3. Graphing method.
Now, let the number be x and y.
Therefore, the sum of two numbers is twenty-four, x+y= 24
Now, given that,
3x-4=2y-12
3x-2y=-8
2y-3x=8
Now, substituting the value of y in the equation
2(24-x)-3x=8
40=5x
x=8
Therefore, the two numbers are 8 and 16.
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you have a score of x = 65 on an exam. which set of parameters would give you the best grade on the exam?
If the class has a mean (μ) of 60 and a standard deviation (σ) of 10, and your score is 65, then you would be above the mean but still within one standard deviation. In this case, the set of parameters μ = 60 and σ = 10 would likely give you a relatively good grade.
To determine which set of parameters would give you the best grade on the exam, we need to understand the grading scheme and how your score is compared to the rest of the class. Specifically, we need to know the mean (μ) and standard deviation (σ) of the exam scores for the entire class.
If the grading scheme involves a curve, where your score is compared to the mean and standard deviation of the class, then the set of parameters that would give you the best grade would depend on the distribution of scores in the class.
If the class has a mean (μ) of 60 and a standard deviation (σ) of 10, and your score is 65, then you would be above the mean but still within one standard deviation. In this case, the set of parameters μ = 60 and σ = 10 would likely give you a relatively good grade.
However, if the class has a different mean and standard deviation, or if the grading scheme does not involve a curve, then a different set of parameters might give you the best grade.
Without more specific information about the grading scheme and the distribution of scores in the class, it is difficult to determine the exact set of parameters that would result in the best grade for you.
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A 6 square units
B 8 square units
C 10 square units
D 16 square units
Hello everyone!
\( \huge \tt\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }\)
Can someone try to do this question using the derivative rule for quotient? No spam please!
Hello Carousel!
\( \huge \boxed{\mathfrak{Question} \downarrow}\)
\( \huge \tt\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }\)
\( \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}\)
\(\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) } \\ \)
To start solving this question, note that ⇨ for any 2 differentiable functions, the derivative of the quotient of the 2 functions will be the denominator multiplied by the derivative of the numerator minus the numerator again multiplied by the derivative of the denominator whole divided by the denominator². By doing all these steps, we'll get it as..
\(\frac{\left(x^{1}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}-2)-\left(3x^{2}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-5)}{\left(x^{1}-5\right)^{2}} \\ \)
Remember that the derivative of the polynomial will be the sum of the derivatives of its terms. We know that, the derivative of a constant term is 0 & the derivative of \(ax^{n}\) is \(nax^{n-1}\). So..
\(\frac{\left(x^{1}-5\right)\times 2\times 3x^{2-1}-\left(3x^{2}-2\right)x^{1-1}}{\left(x^{1}-5\right)^{2}} \\ \)
Now, simplify it..
\(\frac{\left(x^{1}-5\right)\times 2\times 3x^{2-1}-\left(3x^{2}-2\right)x^{1-1}}{\left(x^{1}-5\right)^{2}} \\ = \frac{\left(x^{1}-5\right)\times 6x^{1}-\left(3x^{2}-2\right)x^{0}}{\left(x^{1}-5\right)^{2}} \\ = \frac{x^{1}\times 6x^{1}-5\times 6x^{1}-\left(3x^{2}x^{0}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{1+1}-5\times 6x^{1}-\left(3x^{2}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{2}-30x^{1}-\left(3x^{2}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{2}-30x^{1}-3x^{2}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{\left(6-3\right)x^{2}-30x^{1}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{3x^{2}-30x^{1}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{3x^{2}-30x-\left(-2\right)}{\left(x-5\right)^{2}} \\ = \large \boxed{\boxed{ \bf \frac{3x^{2}-30x + 2}{\left(x-5\right)^{2}} }}\)
You can further simplify the answer to \(\underline{\underline{\frac{146}{\left(x-5\right)^{3}}}}\\\)__________________
Hope it'll help you!
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PLEASE PLEASE HELP!! WHAT EXACTLY SHOULD I TYPE DOWN IN THE BOX?
The area of a triangle is 1/2 x base x height.
The base is 2 , the height is 4.
Area = 1/2 x 2 x 4 = 4 square cm.
The area of a square is s^2
S is the length of a side, which is 2
Area = 2^2 = 4 square cm
Both shapes have an area of 4 square cm
Answer: 4.5.
Step-by-step explanation for the triangle:
A = 1/2 ⋅ b ⋅ h A = 1/2 ⋅ 3 ⋅ 3 A= 4.5
OR
Step-by-step explanation for the square:
A = a^2 = 2^2 = 4 = 4.5
The average high temperatures in degrees for a city are listed.
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
If a value of 101° is added to the data, how does the mean change?
The mean decreases by 1.6°.
The mean increases by 1.6°.
The mean decreases by 8.4°.
The mean increases by 8.4°.
Answer: The mean increases by 1.6°.
Step-by-step explanation:
The mean is the average or the sum of all the data divided by the amount of data.
58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57 = 965
There are 12 data points so...
965 / 12 ≈ 80.4
To find the new mean add 101 to 965, then divide by 13.
965 + 101 = 1066
1066 / 13 = 82
The mean increases by 1.6°.
Hope this helps!
The product of rational numbers can always br written as ?
The product of rational numbers can always be expressed as the ratio of two integers, where the denominator is not zero.
The product of rational numbers can always be written as a rational number. A rational number is defined as the quotient of two integers, where the denominator is not zero. When we multiply two rational numbers, we are essentially multiplying the numerators and denominators separately.
Let's consider two rational numbers, a/b and c/d, where a, b, c, and d are integers and b, d are not equal to zero. The product of these rational numbers is (a/b) * (c/d), which can be simplified as (a * c) / (b * d). Since multiplication of integers results in another integer, both the numerator and denominator are integers.
Furthermore, as long as the denominators b and d are not zero, the product remains a valid rational number.
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alex has $23.25 in his pocket. he wants to buy apples, limes, and kale to make juice. the kale and limes all together cost over $14. which inequality represents the dollar amount, a, that alex can spend on apples?
The inequality that represents the dollar amount, a, that Alex can spend on apples is a < $9.25.
To determine the inequality, we need to consider that Alex has $23.25 in his pocket and wants to buy apples, limes, and kale. The kale and limes together cost over $14, which means their combined cost is greater than $14. To find the maximum amount Alex can spend on apples, we subtract the amount spent on kale and limes from the total amount he has.
Let's represent the amount spent on kale and limes as k + l, where k represents the cost of kale and l represents the cost of limes. The inequality representing the dollar amount, a, that Alex can spend on apples is given by:
a < $23.25 - (k + l)
Since the cost of kale and limes is greater than $14, we have k + l > $14. Substituting this into the inequality, we get:
a < $23.25 - $14
Simplifying, we have:
a < $9.25
Therefore, the inequality that represents the dollar amount, a, that Alex can spend on apples is a < $9.25.
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calculate the surface area and then the volume
Answer:
46
Step-by-step explanation:
length x width x height
7 x 5 x 3
Answer: surface area = 142
Volume = 105
* make sure to add labels (units^2, etc.)
Step-by-step explanation:
Area = length x height
Volume = length x width x height
What is the linear distance traveled in one revolution of a 36-in diameter wheel.
The linear distance traveled in one revolution of a wheel can be calculated using the formula:
Circumference = π * Diameter
Given that the diameter of the wheel is 36 inches, we can substitute the value into the formula:
Circumference = π * 36 inches
Using an approximate value of π as 3.14159, we can calculate the circumference:
Circumference ≈ 3.14159 * 36 inches
Circumference ≈ 113.09724 inches
Therefore, the linear distance traveled in one revolution of a 36-inch diameter wheel is approximately 113.09724 inches.
Select all the expressions that are equivalent to 4 – x. x – 4 4 + -x -x + 4 -4 + x 4 + x Use the distributive property to write an expression that is equivalent to 5(-2x – 3). If you get stuck, use the boxes to help organize your work.
Answer:
B
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
In the diagram, point D divides line segment AB in the ratio of 5:3. If line segment AC is vertical and line segment CD is horizontal, what are the
coordinates of point C?
B = (10,2)
C = (?)
D
A = (2-6)
A.(5,-3). B.(2,-3) c.(7,-1) d.(2,-1)
Answer:
Step-by-step explanation:
2,-1
The coordinated of the point C are (2, -1)
What are coordinated?Coordinates are a set of values which helps to show the exact position of a point in the coordinate plane.
Given: In the diagram, point D divides line segment AB in the ratio of 5:3.
Since, line segment AC is vertical then the x coordinate of C will be same as A (2) and line segment CD is horizontal then the y coordinate of C will be same as D.
Coordinates of A = (2,-6)
Coordinates of B = (10,2)
Section formula:
If a point P(x, y) divides a line segment MN with in ration m:n, then,
x coordinate = mx₂+nx₁ / m+n
y coordinate = my₂+ny₁ / m+n
Using section formula, we have
The y coordinate of C = 5(2)+3(-6) / 8 = (2, -1)
Hence, the coordinated of the point C are (2, -1)
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pls help if you can asap!!!!
Answer: A
Step-by-step explanation: I would say A because the angle is greater than 90 degrees
Answer:
We have supplementary angles.
76 + 3x + 2 = 180
3x + 78 = 180
3x = 102
x = 34
A. In the computer game Civilization, a character representing a world leader who advocates non-violence is uncharacteristically aggressive. In the first version of the game, this was an unintentional effect caused by a bug.
In the game, world leader characters are assigned an ‘aggression’ score and this is increased or decreased according to changes in circumstance. For example, if a leader adopts democracy, their aggression score is decreased by 2.
This particular world leader was given the lowest possible aggression of 1 by the game makers, to reflect that they thought the leader was unlikely to declare war on another character.
Given that the aggression score is stored as an 8-bit unsigned integer representation, do the following:
i. Write down the largest binary integer that can be represented in an 8-bit unsigned integer and by converting this to decimal, show that the largest possible aggression score is 255.
ii. Name the problem that occurs if the initial aggression score is decreased by 2.
iii. Explain, using the relevant binary numbers, why the initial aggression score becomes the maximum value of 255 when it is reduced by 2. The maximum word limit for Question 1(a)(iii) is 40 words
Therefore, the largest possible aggression score is 255.
This results in the aggression score wrapping around to the maximum value, causing an unintended behavior.
This is because an 8-bit unsigned integer has a range from 0 to 255, and subtracting 2 from the minimum value wraps around to the maximum value due to the overflow.
i. The largest binary integer that can be represented in an 8-bit unsigned integer is 11111111. Converting this binary number to decimal, we get 255. Therefore, the largest possible aggression score is 255.
ii. The problem that occurs if the initial aggression score is decreased by 2 is called an overflow. When the initial aggression score, which is already at its minimum value, is decreased by 2, it exceeds the maximum value that can be represented by an 8-bit unsigned integer. This results in the aggression score wrapping around to the maximum value, causing an unintended behavior.
iii. When the initial aggression score, which is represented as 00000001 in binary, is decreased by 2, we subtract 2 from it. In binary subtraction, 1 subtracted by 2 results in a borrow from the next bit, and the result is 11111111, which represents 255 in decimal. This is because an 8-bit unsigned integer has a range from 0 to 255, and subtracting 2 from the minimum value wraps around to the maximum value due to the overflow.
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Indicate below whether the equation in the box is true or false 1/8 = to 2/4 true or false
Answer:
false
Step-by-step explanation:
1/8 = .125
2/4 = .5
therefore 1/8 does not = 2/4
help help help ! (ITS DUE)
Answer:
y÷8
Step-by-step explanation:
The quotient is the result of division between two numbers.
SOOO its y÷8 because a quotient is a division problem and y and 8 are the variables in the equation!
Make cos C the subject of the formula
C?= a? + b? - 2ab cos C
Work Shown:
\(c^2 = a^2 + b^2 - 2ab\cos(C)\\\\c^2 + 2ab\cos(C) = a^2 + b^2\\\\2ab\cos(C) = a^2 + b^2 - c^2\\\\\boldsymbol{\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}}\\\\\)
Find the x - and y -intercepts of the graph of the linear equation 2 x plus 3 y is equal to 2x+3y=12
Answer:
x-intercept is 6 [(6, 0) is the point]
y-intercept is 4 [(0, 4) is the point]
Step-by-step explanation:
To find the x-intercept, we can set y equal to 0 and solve for x. This works because the x-intercept is the point of the graph where the line crosses the x-axis (meaning y is equal to 0 at this point. So, if we set y=0 and solve, we get this:
\(2x+3(0)=12\\2x=12\\x=6\)
So, the x-intercept is 6; if it is written as a point, it is (6, 0) since y is 0
To find the y-intercept, we can set x equal to 0 and solve for y. This works the same way - x is equal to 0 at the point where the line crosses the y-axis. If we set x=0 and solve, we get this:
\(2(0)+3y=12\\3y=12\\y=4\)
So, the y-intercept is 4; if it is written as a point, it is (0, 4) since x is 0
ILL MARK BRAINIEST IF YOU CAN DO THE RIGHT!!!
Answer:
5/2
Step-by-step explanation:
We first find the equation of the line. The slope-intercept form of a line is y = mx + b, where b is the y-intercept and m is the slope. We already know the y-intercept, which is (0,5). Therefore, our current equation is:
y = mx + 5.
We use (0,5) and (1,3) to find the slope. We subtract the two y-coordinates, 5 - 3 = 2. We subtract the x-coordinates, 0 - 1 = - 1. The slope is given as rise/run, which is (y-coordinate difference)/(x-coordinate difference. Thus, the slope is 2/-1, which is just -2. Now, we put -2 in place of m:
y = -2x + 5.
The x-intercept is when the y-coordinate is equal to 0. So we substitute 0 for y:
0 = -2x + 5
-5 = -2x
x = 5/2
Brooke and lamont ran a half marathon. Brooke finished in 1 hour 50 minutes, lamont finished in 100 minute. Who had the faster time
Given:
Brooke and Lamont ran a half marathon. Brooke finished in 1 hour 50 minutes, Lamont finished in 100 minute.
To find:
Who had the faster time?
Solution:
We know that,
1 hour = 60 minutes
Time taken by Brooke = 1 hour 50 minutes
= 60 minutes + 50 minutes
= 110 minutes
Time taken by Lamont = 100 minutes.
Since the time taken by Lamont is less than the Brooke, therefore Lamont had the faster time.
find the equation of the tangent line to the curve 3x3 3y2−11=4xy−x at the point (1,−1).
The equation of the tangent line to the curve 3x3, 3\(y^{2}\) - 11 = 4xy - x at the point (1,−1) , then the slope = \(-\frac{3}{2}\).
The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).
A slope graph looks like a line graph, but with an important difference: there are only two data points for each line.
Given that,
The equation of the tangent line to the curve 3x3 ,
3\(y^{2}\) - 11 = 4xy - x
3\(y^{2}\) - 4xy + x - 11 = 0
Differentiate both side with respect to 'x'
6y \(\frac{dy}{dx}\) - 4 ( x \(\frac{dy}{dx}\) + y ) - 1 = 0
For slope at point ( 1 , -1 ) put x = 1 and y = -1
6(1) \(\frac{dy}{dx}\) - 4 ( 1 \(\frac{dy}{dx}\) - 1 ) -1 = 0
6 \(\frac{dy}{dx}\) - 4 \(\frac{dy}{dx}\) + 4 -1 = 0
2 \(\frac{dy}{dx}\) + 3 = 0
2 \(\frac{dy}{dx}\) = 0 - 3
2 \(\frac{dy}{dx}\) = - 3
\(\frac{dy}{dx}\) = -3/2
slope = \(-\frac{3}{2}\)
Therefore,
The equation of the tangent line to the curve 3x3, 3\(y^{2}\) - 11 = 4xy - x at the point (1,−1) , then the slope = \(-\frac{3}{2}\).
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Find the exact values of the six trigonometric functions functions of the angles 0 shown in the figure. Sin, cos, tan, csc, sec and cot (Use the pythagorean theorem to find the third side of the triangle)
Answer:
\(undefined\)Explanation:
Let x represent the length of the third side of the given triangle.
We can go ahead and determine the value of x using the Pythagorean theorem as seen below;
\(\begin{gathered} 41^2=x^2+40^2 \\ 1681=x^2+1600 \\ x^2=1681-1600 \\ x^2=81 \\ x=\sqrt[]{81} \\ x=9 \end{gathered}\)So the length of the third side of the triangle is 9
We can now determine the value of sine theta as seen below;
\(\begin{gathered} \sin \theta=\frac{opposite\text{ side to angle }\theta\text{ }}{\text{hypotenuse}}=\frac{40}{41} \\ \therefore\sin \theta=\frac{40}{41} \end{gathered}\)We can see that sine theta is 40/41
Let's determine the value of cosine theta as seen below;
\(\begin{gathered} \cos \theta=\frac{\text{adjacent side to angle }\theta}{\text{hypotenuse}}=\frac{9}{41} \\ \therefore\cos \theta=\frac{9}{41} \end{gathered}\)So cosine theta is 9/41
Let's determine the value of tangent theta as seen below;
\(\begin{gathered} \tan \theta=\frac{opposite\text{ side to angle }\theta}{\text{adjacent side to angle }\theta}=\frac{40}{9} \\ \tan \theta=\frac{40}{9} \end{gathered}\)So tangent theta is 40/9
Let's now determine the value of cosecant theta as seen below;
\(\begin{gathered} \csc \theta=\frac{1}{\sin\theta}=\frac{1}{\frac{40}{41}}=1\div\frac{40}{41}=1\times\frac{41}{40}=\frac{41}{40} \\ \therefore\csc \theta=\frac{41}{40} \end{gathered}\)So the value of cosecant theta is 41/40
Let's determine the value of secant theta as seen below;
\(\begin{gathered} \sec \theta=\frac{1}{\cos\theta}=\frac{1}{\frac{9}{41}}=1\div\frac{9}{41}=1\times\frac{41}{9}=\frac{41}{9} \\ \therefore\sec \theta=\frac{41}{9} \end{gathered}\)So the value of secant theta is 41/9
Let's determine the value of cotangent theta as seen below;
\(\begin{gathered} \cot x=\frac{1}{\tan x}=\frac{1}{\frac{40}{9}}=1\div\frac{40}{9}=1\times\frac{9}{40}=\frac{9}{40} \\ \cot x=\frac{9}{40} \end{gathered}\)So the value of cotangent theta is 9/40