find the slope of the tangent line to the polar curve at r = sin(4theta).
Answers
The slope of the tangent line to the polar curve at
`r = sin(4θ)` is:
`dy/dx = (dy/dθ)/(dx/dθ)`
at `r = sin(4θ)`= `(4cos(4θ)sin(θ) + sin(4θ)cos(θ)) / (4cos(4θ)cos(θ) - sin(4θ)sin(θ))`
To find the slope of the tangent line to the polar curve at
`r = sin(4θ)`,
we can use the polar differentiation formula, which is:
`dy/dx = (dy/dθ)/(dx/dθ)`
For a polar curve given by
`r = f(θ)`,
we can find
`(dy/dθ)` and `(dx/dθ)`
using the following formulas:
`(dy/dθ) = f'(θ)sin(θ) + f(θ)cos(θ)` and `(dx/dθ) = f'(θ)cos(θ) - f(θ)sin(θ)`
where `f'(θ)` represents the derivative of `f(θ)` with respect to `θ`.
For the given curve,
`r = sin(4θ)`,
we have
`f(θ) = sin(4θ)`.
So, we first need to find `f'(θ)` as follows:
`f'(θ) = d/dθ(sin(4θ)) = 4cos(4θ)`
Now, we can substitute
`f(θ)` and `f'(θ)` in the above formulas to get
`(dy/dθ)` and `(dx/dθ)`
:
`(dy/dθ) = f'(θ)sin(θ) + f(θ)cos(θ)`` = 4cos(4θ)sin(θ) + sin(4θ)cos(θ)`
and
`(dx/dθ) = f'(θ)cos(θ) - f(θ)sin(θ)`` = 4cos(4θ)cos(θ) - sin(4θ)sin(θ)
Now, we can find the slope of the tangent line using the polar differentiation formula:
`dy/dx = (dy/dθ)/(dx/dθ)`
at
`r = sin(4θ)`
So, the slope of the tangent line to the polar curve at
`r = sin(4θ)` is:
`dy/dx = (dy/dθ)/(dx/dθ)`
at `r = sin(4θ)`= `(4cos(4θ)sin(θ) + sin(4θ)cos(θ)) / (4cos(4θ)cos(θ) - sin(4θ)sin(θ))`
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Related Questions
Which values represent the independent variable? (–2, 4), (3, –2), (1, 0), (5, 5) A. {–2, 3, 1, 5} B. {4, –2, 0, 5} C. {–2, 4, 3, –2} D. {–2, –1, 0, 5} Please select the best answer from the choices provided A B C D
Answers
Answer:
The independent variable is the variable that is manipulated or changed during an experiment. In this case, the independent variable is represented by the x-values of the given points.
So, the answer would be option A: {-2, 3, 1, 5}
Step-by-step explanation:
brainliest Plsssss
2x+5<3x+4 solve inequality
Answers
Please help me with this trigonometric ratio
Answers
Answer: sin = 3/5
cos = 4/5
tan = 3/4
Step-by-step explanation:
sin = opposite/hypotenuse
cos = adjacent/hypotenuse
tan = opposite/adjacent
d^2=41 as a verbal expression
Answers
\(\huge\text{Hey there!}\)
\(\huge\boxed{\mathsf{d^2 = 41}}\)
\(\huge\boxed{\text{TAKE the SQUARE ROOT}}\)
\(\huge\boxed{\mathsf{d = \pm \sqrt{41}}}\)
\(\huge\boxed{\textsf{SIMPLIFY it}}\)
\(\huge\boxed{\mathsf{d = \pm \sqrt{41}\ or \sqrt{-41}}}}\)\(\huge\boxed{\mathsf{Answer: d = \bf \sqrt{41} \ or \ \sqrt{-41}}}\huge\checkmark\)
\(\huge\text{Good luck on your assignment \& enjoy your day!}\)
~\(\huge{\boxed{\frak{Amphitrite1040:)}}}\)
Work out the size of angle x.
Answers
Answer:
x = 13°
Step-by-step explanation:
angle p + 98° = 180° [angles on a straight line add up to 180°]
p = 180° - 98°
p = 82°
angle q = 85° [alternate angles with the 85° angle]
Inside the yellow triangle:
p + q + x = 180° [angles inside a triangle add up to 180°]
82° + 85° + x = 180°
x = 180° - 82° - 85°
∴ x = 13°
Integrate the ODE
dy/dx = x² √y, 0 < x < 2, y(0) = 1
using Euler's method (Δx = 0, 2) to compute y(2). Obtain analytical solution to the ODE and compare y(2) obtained using Euler's method with that obtained analytically.
Answers
we find that the numerical approximation using Euler's method gives y(2) ≈ 1.865, while the analytical solution gives y(2) = 2.5.
Using the formula y(n+1) = y(n) + Δx * f(x(n), y(n)), where f(x, y) = x² √y, we can calculate the values of y at each step. Here's the step-by-step calculation:
Step 1: For x = 0, y = 1 (initial condition).
Step 2: For x = 0.2, y = 1 + 0.2 * (0.2)² * √1 = 1.008.
Step 3: For x = 0.4, y = 1.008 + 0.2 * (0.4)² * √1.008 = 1.024.
Step 4: For x = 0.6, y = 1.024 + 0.2 * (0.6)² * √1.024 = 1.052.
Step 5: For x = 0.8, y = 1.052 + 0.2 * (0.8)² * √1.052 = 1.094.
Step 6: For x = 1.0, y = 1.094 + 0.2 * (1.0)² * √1.094 = 1.155.
Step 7: For x = 1.2, y = 1.155 + 0.2 * (1.2)² * √1.155 = 1.238.
Step 8: For x = 1.4, y = 1.238 + 0.2 * (1.4)² * √1.238 = 1.346.
Step 9: For x = 1.6, y = 1.346 + 0.2 * (1.6)² * √1.346 = 1.483.
Step 10: For x = 1.8, y = 1.483 + 0.2 * (1.8)² * √1.483 = 1.654.
Step 11: For x = 2.0, y = 1.654 + 0.2 * (2.0)² * √1.654 = 1.865.
Therefore, using Euler's method with a step size of Δx = 0.2, we approximate y(2) to be 1.865.
To obtain the analytical solution to the ODE, we can separate variables and integrate both sides:
∫(1/√y) dy = ∫x² dx
Integrating both sides gives:
2√y = (1/3)x³ + C
Solving for y:
y = (1/4)(x³ + C)²
Using the initial condition y(0) = 1, we can substitute x = 0 and y = 1 to find the value of C:
1 = (1/4)(0³ + C)²
1 = (1/4)C²
4 = C²
C = ±2
Since C can be either 2 or -2, the general solution to the ODE is:
y = (1/4)(x³ + 2)² or y = (1/4)(x³ - 2)²
Now, let's evaluate y(2) using the analytical solution:
y(2) = (1/4)(2³ + 2)² = (1/4)(8 + 2)² = (1/4)(10)² = 2.5
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Following the order of operations, describe how the graph of y = x² has been transformed to create a new graph given the equation: y = \(-\frac{1}{2}\)(x + 4)² - 5
Answers
Answer:
Step-by-step explanation:
First we replace the original 'x' with 'x + 4,' which causes translation of the original parabolic graph 4 units to the left. Next, reflect this translated graph about the x-axis (which is the effect of multiplying by -1/2 as shown). Finally, shift the entire graph (the most recent one) down 5 units.
Consider the degree of each polynomial in the problem. The first factor has a degree of . The second factor has a degree of . The third factor has a degree of . The product has a degree of
Answers
The first factor of the expression has a degree of 2.
The second factor has a degree of 3.
The third factor has a degree of 2.
The product has a degree of 7.
\((a^{2} ) (2a^{3} )(a^{2}-8a+9)\)
The given expression of this problem is:
The degree of an expression is deduct by the exponent of each power.
So, the first factor of the expression has a degree of 2, because that's the exponent.
The second factor has a degree of 3.
The third factor has a degree of 2.
Now, to know the degree of the product, we have to solve the expression, and see what is the degree of the resulting polynomial expression:
\((a^{2})(2a^{3})(a^{2} -8a+9)\)
\(2a^{5} (a^{2} -8a+9)\)\(\\2a^{7} -16a^{6}+18a^{5}\)
so, as you can see, the product has a degree of 7.
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i need help, hope you can help me quickly. thanks!
Answers
The function y=x²+5x-6 has a minimum value. All real numbers are included in the function's domain. The function has a range of y ≤ 12.25, the correct option is (c).
The parabola has a minimum value and widens upwards since the x² term's coefficient is positive.
To find the minimum value, we can use the formula for the x-coordinate of the vertex:
x=-b/2a=-5/(2 × 1)=-2.5.
Plugging this value into the equation, we get
y=(-2.5)²+5(-2.5)-6=-12.25.
Therefore, the function has a minimum value of -12.25.
The function is a polynomial, which means it is defined for all real numbers.
Since the function has a minimum value of -12.25, the range of the function is all real numbers less than or equal to -12.25, i.e., {y ≤ 12.25}.
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The complete question is:
Consider the equation y=x^(2)+5x-6. Determine whether the function has a maximum or minimum value. State the maximum or minimum value. What are the domain and range of the function
a. minimum; 0; D: (all real numbers); R: {all real numbers}
b. maximum; 0; D (all real numbers); R: {y <-0}
c. minimum; -12.25; D (all real numbers); R: {y <-12.25}
d. maximum; -12.25; D {x <- 2.5}; R: {all real numbers}
data set 1 has a mean of 54 and a mad of 4. data set 2 has a mean of 60 and a mad of 2. what can be concluded about the two distributions? select each correct answer. responses the means-to-mad ratio is 3. the means-to-mad ratio is 3. the distributions are somewhat similar. the distributions are somewhat similar. the means-to-mad ratio is 1.5. the means-to-mad ratio is 1.5. the distributions are similar.
Answers
The conclusions that can be made about the two distributions are:
The means-to-MAD ratio is 3. The distributions are similar.Options A and D are correct.
How do we calculate?The means-to-MAD ratio is found by dividing the mean of a dataset by its Mean Absolute Deviation (MAD).
We have that in Data Set 1, the means-to-MAD ratio is 54/4 = 13.5, and in Data Set 2, the means-to-MAD ratio is 60/2 = 30.
Since the means-to-MAD ratio in Data Set 1 is 13.5 and in Data Set 2 is 30, we can conclude that the two distributions are not similar.
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Need help again... :>
(using this as an excuse to give out brainliest :P)
Answers
Answer:
The equation of a line in slope-intercept form is y=mx+b. “m” is the slope and “b” is the y-intercept. To find the slope, take two points and plug them into the slope formula. The slope formula is Y2-Y1/X2-X1.
You can use any two points on the graph but I will use (3,-2) and (6,-3).
-3-(-2)/6-3
-3+2/3
-1/3
The slope is -1/3. To find the y-intercept, look the y-value where the line touches the y-axis. In this graph, the line touches the y-axis at (0,-1).
So, the equation of the line in slope-intercept form is y=-1/3-1.
Hope this helps! :)
Jose's school was closed due to the frigid weather. The high temperature for the play was 4 °F.and the low temperature for the day was
-9 °F.Write the compound inequality for the graph showing this daily temperature range.
Answers
Answer:
-36
Step-by-step explanation:
4 Which number is irrational?
A (1.5)
B 741
C 749
D (15)
Answers
An irrational number is a number that cannot be expressed as a fraction for any integers and. . Irrational numbers have decimal expansions that neither terminate nor become periodic.
4 Here are two squares, A and B. A B The length of each side of square B is 4 cm greater than the length of each side of square A. The area of square B is 70 cmº greater than the area of square A. Find the area of square B. Give your answer correct to 3 significant figures. You must show all your working.
Answers
Answer:
116 centimeter squared
Step-by-step explanation:
La²+8La+16 = La²+70
Step-by-step explanation:
8La+16 = 70
8La=70-16
La=54/8
La= 6.75
Area b = (6.75+4)*(6.75+4)
Area b = 10.75*10.75
Area b =115.563
2x + 9 > 21
solve the inequality
Answers
Answer:
x>6
Step-by-step explanation:
2x + 9 > 21
2x > 21-9
2x>12 dividing by 2
x>12/2
x>6
hope this helps
Answer:
x > 6
X has to be smaller than 6
plz help me asnswer this question. Im on a time limit!!!!!!!!!
Answers
Answer:
\(100\)
Answer:
110 pounds
Step-by-step explanation:
This graph shows a positive correlation between the weight of the cheetah and the length.
Therefore, if the cheetah is 6ft long, the weight will be more than the previous one, but not extremely heavier. So 110lb seems like the closest.
Thanks
What is the slope?
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Submit
Answers
Answer:
\(\displaystyle m=2\)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Slope Formula: \(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)Step-by-step explanation:
Step 1: Define
Find points from graph
Point (3, 0)
Point (4, 2)
Step 2: Find slope m
Simply plug in the 2 coordinates into the slope formula to find slope m
Substitute [SF]: \(\displaystyle m=\frac{2-0}{4-3}\)Subtract: \(\displaystyle m=\frac{2}{1}\)Divide: \(\displaystyle m=2\)Prove f(x)= 3x-7 and g(x)= -1/3x-7/3 are tiger inverses or not inverses of each other.
Answers
The functions f(x) = 3x - 7 and g(x) = -1/3x - 7/3 are not inverse functions of each other.
In order to determine if two functions are inverses of each other, we need to check if the composition of the functions results in the identity function.
To find the composition of f(g(x)), we substitute g(x) into f(x):
f(g(x)) = f(-1/3x - 7/3) = 3(-1/3x - 7/3) - 7 = -x - 7 + 7 = -x
Similarly, to find the composition of g(f(x)), we substitute f(x) into g(x):
g(f(x)) = g(3x - 7) = -1/3(3x - 7) - 7/3 = -x + 7/3 - 7/3 = -x
Both f(g(x)) and g(f(x)) result in -x, which is not equal to the identity function x. Therefore, f(x) = 3x - 7 and g(x) = -1/3x - 7/3 are not inverse functions of each other.
In order for two functions to be inverses, the composition of one function with the other should result in the identity function. Since that is not the case here, f(x) and g(x) are not inverses of each other.
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A water tank contains 12 1/2 ltres of water. Two-fifth of it was (1 mark) consumed. How much of it was left?
Answers
Answer:
7 1/2
Step-by-step explanation:
A water tank contains 12 1/2 liters
= 25/2
Two fifth of it was consumedd
= 25/2 ×2/5
= 50/10
= 5
12.5-5
= 7.5
= 7 1/2
Which of the following best describes the slope of the line below?
Answers
Answer:
D. Negative
Step-by-step explanation:
Always read the graph from left to right
The graph is going down thus it is negative
Answer: Negative.
Step-by-step explanation: The graph of the line falls from left to right which makes it negative.
Fill in the blanks so that the resulting statement is true. If f is a polynomial function and f(a) and f(b) have opposite signs, then there must be at least one value of c between a and b for which f(c)equals_______ This result is called the _______ Theorem.
Answers
Answer:
f(c) equals zero
Intermediate Value Theorem
Step-by-step explanation:
Intermediate value theorem is one which states that f is a continuous function whose domain contains intervals which are a and b. It takes on any value between f(a) and f(b) at some point within interval. If we know the two values, we can pick any number between those two values and determine its function.
which hypothesis or hypotheses about the relation between life satisfaction and job satisfaction has received the most empirical support?
Answers
Recent research has confirmed the Spillover Hypothesis by demonstrating a favourable association between life and job happiness.
According to the spillover hypothesis, behaviour or affect in a family system might flow immediately from one place or connection to another. Transfer takes place in the same valence, which means that one subsystem's negative affect is connected to another's negative affect, or stress at work spills over and intensifies stress at home. The compensatory hypothesis offers a different perspective by arguing that transfer between family subsystems occurs in the opposite valence, leading a person to seek fulfilment in one relationship or environment in order to make up for shortages in another.
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Please help I’ll give brainliest
Answers
Answer:
$48
Step-by-step explanation:
$160 * 0.05 = $8
$8 * 6 = $48
Hope this helps
Answer:
48
Step-by-step explanation:
Please Mark me brainlist
ANSWER FAST PLEASEEE
A company set aside a certain amount of money in the year 2000. The company spent exactly the same amount from that fund each year on perks for its employees. In 2003 , there was still $790,000left in the fund. In 2005, there was $702,000 left.
Let x be the number of years since 2000, and let y be the amount of money, in dollars, left in the fund that year. Use a linear equation to model the amount of money left in the fund after so many years.
The linear model’s slope-intercept equation is _____
In the year 2009, there was _____ left in the fund.
In the year _____, the fund will be empty.
Answers
Answer:
Linear Equation: y=-44000x+88922000
Year 2009: $526000
By year 2021, they have no more fund
Step-by-step explanation:
Slope formula is difference of y/difference of x. 79000-70200/2003-2005=
-88000/2=-44000.
To find the y intercept, x=0.
It should be year 0. 44000x2003=88132000+790000=88922000
Now put it into y=mx+b
y=-44000x+88922000
Plug in 2009.
It should equal 526000.
Can two numbers have 16 as their HCF and 380 as their LCM? Give reason
Answers
Yes, two numbers could have 16 as their HCF and 380 as their LCM.
Right here's how you could find those two numbers:
Step 1: prime Factorization
Write the high factorization of the LCM, 380, because the manufactured from its prime elements:
\(380 = 2^2 * 5 * 19\)
Step 2: HCF Calculation
The HCF of numbers is the highest common element that divides each numbers evenly. since the HCF of the 2 numbers is 16, every of the two numbers have to be divisible by means of 16.
Step 3: number Formation
permit the two numbers be 16a and 16b, in which a and b are co-top (meaning they haven't any common elements aside from 1).
Step 4: LCM Calculation
The LCM of numbers is the smallest variety this is divisible by way of each numbers. because the LCM of the two numbers is 380, we will write:
LCM(16a, 16b) =\(2^2 * 5 * 19\)
To find the values of a and b, we need to discover the smallest viable values of a and b such that their product is identical to 19. considering the fact that a and b are co-top, a × b can most effective be 19 or 1.
Case 1: a × b = 1
If a × b = 1, then a = 1 and b = 1, which means that that the 2 numbers are 16 and 16. but, 16 isn't divisible by using 19, so this case is not valid.
Case 2: a × b = 19
If a × b = 19, then the two numbers are 16 × 1 and 16 × 19, which can be 16 and 304, respectively.
Step 5: Verification
We will verify that the two numbers, 16 and 304, have 16 as their HCF and 380 as their LCM as follows:
HCF(16, 304) = 16 (seeing that 16 is the largest common factor of 16 and 304)
LCM(16, 304) = 3040/sixteen = 380 (due to the fact 3040 is the smallest multiple of 16 and 304)
Consequently, the 2 numbers with 16 as their HCF and 380 as their LCM are 16 and 304.
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Please helppp i will mark you brainliest
Answers
Answer:
c
Step-by-step explanation:
Select the three ratios that describe the model
Answers
Answer:
1= 4/7 of the fruit are apples 2 = the ratio of apples to oranges is 4 to 3 3= the ratio of oranges to apples 3 to 4
PLZ HELP ASAP FOR BRAINLIEST
Answers
Answer:
1/4
Step-by-step explanation:
You can set up ratios or fractions for the matching sides.
A/B
1/4 = 1/4
1.1/ 4/4= 1/4
1.5/6 = 1/4
1/4 is the correct answer as from figure a to b got smaller
17.(A.O.I Topic 12 Algebra 1) A group of teachers use green and red pens. For every green pen used up, three red pens get used up. The green and red pens are packed in packs of 30 pieces and 50 pieces respectively. One green pen costs sh2000 and one red pen costs sh1000. Sh 1,050,000 is to be used to buy the pens so that they get finished at the same time. They want to know the number of packs of each color of pen. Knowledge: Forming and solving simple equations. Tasks: Help find the number of packs of each color of pen.
Answers
Based on the calculation below, the number of packs of green pens is 7, while the number of packs of red pens is 12.60.
How do we use the ratio to solve a problem?Let g and r represent the number of green and red pens respectively.
Therefore, we have the following ratio:
g = 1:3 = r
Since one green pen costs sh2000 and one red pen costs sh1000; the ratio 1:3 above implies that for every sh2000 spent on green, sh3000 (i.e. 3 * sh1000) is spent on red pen.
To ensure that the pens get finished at the same time, the total budget of sh1,050,00 is shared between the two pens as follows:
Amount to spend on green pen = (2000 / (2000 + 3000)) * sh1,050,00 = sh420,000
Amount to spend on red pen = (3000 / (2000 + 3000)) * sh1,050,00 = sh630,000
Therefore, we have:
Number of green pen = Amount to spend on green pen / Price per green pen = sh420,000 / sh2000 = 210
Number of red pen = Amount to spend on red pen / Price per red pen = sh630,000 / sh1000 = 630
Finally, we have:
Number of packs of green pen = 210 / 30 = 7
Number of packs of red pen = 630 / 50 = 12.60
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A backpack that normally sells for $39 is on sale for $25. Find the
percent of change.
Answers
Answer: To find the discount, simply multiply the original selling price by the %discount:
ie: 39 x 33/100= $12.87
So, the discount is $12.87.
Step-by-step explanation: To find the sale price, simply minus the discount from the original selling price:
ie: 39- 12. 87= 26.13
So, the sale price is $26.13