Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y 2 ( y 2 − 4 ) = x 2 ( x 2 − 5 ) , ( 0 , − 2 ) (devil's curve) y2(y2-4)=x2(x2-5), (0,-2) (devil's curve)
Answers
Answer:
y = -2
Step-by-step explanation:
To find the equation of the tangent we apply implicit differentiation, and then we take apart dy/dx
The equation is
implicit differentiation give us
But we know that
Hence, for the point (0,-2) and by replacing for dy/dx
Hence m=0, that is, the tangent line to the point is a horizontal line that cross the y axis for y=-2. The equation is:
y=(0)x+b = -2
HOPE THIS HELPS!!
In order to find the equation of the tangent line to the curve y²(y² - 4) = x²(x² - 5) at the point (0, -2), we will use the method of implicit differentiation. Here are the steps:
Step 1: Differentiate Each Side of the Given Equation with Respect to x
Applying the chain rule to differentiate y²(y² - 4) with respect to x gives:
2y*y'(y² - 4) + y²*2y*y' = d/dx [y²(y² - 4)]
The chain rule is also applied to differentiate x²(x² - 5) with respect to x, yielding:
2x(x² - 5) + x²*2x = d/dx [x²(x² - 5)]
Step 2: Equate the Two Expressions Found from Step 1 and Solve for y'
2y*y'(y² - 4) + y²*2y*y' = 2x(x² - 5) + x²*2x
This equation can be solved by isolating y' (the derivative of y with respect to x), which represents the slope of the tangent line.
Step 3: Use the Given Point (0, -2) to Find the Slope of the Tangent Line
Substitute x = 0 and y = -2 into the equation found in Step 2 to get the specific value for the slope at the given point.
Step 4: Use the Point-Slope Form of the Line to Write the Equation of the Tangent Line
The point-slope form of the line y - y₁ = m(x - x₁) can be used to write the equation of the tangent line. We substitute for x₁ and y₁ with the coordinates of the given point (0, -2), and m with the slope found from Step 3.
The resulting equation represents the tangent line to the curve at the given point (0, -2). Please note that the full calculation may result in a complex slope due to the nature of the given curve equation. Nonetheless, this process illustrates the application of implicit differentiation and the point-slope form of a line in finding the equation of a tangent line to a curve.
#SPJ3
Related Questions
a car is purchased for 26,000 after each year the resale decrease by 35% what will the resale value be
) A patient drank 12 ounces of orange juice. How many milliliters did the patient drink?
Given f(x)=5x+5f(x)=5x+5, find f(-4)f(−4).
Solve for x x/10+5=−1x=
Answers
Answer:
A = 145cm
Step-by-step explanation:
A+B+C = 445cm
b = 1/2c
a = b+45 = 1/2c + 45
445 = c + (0.5c) + (0.5c + 45)
445 = 2c + 45
445 - 45 = 2c
400/2 = c
c = 200cm
a = 1/2c + 45 = 1/2*200 + 45 = 100 + 45 = 145cm
Answers
Answer:
4000grams in each bag
Step-by-step explanation:
Assume he put x amount of clay into each bag, he then put 6x into 6 bags.(since the question didn’t mention, I assume he put every clay he has into the bags)
6x=24kg
x=4kg
Since the question is asking for grams not kilograms:
4kg=4000g
Answers
Step-by-step explanation:
1) angle 2 and 4
2)angle 2 and 3
3)angle 1 and 4
Hope it helps
Answers
Answer:
18,375
Step-by-step explanation:
7 minutes, 1730.4 gallons of water flowed from a 4-inch pipe.
Based on Janice's data, what is the difference in flow rate
between a 2-inch and 4-inch pipe?
Answers
Answer:
208.6 gal/min
Step-by-step explanation:
For 2" pipe,
Given Volume = 463.2 gal, time = 12 min
flow rate for 2" pipe
= Volume ÷ time
= 463.2÷12
= 38.6 gal/min
For 4" pipe,
Given Volume = 1730.4 gal, time = 7 min
flow rate for 4" pipe
= Volume ÷ time
= 1730.4÷7
= 247.2 gal/min
Difference in flow rate = 247.2 - 38.6 = 208.6 gal/min
Final answer:
The difference in flow rate between a 2-inch and 4-inch pipe, based on Janice's data, is 208.6 gallons per minute.
Explanation:
To determine the difference in flow rate between a 2-inch and a 4-inch pipe, we first need to calculate the flow rate for each pipe. This can be done by dividing the amount of water that flowed within a given time by that time.
For the 2-inch pipe: 463.2 gallons flowed in 12 minutes, so the flow rate is 463.2 / 12 = 38.6 gallons per minute.
For the 4-inch pipe: 1730.4 gallons flowed in 7 minutes, so the flow rate is 1730.4 / 7 = 247.2 gallons per minute.
Now, to find the difference in flow rate between the two pipes, we subtract the smaller flow rate from the larger one. Thus, 247.2 - 38.6 = 208.6 gallons per minute.
Therefore, the difference in flow rate between a 2-inch and 4-inch pipe, based on Janice's data, is 208.6 gallons per minute.
Learn more about Flow Rate here:
#SPJ3
Answers
Answer:
closed questions are questions with fixed answer options
A. Closed questions allow the respondent to go in-depth with their answers.
- no, it is relevant to open questions
B. It is possible to automate the collection of results for closed questions.
- yes
C. Closed questions allow for new solutions to be introduced.
- no, it allows to collect statistical data but not good for new solutions, it better works with open questions when new ideas, solutions may appear
D. It is easy to quantify and compare the results of surveys with closed questions.
- yes