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zimsam
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How would you solve this Related Rates problem?
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In summary, the conversation discusses using the derivative to calculate $\dfrac{dy}{dt}$ when given $\dfrac{dx}{dt}$ and the position of a planet. The hint given is to use the equation $\theta = \arctan\left(\dfrac{y}{x}\right)$. The speaker initially makes a mistake in their implicit differentiation, resulting in an incorrect answer of $\dfrac{dy}{dt}=0$. However, it is clarified that this is the correct value at the endpoint of the minor axis, as y changes from increasing to decreasing with respect to time at that position.
- #2
skeeter
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(a) $\dfrac{d}{dt}\bigg[(x-1)^2+ 2y^2=2 \bigg]$
you are given $\dfrac{dx}{dt}$ and the position of the planet.
use the derivative to calculate $\dfrac{dy}{dt}$
(b) hint …
$\theta = \arctan\left(\dfrac{y}{x}\right)$
you are given $\dfrac{dx}{dt}$ and the position of the planet.
use the derivative to calculate $\dfrac{dy}{dt}$
(b) hint …
$\theta = \arctan\left(\dfrac{y}{x}\right)$
- #3
zimsam
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skeeter said:
When I find dy/dt, it comes out to =0 for me. dy/dx=(-dx/dt(x-1))/2y. Surely the rate of change of y is not zero...
- #4
skeeter
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zimsam said:
is that so … ?
- #5
zimsam
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skeeter said:
I already know that dy/dt must be changing as well...
How did I make a mistake in my implicit differentiation?
- #6
skeeter
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zimsam said:
you didn’t make a mistake …
dy/dt = 0 at (1,1) which is at the endpoint of the minor axis
y is changing from increasing to decreasing w/respect to time at that position
1. How do you identify the related rates in a problem?
In order to identify the related rates in a problem, you need to look for variables that are changing with respect to time. These variables are typically represented by letters such as "x" or "y" and are usually given in units of distance, volume, or area.
2. What is the first step in solving a related rates problem?
The first step in solving a related rates problem is to clearly define the variables and their relationship to each other. This can be done by drawing a diagram and labeling the given and unknown variables.
3. How do you set up an equation for a related rates problem?
To set up an equation for a related rates problem, you need to use the chain rule to relate the rates of change of the different variables. This involves taking the derivative of each variable with respect to time and then multiplying them together.
4. What is the importance of units in solving a related rates problem?
Units are crucial in solving a related rates problem because they help to ensure that the rates of change are consistent and can be properly multiplied and divided. It is important to keep track of units throughout the problem and to convert them as needed.
5. How do you know if your solution to a related rates problem is correct?
You can check the correctness of your solution by plugging in the given values and comparing the resulting rate of change with the given rate of change. If they match, then your solution is likely correct. It is also a good idea to double check your calculations and units to ensure accuracy.
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