- #1
zimsam
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MHB Why Does dy/dt Equal Zero at the Endpoint of the Minor Axis?
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SUMMARY
The discussion centers on the calculation of the derivative dy/dt at the endpoint of the minor axis for the equation (x-1)² + 2y² = 2. Participants confirm that dy/dt equals zero at the point (1,1), indicating a transition in the rate of change of y from increasing to decreasing. The implicit differentiation process is correctly applied, leading to the conclusion that while dy/dt is zero at this specific point, it does not imply that y is constant over time. The relationship dy/dx = (-dx/dt(x-1))/(2y) is also highlighted.
PREREQUISITES- Understanding of implicit differentiation
- Familiarity with derivatives and their applications in physics
- Knowledge of trigonometric functions, specifically arctangent
- Basic concepts of planetary motion and its mathematical modeling
- Study implicit differentiation techniques in calculus
- Explore the application of derivatives in physics, particularly in motion analysis
- Learn about the geometric interpretation of derivatives in relation to conic sections
- Investigate the behavior of functions at critical points and endpoints
Students and educators in calculus, physicists analyzing motion, and mathematicians interested in conic sections and their properties.
- #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
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