Velocity Vector Versus Tangent Line

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SUMMARY

The discussion centers on the relationship between the tangent line to a curve defined by the position function r(t) = (t^2)i + (4t)j and the velocity vector at t=3. The slope of the tangent line, calculated as dy/dx, is confirmed to be equivalent to the derivative dr/dt, which represents the velocity vector at that point. While both share the same slope at t=3, they are fundamentally different entities; the tangent line provides a linear approximation of the position function, while the velocity vector indicates the rate of change of position without a specific location context.

PREREQUISITES
  • Understanding of calculus concepts such as derivatives and slopes
  • Familiarity with vector notation and functions
  • Knowledge of position functions and their graphical representations
  • Ability to compute derivatives of functions
NEXT STEPS
  • Study the concept of derivatives in calculus, focusing on their geometric interpretations
  • Learn about vector functions and their applications in physics
  • Explore the relationship between position, velocity, and acceleration in motion
  • Investigate the use of tangent lines in approximating curves in calculus
USEFUL FOR

Students studying calculus, particularly those focusing on vector functions and their applications in motion analysis. This discussion is beneficial for anyone seeking to deepen their understanding of the relationship between position functions and their derivatives.

lovelylila
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I have encountered a problem in my Calculus homework.

I have a position function, r(t)= (t^2)i + (4t)j and in my homework, I am asked to find the tangent to this curve at the point t=3. I did this by finding dy/dx, or 2t/4 @ t=3 is 6/4. However, I am also asked to relate this to the velocity vector for the position function @ t=3, but I don't understand the relationship. Would they share the same slope? Any help is very much appreciated! :-)
 
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lovelylila said:
Would they share the same slope? Any help is very much appreciated! :-)

Well you have the position's function. The tangent line necessarily has a slope of dr/dt, which is equivalent to the velocity so yes.
 
Oh that makes sense! Thank you very much :-) But they're not the same line...are they?
 
The velocity has no inherent sense of position, so you can't really compare velocity and a tangent line even at the same time t. A tangent line more or less answers the question "where would you be at some time you knew you were at position r(t) at time t and maintained a constant velocity for all time?", but don't read too far into this.
 

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