Solving Calculus Equation: a=dv/dt =>adt=dv

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In summary, the conversation discusses the use of differentials in calculus, specifically in the equation a = dv/dt. It is noted that dt and dv are differentials, not numbers, and are used in the study of differential equations. The conversation also highlights the use of differentials in finding the displacement of an object with constant acceleration.
  • #1
Calculus 142
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Hi,
I am new to calculus, and in some books I read
a= dv/dt
=>adt=dv.

If dt means with respect to t, how is it possible multiply both sides of the equation by dt?
Is there a theorem stating this?

Thankyou.
 
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  • #2


adt=dv means nothing but a= dv/dt; just a prevalent abuse of notation. Also, dt is not a number which can be multiplied.
 
  • #3


dv and dt are differentials. For more information, see http://en.wikipedia.org/wiki/Differential_(infinitesimal ).

Differentials come up in the study of differential equations, a simple example of which is a = dv/dt. If a is a constant, we can separate this equation to dv = a dt, and integrate both sides with respect to t, to get v = at + C, where C is an arbitrary constant.

If we realize that v = ds/dt, the time rate of change of position, then we have ds/dt = at + C, which implies that ds = (at + C)dt. Integrating again with respect to t, we get s = (1/2)at^2 + Ct + D, which gives us the displacement of an object moving with a constant acceleration as a function of t.
 
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  • #4


Thank you for the response.
 

FAQ: Solving Calculus Equation: a=dv/dt =>adt=dv

What is the purpose of solving calculus equations?

Calculus is a branch of mathematics that is used to solve problems involving rates of change. By solving calculus equations, we can determine the behavior of a system over time and make predictions about its future state.

How do you solve equations involving derivatives?

To solve equations involving derivatives, we use various mathematical techniques such as the chain rule, product rule, and quotient rule. We also use the fundamental theorem of calculus to relate derivatives to integrals.

What does the equation a=dv/dt represent?

This equation represents the relationship between acceleration (a) and the rate of change of velocity (dv/dt). It states that the acceleration of an object is equal to the derivative of its velocity with respect to time.

Can calculus equations be applied to real-world situations?

Yes, calculus equations are commonly used in various fields such as physics, engineering, economics, and biology to model and analyze real-world phenomena. They can be used to solve problems related to motion, growth, optimization, and many other situations.

What is the process for solving the equation adt=dv?

The process for solving this equation involves using algebraic manipulation to isolate the variable of interest. In this case, we would divide both sides by dt to get a=dv/dt, which is the same as the original equation. This shows that the equation is already solved and does not have a unique solution.

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