Differentiation (1 Viewer)

[ISAM77]

Just doing some practice on Khan Academy.

I'm just wondering why the working out on the right is wrong. If I use dV/dt = dV/dh * dh/dt , it also gives a different wrong answer. Why are these methods wrong?

 

[WeiWeiMan]

View attachment 46181

Just doing some practice on Khan Academy.

I'm just wondering why the working out on the right is wrong. If I use dV/dt = dV/dh * dh/dt , it also gives a different wrong answer. Why are these methods wrong?

Height isn’t constant relative to radius

 

[ISAM77]

Height isn’t constant relative to radius

I get that they're both changing over time. But why does that make them dependent on each other? e.g. you can increase the radius of a cone without changing the height of it. So the rates at which they are changing are independent are they not? Pls help me understand

 

[C2H6O]

I get that they're both changing over time. But why does that make them dependent on each other? e.g. you can increase the radius of a cone without changing the height of it. So the rates at which they are changing are independent are they not? Pls help me understand

aight so in the calculation on the right, you haven’t taken into account that the height is also changing, which would obviously affect the rate of change of V. If height were constant, then the calculation on the right works. You are right that height and radius are not dependent on each other, but volume is dependent on both so you must account for both their rates of change. So with the working on the left you have
Since both r and h are dependent of t, you should use product rule, as the correct solution does. I hope this makes sense

 

[ISAM77]

aight so in the calculation on the right, you haven’t taken into account that the height is also changing, which would obviously affect the rate of change of V. If height were constant, then the calculation on the right works. You are right that height and radius are not dependent on each other, but volume is dependent on both so you must account for both their rates of change. So with the working on the left you have
Since both r and h are dependent of t, you should use product rule, as the correct solution does. I hope this makes sense

Ok

So what you're saying is, in this circumstance, dV/dt =/= dV/dr * dr/dt . Is that correct?

Would this statement be true for the question above? (I know you can't solve the question using this statement. I just want to know if the statement is true)

 

[liamkk112]

Ok

So what you're saying is, in this circumstance, dV/dt =/= dV/dr * dr/dt . Is that correct?

Would this statement be true for the question above? (I know you can't solve the question using this statement. I just want to know if the statement is true)
View attachment 46185

this is where you start getting into multivariable calculus. here volume is a function of both r and h (V(r,h)) but both of these variables also depend on time (r(t), h(t)). you can represent that with a diagram like this:

now the multivariable chain rule basically says that in a diagram like this, to get dV/dt, you follow all paths to t and sum them up. in other words you do this:

 

[ISAM77]

this is where you start getting into multivariable calculus. here volume is a function of both r and h (V(r,h)) but both of these variables also depend on time (r(t), h(t)). you can represent that with a diagram like this:
View attachment 46187
now the multivariable chain rule basically says that in a diagram like this, to get dV/dt, you follow all paths to t and sum them up. in other words you do this:
View attachment 46188

Ok, awesome! I'll keep this in mind, and will look forward to learning multivariable calculus in the future Thanks very much