Calculus - finding the minimal length between two functions

In summary: Anyway, you're not solving an equation that is equal to zero here - you're calculating the derivative of a function.The derivative of a function can only take two forms: one where the denominator is the same and one where it's not.In this case, since the denominator is not the same, you can just get rid of it and get the right answer. Hmm, then how come in this case I dispose of the denominator and get the right... solution?I think because you're multiplying by x instead of dividing by x. The derivative of a function can only take two forms: one where the denominator is the same and one where it's not.In this case,
  • #36
I'll have to do some backtracking to see how this all makes sense. I lost the string of the actual problem with all the math mishaps. Thank you all though! Much appreciate ILS for going for the kill! :)
 
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  • #37
Femme_physics said:
I'll have to do some backtracking to see how this all makes sense. I lost the string of the actual problem with all the math mishaps. Thank you all though! Much appreciate ILS for going for the kill! :)

The basic idea of how to answer this question - without all the maths - is we first start with some function f(x) and another g(x), these functions don't intersect by looking at their graphs (we could also have proven this by setting f(x)=g(x) and shown there are no real solutions too, but it wasn't necessary here) and now to find the minimal distance in the y-direction between each function, we consider that the function f(x)-g(x) describes their differences in their height y at each point x, so for example if we took the value x=3, then f(3)=1/4 and g(3)=-4/3, which means their height difference is f(2)-g(2)=1/4-(-4/3)=19/12. So the function f(x)-g(x) is their height difference, and finding the minimal height means finding the lowest point on this graph (it will be above the y-axis, because we've already argued that the graphs don't intersect so f(x)-g(x)[itex]\neq[/itex]0) and this is where calculus comes into it. Since when we find the derivative of a function, it tells us the gradient of the tangent at each x value, well we know the lowest point on the function has a tangent gradient of 0, so we find the derivative and set it to zero (this says "tangent gradient = 0") then solving for x, it will tell us the x-value where lowest point is. Now substituting x back into the derivative equation obviously yields 0, because we just solved the equality and found the value of x that makes the derivative 0. We need to substitute the x-value back into the function f(x)-g(x) which will give us the y-value, which is the height difference between the functions. This is it.
 
<h2>1. What is the purpose of finding the minimal length between two functions in calculus?</h2><p>Finding the minimal length between two functions in calculus is important because it allows us to determine the shortest distance between two points on a curve. This is useful in various real-world applications, such as optimization problems in engineering and physics.</p><h2>2. How is the minimal length between two functions calculated?</h2><p>The minimal length between two functions is calculated using the concept of arc length. This involves integrating the square root of the sum of the squares of the derivatives of the two functions over the given interval. The resulting value is the minimal length between the two functions.</p><h2>3. Can the minimal length between two functions be negative?</h2><p>No, the minimal length between two functions cannot be negative. This is because the length of a curve is always a positive value and cannot be less than zero. If the resulting value from the calculation is negative, it means that an error has been made in the calculation.</p><h2>4. Are there any practical applications of finding the minimal length between two functions?</h2><p>Yes, there are many practical applications of finding the minimal length between two functions. Some examples include finding the shortest distance between two points on a road or a river, determining the shortest possible flight path between two cities, and minimizing the amount of material needed to construct a bridge between two points.</p><h2>5. What is the difference between finding the minimal length and the shortest distance between two functions?</h2><p>The minimal length between two functions is the actual length of the curve connecting the two points, while the shortest distance between two functions is the straight line distance between the two points. In other words, the minimal length takes into account the shape of the curve, while the shortest distance does not.</p>

1. What is the purpose of finding the minimal length between two functions in calculus?

Finding the minimal length between two functions in calculus is important because it allows us to determine the shortest distance between two points on a curve. This is useful in various real-world applications, such as optimization problems in engineering and physics.

2. How is the minimal length between two functions calculated?

The minimal length between two functions is calculated using the concept of arc length. This involves integrating the square root of the sum of the squares of the derivatives of the two functions over the given interval. The resulting value is the minimal length between the two functions.

3. Can the minimal length between two functions be negative?

No, the minimal length between two functions cannot be negative. This is because the length of a curve is always a positive value and cannot be less than zero. If the resulting value from the calculation is negative, it means that an error has been made in the calculation.

4. Are there any practical applications of finding the minimal length between two functions?

Yes, there are many practical applications of finding the minimal length between two functions. Some examples include finding the shortest distance between two points on a road or a river, determining the shortest possible flight path between two cities, and minimizing the amount of material needed to construct a bridge between two points.

5. What is the difference between finding the minimal length and the shortest distance between two functions?

The minimal length between two functions is the actual length of the curve connecting the two points, while the shortest distance between two functions is the straight line distance between the two points. In other words, the minimal length takes into account the shape of the curve, while the shortest distance does not.

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