Point on a Curve: Solving Related Rates Problems

In summary, the points where x and y are changing at the same rate on the curve y = 4 - x^2 are (-1/2, 15/4) and (1/2, 15/4). This can be determined by setting the derivative of y with respect to x equal to 1 and solving for x, which yields two solutions. These points can be verified by plugging them back into the original equation and confirming that the derivative at those points is indeed 1.
  • #1
jgens
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Homework Statement



A particle moves along a path described by y = 4 - x^2. At what point along the curve are x and y changing at the same rate

Homework Equations



Simple equations regarding derivatives.

The Attempt at a Solution



It's been a while before I've done any related rates problems, could someone please let me know if this is correct:

Since, x and y must be changing at the same rate (presumably with respect to time) x' = y' and y' = -2xx'. Therefore, -2x = 1 and x = -1/2. Placing my x value into the original equation yields 15/4. Hence, the point is (-1/2, 15/4).

Thanks.
 
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  • #2
SEEMS correct...
 
  • #3
Of course, it's right. What could go wrong?
 
  • #4
Plenty, I could have made an incorrect assumption ultimately leading to false conclusions.
 
  • #5
Good answer but you do not need to assume that x and y are varying wrt an external parameter. The derivative y'(x) = dy/dx of y wrt x expresses the instantaneous rate of change of y wrt a change in x.

The points where y and x are changing at the same rate are those where y'(x)=1.
 

FAQ: Point on a Curve: Solving Related Rates Problems

1. What is a "point on a curve" in related rates problems?

A point on a curve refers to a specific location on a curve or graph that is being analyzed in a related rates problem. It is usually represented by a set of coordinates, such as (x,y), and can change over time as the variables in the problem change.

2. How do you solve a related rates problem involving a point on a curve?

To solve a related rates problem involving a point on a curve, you will need to first identify the variables and their rates of change. Then, you can use the chain rule and implicit differentiation to find the relationship between the variables. Finally, you can plug in the given values and solve for the desired rate of change at the specific point on the curve.

3. What is the chain rule and how is it used in related rates problems?

The chain rule is a calculus rule that allows us to find the derivative of a composite function. In related rates problems, we often have a relationship between multiple variables that are changing with respect to time. The chain rule allows us to find the rate of change of one variable with respect to another variable.

4. Can you provide an example of a related rates problem involving a point on a curve?

Sure, an example of a related rates problem involving a point on a curve is a ladder leaning against a wall. The base of the ladder is sliding away from the wall at a constant rate, while the top of the ladder is sliding down the wall at a different rate. The question may ask for the rate at which the angle between the ladder and the wall is changing at a specific point on the ladder.

5. What are some tips for solving related rates problems involving a point on a curve?

Some tips for solving related rates problems involving a point on a curve include: clearly identifying the variables and their rates of change, drawing a diagram to visualize the problem, using the chain rule and implicit differentiation, and carefully plugging in values and units to ensure the correct answer. It is also important to check your answer for reasonableness and to practice with different types of related rates problems to improve your problem-solving skills.

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