X- and y-coordinate's rate of change

• gunch
In summary, The problem is to find the point on the parabola y=x^2 where the x- and y-coordinate are moving at the same rate. According to the book, the answers are 1/2 and 1/4, but the person asking the question only got 1/2. To solve the problem, they set the rates equal to each other and differentiated with respect to x and y. They eventually found the correct answer by setting the x-value to 1/2 and realizing that the corresponding y-value must be 1/4.
gunch
Hi, I have a problem with an exercise in the book "A First Course in Calculus, fifth edition". The problem is stated as follows:
"A particle moves differentiably on the parabola $$y=x^2$$. At what point on the curve are its x- and y-coordinate moving at the same rate? (You may assume dx/dt and dy/dt is not equal to 0 for all t)"
According to the book the answers should be 1/2 and 1/4 however I only get 1/2.

I solve it, as follows:
Their rate is the same when
$$\frac{dx}{dy}=\frac{dy}{dx}$$
so I differentiate the equation with respect to x.
$$\frac{dy}{dx}=2x$$
And then with respect to y.
$$1=2x\frac{dx}{dy}$$
$$\frac{dx}{dy}=\frac{1}{2x}$$

Then to find when their rate of change is the same I write:
$$\frac{1}{2x}=2x$$

$$4x^2=1$$

$$x=\sqrt{1/4}$$

$$x=\frac{1}{2}$$

So, what have I done wrong? I'm not all that familiar with this stuff yet so I may have misunderstood the exercise.

Last edited:
Maybe that 1/4 is the "y" value for the 1/2 "x" value ?

Daniel.

dextercioby said:
Maybe that 1/4 is the "y" value for the 1/2 "x" value ?

Daniel.

Thanks, that must be it. I can't believe I failed to notice that.

1. What is the definition of rate of change for x- and y-coordinates?

The rate of change for x- and y-coordinates refers to the speed at which the values of the x- and y-coordinates are changing in relation to each other. This is also known as the slope of a line.

2. How do you calculate the rate of change for x- and y-coordinates?

The rate of change for x- and y-coordinates can be calculated by finding the change in y-values divided by the change in x-values. This is represented by the formula (y2-y1)/(x2-x1). This will give you the slope or rate of change of the line connecting the two points.

3. What does a positive rate of change for x- and y-coordinates indicate?

A positive rate of change for x- and y-coordinates indicates that the values of both coordinates are increasing in a positive direction. This means that the line connecting the two points is going up and to the right, and has a positive slope.

4. What does a negative rate of change for x- and y-coordinates indicate?

A negative rate of change for x- and y-coordinates indicates that the values of both coordinates are decreasing in a negative direction. This means that the line connecting the two points is going down and to the left, and has a negative slope.

5. How does the rate of change for x- and y-coordinates affect the shape of a graph?

The rate of change for x- and y-coordinates affects the steepness of a line on a graph. A higher rate of change results in a steeper line, while a lower rate of change results in a flatter line. This can also indicate the speed at which a quantity is changing over time.

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