Finding the slope of a curve at a point.

Keep up the good work. In summary, the surface x = x^2 - y^2, when cut by the plane y = 3x, produces a curve in the plane with a slope of -17 at the point (1,3,-8). The derivative of x = -8x^2 - x = 0 is -16x - 1.
  • #1
devin8721
1
0
The surface given by x = x^2 - y^2 is cut by a plane give by y = 3x. producing a curve in the plane. Find the slope of this curve at the ppoint (1,3,-8)

A) 3
B) -16
C) - 8sqrt(2/5)
D) 0
E) 18/sqrt(10)

So we want to look at this curve when y = 3x. Then x = x^2 -y^2 becomes x = x^2 - (3x)^2 which gives x = x^2 - 9x^2.

So we have x = -8x^2 or -8x^2 - x = 0.

The derivative is then - 16x - 1. At the point (1,3,-8) we should should have the derivative equal to
-17 which is not a choice..

Can someone tell me if I did something wrong.
 
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  • #2


Hey there! You did everything correctly up until finding the derivative. The derivative of x = -8x^2 or -8x^2 - x = 0 actually gives us -16x - 1, not -17. So at the point (1,3,-8), the slope of the curve should be -17.

To double check, we can plug in the values of x and y into the original equation x = x^2 - y^2. We get 1 = 1 - 9, which is true. This confirms that our point (1,3,-8) lies on the curve.

Hope that helps!
 

1. What is the slope of a curve at a point?

The slope of a curve at a point is the measure of its steepness at that specific point. It tells us how much the curve is rising or falling at that point.

2. How is the slope of a curve at a point calculated?

The slope of a curve at a point can be calculated using the derivative of the function at that point. This involves finding the limit of the change in y divided by the change in x as the interval approaches zero.

3. Why is finding the slope of a curve at a point important?

Finding the slope of a curve at a point is important because it helps us understand the behavior of the curve at that point. It can also help us find the maximum and minimum points of a curve, as well as the rate of change of the curve.

4. Can the slope of a curve at a point be negative?

Yes, the slope of a curve at a point can be negative. A negative slope indicates that the curve is decreasing at that point, while a positive slope indicates that the curve is increasing at that point.

5. How is the slope of a curve at a point represented mathematically?

The slope of a curve at a point is represented by the derivative of the function at that point. It can be written as dy/dx or f'(x), where dy represents the change in y and dx represents the change in x.

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