Why is the Coordinate of Point P (3/2, 0) in this Differentiation Problem?

When x = -2, the derivative is -1/2. The normal of the curve at this point would have a gradient of 2. To find the coordinates of P, we can use the equation of the normal line, which is y = 2x + b. Since the normal line meets the x-axis at P, we know that y = 0 at this point. Substituting this into the equation gives us 0 = 2x + b. Plugging in our known point (-2, 0), we can solve for b and get b = 4. Therefore, the coordinates of P are (3/2, 0). In summary, the equation of the curve is x^2y
  • #1
david18
49
0
"the equation of a curve is x^2y=x-6 (x^2y is x squared times y)

The normal of the curve at the point where x=-2 meets the x-axis at p.

Find the coordinates of P"


The method I used was to find y by rearranging the given equation and differentiate it.
This gave me f ' (x)= 1+12x^-3

Then I subbed x=-2 in so that I could find the gradient of the tangent at that point. It worked out as -1/2, so then the gradient of the normal would be 2.

After finding the equation of the line and making y=0, I got the wrong answer: x=-1 and y=0

apparently the answer is (3/2, 0)


any help would be appreciated
 
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  • #2
your derivative is wrong
get it right , you will have your answer.
 
  • #3
If you rearrange the equation to y = (x - 6)/x^2, the derivative is (12-x)/x^3.
 

FAQ: Why is the Coordinate of Point P (3/2, 0) in this Differentiation Problem?

What is differentiation?

Differentiation is a mathematical concept that refers to finding the rate of change of a function with respect to its independent variable. It is commonly used in calculus to find the slope of a curve at a given point.

Why is differentiation important?

Differentiation is important because it allows us to analyze and understand the behavior of mathematical functions. It is also used in many real-world applications, such as physics, economics, and engineering.

How is differentiation different from integration?

Differentiation and integration are inverse operations: differentiation finds the rate of change of a function, while integration finds the accumulation of a function. In other words, differentiation is the process of finding the slope of a curve, while integration is the process of finding the area under a curve.

What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of x^n is nx^(n-1), the product rule states that the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x), the quotient rule states that the derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x))/g(x)^2, and the chain rule states that the derivative of f(g(x)) is f'(g(x))g'(x).

How is differentiation used in real life?

Differentiation is used in many real-life scenarios, such as calculating velocity and acceleration in physics, determining marginal cost and revenue in economics, and finding optimal solutions in engineering. It is also used in fields such as biology, chemistry, and medicine to model and analyze various processes.

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