need help with Calculus

Question

normal to the curve y=x^2+1 at the point where x=2 cuts the curve again at point P. Find the coordinates of P

Answer 1

The normal to a curve at a given point is the line perpendicular to the line tangent to the curve at that point.

The first thing we should do is find the slope of the line tangent to the curve at the given point (i.e., when x=2). To do so, we need to find the derivative of the function that defines the curve:

f(x) = x² + 1

f'(x) = 2x

Now, we evaluate the derivative at x=2:

f'(2) = 2(2) = 4

Thus, the slope of the line tangent to the curve is m_t=4. (Let us similarly call the slope of the normal to the curve m_n.)

Recall that the product of the slopes of two perpendicular lines is -1.

Thus,

(m_t)*(m_n) = -1

4m_n = -1

m_n = -1/4

At this point we have the slope of the normal to the curve. Recall that a line can be defined by a slope and one point on the line. So, we will now find a point on the normal and we will be able to find the equation that defines the line.

We know the normal passes through the given curve at x=2. We will find the y-value using the equation for the curve:

f(x) = x² + 1

f(2) = 2² + 1

f(2) = 5

Thus, there is a point (2,5) on the normal.

Now we can use point-slope form to find the equation for the normal, using m=-1/4 and the point (2, 5).

y - y₁ = m(x - x₁)

y - 5 = (-1/4)(x - 2)

y - 5 = (-1/4)x + 1/2

y = (-1/4)x + 11/2

The point P is the intersection of the curve and the normal to the curve. By definition, the two functions are equal at the point of their intersection:

x² + 1 = (-1/4)x + 11/2

0 = x² + (1/4)x - (9/2)

Solving using the quadratic formula, we get two answers:

x = 2 or x = (-9/4)

The point at which x = 2 was already given as a point of intersection earlier in the problem. Thus, the x-value of the point P must be the second value:

x = (-9/4)

Solving the equation for the curve at x = (-9/4) yields:

f(x) = x² + 1

f(-9/4) = (-9/4)² + 1

f(-9/4) = (81/16) + 1

f(-9/4) = 97/16

Thus, finally, we have the coordinates for point P:

P --> (-9/4, 97/16)