# Math Help - Differentiation

1. ## Differentiation

Find the coordinates of the point where the tangent to the curve $y = x^2 + 1$ at the point (2, 5) meets the normal to the same curve at the point (1, 2).

2. Given the curve $y = x^2 +1$ the slope of the tangent at $(2,5)$ is $4$ and the slope of the normal at $(1,2)$ is $\frac{-1}{2}$.
Now write the equations of the two lines.
Find the point of their intersection.

3. Thank you Plato. Problem resolved

4. So $y-5 = 4(x-2)$ or $y = 4x-3$ (tangent line).

$y = -\frac{1}{2}x$ (normal at point).

$4x-3 = -\frac{1}{2}x, \ x = \frac{2}{3}, y = -\frac{1}{3}$.

5. Originally Posted by Plato
Given the curve $y = x^2 +1$ the slope of the tangent at $(2,5)$ is $4$ and the slope of the normal at $(1,2)$ is $\frac{-1}{2}$.
Now write the equations of the two lines.
Find the point of their intersection.
You are probably wondering how Plato got those slopes.

$f(x) = x^2 +1$

Find the derivative of $f$

$f'(x) = 2x$

Slope at $(2, 5)$ is a simple plug and chug of the x-coordinate, so $f'(2) = 4$

Slope at $(1, 2)$ is just $f'(1) = 2$, so the line normal to this point is perpendicular. Perpendicular slopes are opposite signed reciprocals, so that's how Plato got $\frac{-1}{2}$

6. I’ve another problem.

The curve with equation $y = ax^2 + bx + c$ passes through the point (1, 2). The gradient of the curve is zero at the point (2, 1). Find the values of a, b, c.

How can I solve this?

7. Originally Posted by geton
I’ve another problem.

The curve with equation $y = ax^2 + bx + c$ passes through the point (1, 2). The gradient of the curve is zero at the point (2, 1). Find the values of a, b, c.

How can I solve this?
let f(x) = ax^2 + bx + c.

set up three simultaneous equations. since (1,2) and (2,1) are on the curve, we have:
f(1) = 2 and f(2) = 1. so form two equations from that.

but there are three unknowns, we need at least three equations. so use the information from the derivative. we are also told that f ' (2) = 0, that's your third equation

8. New questions need to start a new thread.
You know that $y(1) = 2,\quad y(2) = 1\quad \& \quad y'(2) = 0$.
Make the substitutions and solve.

9. Originally Posted by Plato
New questions need to start a new thread.
thanks, i forgot to mention that

10. Thank you so much Jhevon & Plato.