Equation of a tangent line and normal line?

Question

1. what's a tangent line
2. What's a nomal line
3. find the equation of the tangent line and normal line to the graph of y= 3x^2- 6x at x= 3

Please answer in detail. I wanna learn the procedure

normal line is the same as the perpendicular line.

Answer 1

I do not know what "normal" line is, but I can help you with the rest.

A tangent line is a line that defines the slope of the curve at a point. For example, a function y = 2x has a slope of 2. However, a slope for a binomial y = x^2 does NOT have a slope that is constant. It keeps changing depending on X, as the slope is not constant.

To find out what the tangent line is for such function, you need to use calculus, in particular, the concept of derivative. Without explaining the detail, understand derrivitive is a slope. For the function y = x^2, the derrivitive (slope) is 2x. As you can see, now the slope is not a fixed number.

The graph of y = 3x^2 - 6x has a derrivitive of 3(2)x - 6(1) which is 6x + 6. We say, y' = 6x + 6

Now, at x=3, the slope is 6x + 6 = 6(3) + 6 = 18 + 6 = 24

At x=3, y is y=3(3)^2 - 6(3) = 27 - 18 = 9
So the coordinate is (3, 9)

Knowing the slope is 24, and the coordinate is (3, 9) you can arrive at a equation for a graph by:

y-y1 = m(x-x1)
y-9 = 24(x-3)
y-9 = 24x + 72
y = 24x + 72 + 9
y = 24x + 81

There is your answer.

Answer 2

A tangent line to a curve has a similar slope as a results of fact the by-made from the curve on the ingredient of tangency. in case you comprehend the tangent ingredient, the technique is to locate the by-made from the curve at that ingredient, then use the spinoff and the ingredient in the ingredient-slope equation of a line to stipulate the line. A line normat to a curve at a ingredient has a slope that's the destructive reciprocal of the by-made from the curve at that ingredient. in case you comprehend the ingredient of intersection, the technique is to locate the by-made from the curve at that ingredient, then use the destructive reciprocal of the spinoff and the ingredient in the ingredient-slope equation of a line to stipulate the line.