Tangent: In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

#### key concept 1

The equation for the slope of the tangent line to
is
.

is the derivative of
.

You can always get the slope of the tangent line by using
.

> `f(x)` is nothing but a function or formula.

#### key concept 2

And a line equation could be represent as:

- the `slope-intercept formula`:
  (Where
  is the slope of the line, and
  is the y-intercept)

> The equation of any straight line, called a `linear equation`, can be written as: `y = mx + b`, where `m` is the `slope of the line` and `b` is the `y-intercept`.

- the `point-slope formula`:
  (This formula uses a point on the line, denoted by
  , and the slope of the line, denoted by
  , to calculate the slope-intercept formula for the line)

> `Point-slope` is the general form `y-y₁=m(x-x₁)` for linear equations. It emphasizes the `slope of the line` and `a point on the line`.

# Question and Answers

#### `Area of a triangle` is
. The triangle is enclosed by a `Cartesian coordinate system` and `a tangent`. The equation of the tangent to the curve
is at the point
. What's the value of
?

> The answer is `64`

##### we should try to understand the question sentence by sentence first.

###### 1.`Area of a triangle` is
:

###### 2. The triangle is enclosed by a `Cartesian coordinate system` and `a tangent`.

- `tangent` is a line
- the `triangle` was created by the `Cartesian coordinate system` and `the tangent line`

###### 3. The equation of the tangent to the curve
is at the point
.

The slope of the tangent line is:

Now we already have a point and a slope, then we can generate `the equation of the line`:

If we could get the x, y in the following picture, we can find an equation of `the area of the triangle`:

So, finally, we have got the value of a, which is `64`.

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