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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