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
, wherem
is theslope of the line
andb
is they-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 formy-y₁=m(x-x₁)
for linear equations. It emphasizes theslope of the line
anda 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 theCartesian coordinate system
andthe 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
.