UNIT 3 – SLOPE OF A
CURVE AND
DIFFERENTIATION
BY: ENGR. RYUICHI T. KISHIMOTO, REE,MSEM
Slope of a Curve
•The tangent is defined as the line intersects the curve at only one
point, while the line that intersects the curve in two or more distinct
points is called a secant.
•The slope m of a line is defined a the tangent of its inclination ϴ
•It is the ratio of the change in vertical distance (rise) to the change in
horizontal distance (run) as the point moves along the line either
direction. Notice that the slope of a line is constant.
Slope of a
Curve
𝒎=
∆𝒚
∆𝒙
= Slope of a curve
Slope of a Curve
The slope of a curve y = f(x) at (x, f(x)) is equal to the slope of its tangent line at (x,f(x)). Hence,
m=
Note: Provided that limit exists.
𝒇 𝒙+∆𝒙 −𝒇(𝒙)
𝐥𝐢𝐦
∆𝒙
𝒙→𝟎
The Normal Line
•The normal line to a curve y=f(x) at a given point is the line perpendicular to a tangent line at
that point.
•The slope of the normal line is equal to the negative reciprocal of the slope if the tangent line.
That is,
mn =
Where mt is the slope of a curve
𝟏
−𝒎
𝒕
Example 1
Find the slope and the equation of the tangent and normal lines to the curve of
y = x3 at x = 1
Graphical
Representation
➢The red curve represents the function
y=x3
➢Green line represents the slope of a
curve.
Graphical
Representation
➢The red curve represents the function
y=x3
➢Green line represents the slope of a
curve.
➢Blue line represents the normal line
Example 2
Find the equation of the tangent line to the curve y = x2 + 1 at x = 0. Also, determine the
equation of the normal line.
Graphical
Representation
➢The red curve represents the function
y=x3
➢Green line represents the slope of a
curve.
Rules for
Differentiation