Download Lecture Notes 11: Differentiability, Tangent Line and Linearization I

Lecture Notes 11: Differentiability, Tangent
Line and Linearization I
Instructor: Anatoliy Swishchuk
Department of Mathematics & Statistics
University of Calgary, Calgary, AB, Canada
MATH 265 ’University Calculus I’
L01 Winter 2017
Outline of Lecture
1. Short Introduction
2. Differentiability
3. Tangent Line
4. Linearization
Short Introduction: Two main Problems in Calculus
In calculus, there are two main (or fundamental) problems: 1)
the problem of slopes, and 2) the problem of areas.
1) The problem of slopes: finding the slope of (the tangent line
to) a given curve at a given point on the curve; the solution of
this problem is the subject of differential calculus (see Chapters
2-4 of our textbook).
2) The problem of areas: finding the area of a plane region
bounded by curves and straight lines; the solution of this problem
is subject of integral calculus (see Chapters 5-7 of our textbook).
Usually we use the following notations: C for the graph of continuous function y = f (x); P for the point (x0, y0) on C, such
that y0 = f (x0).
Differentiability
A function f (x) is differentiable at point a if f 0(a) exists:
f (a + h) − f (a)
0
.
f (a) = lim
x→a
h
If f is differentiable at a, then f is continuous at a.
Tangent Line to a General Curve
We’ll consider the first problem that deals with finding a straight
line L that is tangent to a curve C at a point P.
Tangent Line to a Circle (with a center)
A tangent line meets the circle at any point,
the circle lies on only one side of the line, the tangent is perpendicular to the radius
Tangent Line to a Curve (no center)
a) L meets C only at P but is not tangent to C b) L meets C at
several points but is tangent to C at P c) L is tangent to C at
P but crosses C at P d) Many lines meet C only at P but none
of them is tangent to C at P
Secant Line
If Q is a point on C different from
P and Q is called a secant line to
through P and Q, then we call it line
notation for the coordinates of Q :
P, then the line through
the curve C. If line goes
P Q. We use the following
Q = (x0 + h, f (x0 + h)).
Short Introduction: Slope, Difference Quotient
The slope of the line P Q is:
f (x0 + h) − f (x0)
.
(1)
h
This expression (1) is called the Newton quotient or difference
quotient for f at x0.
Definition of Nonvertival Tangent Lines
If f continuous at x = x0 and
f (x0 + h) − f (x0)
lim
= m = f 0(x0)
h→0
h
exists, then the straight line having slope m and passing through
the point P = (x0, f (x0)) is called the tangent line (or simply
the tangent) to the graph of y = f (x) at P.
The equation of this tangent line is
y = m(x − x0) + y0,
where y = f (x),
y0 = f (x0).
(1)
Example: An Equation of the Tangent Lines
Find an equation of the tangent line to the curve y = x2 at the
point (1, 1).
Sol. Here, f (x) = x2,
x0 = 1,
y0 = f (1) = 1. The slope is:
f (1+h)−f (1)
(1+h)2 )−1
0
f (x0) = limh→0
= limh→0
h
h
2
2
−1
2h+h
=
lim
= limh→0 1+2h+h
h→0
h
h
= limh→0(2 + h) = 2.
According to (1) (see previous slide), we have
y = 2(x − 1) + 1
or
y = 2x − 1.
Example: An Equation of the Tangent Lines (Figure)
Definition of Vertical Tangent Lines
If f continuous at P = (x0, y0), where y0 = f (x0), and if either
f (x0 + h) − f (x0)
f (x0 + h) − f (x0)
= +∞ or
lim
= −∞,
h→0
h→0
h
h
then the vertical line x = x0 is tangent to the graph y = f (x)
at P. If the limit of the difference quotient fails to exist in any
other way than by being +∞ or −∞, the graph y = f (x) has no
tangent line at P.
lim
Vertical Tangent Lines: Example
Here f (x) = x1/3, and slope approaches the infinity:
1/3
(0)
h
limh→0 f (0+h)−f
=
lim
h→0 h
h
1 = +∞.
= limh→0 2/3
h
Example 1: No Tangent Line
(0)
h2/3 = 1 has no limit
Here: f (x) = x2/3 and f (0+h)−f
=
h
h
h1/3
(right is +∞ and left is −∞- cusp situation-infinitely sharp
point).
Example 2: No Tangent Line
Here: y = |x| and there is no unique
limit:
|0 + h| − |0|
|h|
=
= sgnh =
h
h
(
1,
−1,
if
if
h>0
h < 0.
The Slope of a Curve: Definition
The slope of a curve C at a point P is the slope of the tangent
line to C at P if such a tangent line exists. The slope of the
graph of y = f (x) at the point x0 is
f (x0 + h) − f (x0)
.
h→0
h
lim
The Slope of a Curve: Example
x
Find the slope of the curve y = 3x+2
at the point x = −2.
Sol. If x = 2, then y = 1/2, then the slope is:
f 0(−2) = lim
h→0
−2+h
1
−
2
3(−2+h)+2
h
−h
1
= .
h→0 2h(−4 + 3h)
8
= lim
Normals to a Curve: Definition
If a curve C has a tangent line L at point P, then the straight
line N through P perpendicular to L is called the normal to C
at P.
The slope of N is the negative reciprocal of the slope of L :
slope
of
the
normal = −
slope
of
1
the
tangent
.
Normals to a Curve: Example Find equations of the straight
√
lines that are tangent and normal to the curve y = x at the
point (4, 2).
1 . The tangent line is:
Sol. Here, f 0(4) = 4
1
x
(x − 4) + 2 or y = 1 + ,
4
4
and the normal has slope −4, and equation
y=
y = −4(x − 4) + 2
or
y = 18 − 4x.
Normals to a Curve: Example with Figure
Linearization
Idea of Linearization: how to approximate some values of f (x)?
We use the first derivative as an application of the tangent line
to approximate f.
The linear approximation of f near the point a has the following
form:
L(x) ≈ f 0(a)(x − a) + f (a).
References
1) Calculus: Early Transcendental, 2016, An Open Text, by
David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S
1/ucalgary/winter2016/math265/Guichard
− Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf
2) Optional Textbook: Essential Calculus, Early Transcendental,
2013, by J. Stewart, 2nd edition, Brooks/Cole