Alliance School of Applied Mathematics

Semester I
Calculus and Linear Algebra

Module 2: Single-variable Calculus (Differentiation)

Syllabus
Functions of single variable; Limit, continuity and
MODULE 2 differentiability. Rolle’s theorem-geometrical interpretation,
Single-variable
Lagrange’s and Cauchy’s mean value theorems and Application;
Calculus
(Differentiation) Extreme values of functions; Linear approximation;
Indeterminate forms and L' Hospital's rule.

Motivation:

In Differential Calculus, differentiation is implemented in all the field of applied mathematics.

Mean Value Theorem is one of the important topics of calculus. This theorem relates the slope
of the tangent at one point of the arc of the curve to the slope of its secant. In other word mean
value theorem relates the average value of the function with it derivative. Mean value theorem
can be applied in the comparison of instantaneous velocity with the average velocity in its
engineering applications. The mean value theorem is a very important result in Real Analysis
and is very useful for analyzing the behavior of functions in higher mathematics.

The operation of adding infinite terms is called infinite series. In mathematics the enumerated
no of objects when added they give the structure of infinite series which are required for the
study of finite structures with help of generating functions. Infinite series emphasizes methods
for discussing convergence and divergence of series. It also helps in expanding different
functions which can be expressed into convergent series.

Limit
Meaning of 𝒙 → 𝒂:
Let ‘𝑥’ be a variable and ‘𝑎’ be constant. Since 𝑥 is a variable; we can change its value at our
pleasure. It can be changed in such a way that its value comes nearer and nearer to 𝑎. Then we
say that 𝑥 approaches 𝑎 and it is denoted by 𝑥 → 𝑎 .

Definition: Given a number 𝛿 > 0 however small, if 𝑥 takes up values, such that
0 < |𝑥 − 𝑎| < 𝛿. Then 𝑥 is said to tend to 𝑎, and is symbolically written as 𝑥 → 𝑎.

Limit of a function: Let 𝑓(𝑥) be a function defined on (𝑎, 𝑏) except possibly at 𝑐 ∈

(𝑎, 𝑏). We say that lim 𝑓(𝑥) = 𝐿 if, for every real number 𝜖 > 0; there exists a real number
𝑥→𝑐
𝛿 > 0 such that
𝟎 < |𝒙 − 𝒄| < 𝜹 ⇒ |𝒇(𝒙) − 𝑳| < 𝝐
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Important facts on limits:

a) The fact that the limit of 𝑓(𝑥) exists at 𝑥 = 𝑎 means that the graph of 𝑓(𝑥) approaches
the same value from both sides of 𝑥 = 𝑎 .
b) Right hand limit is the limit of the function as 𝑥 approaches 𝑎 from the positive side.
[RHL= lim+ 𝑓(𝑥) = lim 𝑓(𝑎 + ℎ)]
𝑥→𝑎 𝑥→0
c) Left hand limit is the limit of the function as 𝑥 approaches 𝑎 from the negative side.
[LHL= lim− 𝑓(𝑥) = lim 𝑓(𝑎 − ℎ)]
𝑥→𝑎 𝑥→0

Working Rule for the Evaluation of Limit:

Right hand limit of 𝒇(𝒙), when 𝒙 → 𝒂 = 𝐥𝐢𝐦+ 𝒇(𝒙)

𝒙→𝒂
Step I. Put 𝑥 = 𝑎 + ℎ and replace 𝑥 → 𝑎 by ℎ → 0.
+

Step II. Simplify 𝑥→0

lim 𝑓(𝑎 + ℎ).
Step III. The value obtained in step 2 is the right hand limit of 𝑓(𝑥) at 𝑥 = 𝑎.

Similarly for evaluating the left-hand limit put 𝒙 = 𝒂 − 𝒉.

Continuity
The word ‘continuous’ means without any break or gap. If the graph of a function has no break
or gap or jump, then it is said to be continuous. A function which is not continuous is called a
discontinuous function.

Definition: A real valued function 𝒇(𝒙) is said to be continuous at 𝒙 = 𝒄 if

(i) 𝒇(𝒙) is defined.
(ii) 𝐥𝐢𝐦 𝒇(𝒙) exists [i.e., 𝐥𝐢𝐦− 𝒇(𝒙) = 𝐥𝐢𝐦+ 𝒇(𝒙)].
𝒙→𝒄 𝒙→𝒄 𝒙→𝒄
(iii) 𝐥𝐢𝐦 𝒇(𝒙) = 𝒇(𝒄)
𝒙→𝒄
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Semester I
Calculus and Linear Algebra

➢ A real valued function 𝒇(𝒙) is said to be continuous in [𝒂, 𝒃] if 𝒇(𝒙) is continuous

at every point 𝒄 ∈ [𝒂, 𝒃].

Note: All elementary functions such as algebraic, exponential, trigonometric, logarithmic,

and hyperbolic functions are continuous functions. Also, the sum, difference, and product
of continuous functions is continuous. The quotient of continuous functions is continuous at
all those points at which the denominator does not become zero.

Continuity from Left and Right:

A function 𝑓(𝑥) is said to be
(i) Left continuous at 𝑥 = 𝑎 if, lim− 𝑓(𝑥) = 𝑓(𝑎).
𝑥→𝑎
(ii) Right continuous at 𝑥 = 𝑎 if, lim+ 𝑓(𝑥) = 𝑓(𝑎).
𝑥→𝑎
Thus, a function 𝑓(𝑥) is continuous at a point 𝑥 = 𝑎, if it is left continuous as well as right
continuous at 𝑥 = 𝑎.

Continuity of a Function in an Interval:

➢ A function f(x) is said to be continuous in an open interval (a, b), if it is continuous

at every point in (a, b). For example, the functions y = sin⁡ x, y = cos⁡ x, y = ex
are continuous in (−∞, ∞).

➢ A function f(x) is said to be continuous in the closed interval [a, b], if it is:
a) Continuous at every point of the open interval (a, b).
b) Right continuous at x = a, i.e. RHL|x=a = f(a).
c) Left continuous at x = a, i.e. LHL|x=a = f(b).

Reasons of Discontinuity
a) Limit does not exist i.e. 𝐥𝐢𝐦− 𝒇(𝒙) ≠ 𝐥𝐢𝐦+ 𝒇(𝒙)
𝒙→𝒂 𝒙→𝒂
b) 𝒇(𝒙) is not defined at 𝒙 = 𝒂.
c) 𝐥𝐢𝐦 𝒇(𝒙) ≠ 𝒇(𝒂).
𝒙→𝒂

Differentiability
Meaning of a Derivative: The instantaneous rate of change of a function with respect
to the dependent variable is called the derivative. Let 𝑓(𝑥) be a given function of one variable
and let 𝑑𝑥 denote a number (positive or negative) to be added to the number 𝑥. Let df denote
the corresponding change of ‘𝑓’, then
∆𝒇 𝒇(𝒙 + ∆𝒙) − 𝒇(𝒙)
𝒅𝒇 = 𝒇(𝒙 + 𝒅𝒙) − 𝒇(𝒙) ⇒ =
∆𝒙 ∆𝒙
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

∆𝑓
If ∆𝑥 approaches a finite value as 𝑑𝑥 approaches zero, this limit is the derivative of ‘𝑓’ at the
point 𝑥. The derivative of a function ‘𝑓’ is denoted by symbols such as
𝒅𝒇 𝒅
𝒇′ (𝒙); ; (𝒇(𝒙))
𝒅𝒙 𝒅𝒙

Note: If 𝑦 = 𝑓 (𝑥 )
𝑑𝑓 ∆𝑓 𝑓(𝑥+∆𝑥)−𝑓(𝑥)
Then 𝑦 ′ = = 𝑙𝑖𝑚 = 𝑙𝑖𝑚
𝑑𝑥 ∆𝑥→0 ∆𝑥 ∆𝑥→0 ∆𝑥

𝒇(𝒙)−𝒇(𝒙𝟎 )
Definition: A real-valued function 𝒇(𝒙) is said to be differentiable at point 𝒙𝟎 if 𝒍𝒊𝒎
𝒙→𝒙𝟎 𝒙−𝒙𝟎
′ (𝒙) 𝒅𝒇
exists uniquely and it is denoted by 𝒇 or 𝒅𝒙.

➢ The fact that 𝑓(𝑥) is differentiable at 𝑥 = 𝑎 means that the graph is smooth as 𝑥
moves from 𝑎 − to 𝑎 to 𝑎+ . Derivative of an even function is always an odd function.

Important facts on Differentiability:

➢ If a function is not differentiable but is continuous at a point, it geometrically implies

there is a sharp corner or a kink at that point.
➢ If a function is differentiable at a point, then it is also continuous at that point.
➢ If a function is continuous at point 𝑥 = 𝑎, then nothing can be guaranteed about the
differentiability of that function at that point.
➢ If a function 𝑓(𝑥) is not differentiable at 𝑥 = 𝑎, then it may or may not be continuous
at 𝑥 = 𝑎.
➢ If a function 𝑓(𝑥) is not continuous at 𝑥 = 𝑎, then it is not differentiable at 𝑥 = 𝑎.
➢ If the left hand derivative and the right hand derivative of 𝑓(𝑥) at𝑥 = 𝑎 are finite (they
may or may not be equal), then 𝑓(𝑥) is continuous at 𝑥 = 𝑎.
➢ Every differentiable function is necessarily continuous but every continuous function
is not necessarily differentiable i.e. Differentiability ⇒ continuity but continuity ⇏
differentiability
➢ All polynomial, trigonometric, logarithmic and exponential functions are continuous
and differentiable in their domains.
➢ If 𝑓(𝑥) & 𝑔(𝑥) are differentiable at 𝑥 = 𝑎 then the functions 𝑓(𝑥) ± 𝑔(𝑥),
𝑓(𝑥). 𝑔(𝑥) will also be differentiable at 𝑥 = 𝑎 & if 𝑔(𝑎) ≠ 0, then the function
𝑓(𝑥)/𝑔(𝑥) will also be differentiable at 𝑥 = 𝑎.

Differentiability of a Function in an Interval:

➢ A function 𝑓(𝑥) is said to be differentiable over an open interval (𝑎, 𝑏), if it is
differentiable at every point in (𝑎, 𝑏).
➢ A function 𝑓(𝑥) is said to be differentiable over a closed interval [𝑎, 𝑏], if it is:
a) Differentiable at every point of the open interval (𝑎, 𝑏).
b) Right derivative exists at 𝑥 = 𝑎.
c) Left derivative exists at 𝑥 = 𝑎.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Rolle’s Mean value theorems

Introduction: The mean value theorem tells us (roughly) that if we know the slope of the
secant line of a function whose derivative is continuous, then there must be a tangent line
nearby with that same slope. This lets us draw conclusions about the behavior of a function
based on knowledge of its derivative.

Statement: If a function 𝑓(𝑥) is

(i) 𝑓(𝑥) is continuous on the closed interval [𝑎, 𝑏];
(ii) 𝑓 (𝑥) is differentiable on the open interval (𝑎, 𝑏);
(iii) 𝑓(𝑎) = 𝑓(𝑏).
then there exists at least one point 𝑐 in (𝑎, 𝑏) (i.e. 𝑎 < 𝑐 < 𝑏) such that
𝑓 ′ (𝑐 ) = 0
Alternative or Another Statement of Rolle’s Theorem : If a function f ( x) is
(i) continuous in [ a , a + h ]
(ii) differentiable in ( a , a + h )
(iii) f (a) = f (a + h) ,

then there exists at least one real number  such that

f '( a + h ) = 0 , for 0    1.

➢ Geometrical Interpretation of Rolle’s Theorem: If graph of a function 𝑓(𝑥) is a

continuous curve between 𝑥 = 𝑎 and 𝑥 = 𝑏; having a unique tangent at all point in
(𝑎, 𝑏) and 𝑓(𝑎) = 𝑓(𝑏), then there exists at least one point 𝑃 between 𝑥 = 𝑎 and 𝑥 =
𝑎 on the curve, such that the tangent at this point is parallel to 𝑥 − axis i.e., 𝑓 ′ (𝑐) = 0.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Lagrange’s mean value theorem

Statement: Suppose 𝑓(𝑥) be a function that satisfies following conditions:
(i) 𝑓(𝑥) is continuous on the closed interval [𝑎, 𝑏];
(ii) 𝑓(𝑥) is differentiable on the open interval (𝑎, 𝑏).
Then there exists a point 𝑐 ∈ (𝑎, 𝑏) (i.e. 𝑎 < 𝑐 < 𝑏) such that
𝑓(𝑏 )−𝑓(𝑎)
𝑓 ′ (𝑐 ) = .
𝑏−𝑎

Note: Rolle’s Theorem is a particular case of Lagrange’s MVT when the value of
function is same at the end points of [𝑎, 𝑏] i.e., 𝑓(𝑎) = 𝑓(𝑏).

Geometrical Interpretation of L.M.V.T.: If graph of a function 𝒇(𝒙) is a continuous curve

between 𝒙 = 𝒂 and 𝒙 = 𝒃; having a unique tangent at all point in (𝒂, 𝒃) and 𝒇(𝒂) = 𝒇(𝒃),
then there exists at least one point 𝑪 between 𝒙 = 𝒂 and 𝒙 = 𝒂 on the curve, such that the
tangent at this point is parallel to 𝒙 − axis i.e., 𝒇′ (𝒄) = 𝟎.

𝒇(𝒃)−𝒇(𝒂)
Slope of the chord; 𝐭𝐚𝐧 𝜶 =
𝒃−𝒂
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Cauchy’s mean value theorem

Statement: Suppose 𝑓(𝑥) and 𝑔(𝑥) be functions such that:

(i) Both are continuous on the closed interval [𝑎, 𝑏];
(ii) Both are differentiable on the open interval (𝑎, 𝑏).
(iii) 𝑔′ (𝑥) ≠ 0, ∀𝑥 ∈ (𝑎, 𝑏).
Then there exists a point 𝑐 ∈ (𝑎, 𝑏) (i.e. 𝑎 < 𝑐 < 𝑏) such that
𝑓 ′ (𝑐 ) 𝑓(𝑏 )−𝑓(𝑎)
= 𝑔(𝑏)−𝑔(𝑎).
𝑔 ′ (𝑐 )

➢ Cauchy's Mean Value Theorem generalizes Lagrange’s Mean Value Theorem. This
theorem is also called the Extended or Second Mean Value Theorem. It establishes
the relationship between the derivatives of two functions and changes in these functions
on a finite interval.

Geometrical Interpretation of C.M.V.T.: Suppose that a curve 𝛾 is described by the

parametric equations 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), where the parameter t ranges in the
interval [𝑎, 𝑏]. When changing the parameter t, the point of the curve runs from
𝐴(𝑓(𝑎), 𝑔(𝑎)) to 𝐵(𝑓(𝑏), 𝑔(𝑏)). According to the theorem, there is a
point (𝑓(𝑐), 𝑔(𝑐)) on the curve 𝛾 where the tangent is parallel to the chord joining
the ends A and B of the curve.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Exercise
1. Verify⁡Rolle’s⁡theorem⁡for:
sin 𝑥
a) 𝑓(𝑥) = , 𝑖𝑛⁡[0, 𝜋].
𝑒𝑥

b) 𝑓(𝑥) = 1 − 3(𝑥 − 1)2/3 ⁡⁡⁡𝑖𝑛⁡0 ≤ 𝑥 ≤ 2.

c) 𝑓(𝑥) = |𝑥|⁡𝑖𝑛⁡[−1,1].
𝑥 2 + 1, 0 ≤ 𝑥 ≤ 1
d) 𝑓(𝑥) = { .
3 − 𝑥, 1 ≤ 𝑥 ≤ 2
e) 𝑓(𝑥) = tan 𝑥 , 0 ≤ 𝑥 ≤ 𝜋.
𝑥 2 +𝑎𝑏
f) 𝑓(𝑥) = log [(𝑎+𝑏)𝑥] 𝑖𝑛[𝑎, 𝑏], 𝑎 > 0, 𝑏 > 0.
2
g) 𝑓(𝑥) =x3 – 12x in [0,2√3]
1, 𝑤ℎ𝑒𝑛⁡𝑥 = 0
h) f(x) = {
𝑥, 𝑤ℎ𝑒𝑛⁡0 < 𝑥⁡ ≤ 1
𝑎𝑏⁡+𝑥 2
i) 𝑓(𝑥) = 𝑙𝑜𝑔 𝑎𝑥+𝑏𝑥 in [a,b]
𝜋 5𝜋
j) 𝑓(𝑥) = 𝑒 𝑥 (Sin⁡x⁡ − ⁡Cos⁡x) in [4 , ]
4

2. Verify Lagrange’s mean value theorem for the functions:

a) 𝒇(𝒙) = 𝒙𝟐/𝟑 in [−𝟖, 𝟖 ] .
b) 𝒇(𝒙) = 𝟐𝒙𝟐 − 𝟕𝒙 − 𝟏𝟎 in [𝟐, 𝟓] .
𝟏 𝟏
c) 𝒇(𝒙) = 𝒄𝒐𝒕 𝝅𝒙 in [− 𝟐 , 𝟐] .
𝟏
d) 𝒇(𝒙) = 𝒙(𝒙 − 𝟏)(𝒙 − 𝟐) in [𝟎, 𝟐] .
𝟏
e) 𝒇(𝒙) = 𝒍𝒐𝒈⁡𝒙 in [𝟎, 𝟐].

f) 𝒇(𝒙) = 𝒔𝒊𝒏−𝟏 ⁡𝒙 in [0,1]

3. Verify Cauchy’s mean value theorem for

𝜋
a) 𝑓(𝑥) = 𝑠𝑖𝑛⁡𝑥 ; 𝑔(𝑥) = 𝑐𝑜𝑠⁡𝑥 in [0, 2 ].

b) 𝑓(𝑥) = 𝑥2 ; 𝑔(𝑥) = 𝑥3 in [1,2].

1
c) 𝑓(𝑥) = 𝑙𝑜𝑔 𝑥 ; 𝑔(𝑥) = 𝑥 in [1, 𝑒].
d) 𝑓(𝑥) = 𝑒 𝑥 ; 𝑔(𝑥) = 𝑒 −𝑥 in [𝑎, 𝑏].
𝑠𝑖𝑛 𝑏−𝑠𝑖𝑛 𝑎
4. Prove that 𝑐𝑜𝑡 𝑐 = where 𝑐 ∈ (𝑎, 𝑏) using Cauchy’s mean value
𝑐𝑜𝑠 𝑎−𝑐𝑜𝑠 𝑏

theorem.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Extreme values of functions – Maxima and Minima

Definition: A function 𝑓(𝑥) is said to have a maximum at x⁡ = ⁡a, if there exists a small number
′ℎ′, however small, such that f(a) > f(a − h) and f(a) > f(a + h), both.

A function 𝑓(𝑥) is said to have a minimum at x⁡ = ⁡a, if there exists a small number ′ℎ′,
however small, such that f(a) < f(a − h) and f(a) < f(a + h), both.

Note: The maximum and minimum values of a function taken together are called its extreme
values and the points at which the function attains the extreme values are called the turning
points of the function.

Necessary Condition for an Extremum

Definition: The points at which the derivative of the function 𝒇(𝒙) is equal to zero are called
the stationary points.
The points at which the derivative of the function 𝒇(𝒙) is equal to zero or does not
exist are called the critical points of the function. Consequently, the stationary points are a
subset of the set of critical points.

A necessary condition for an extremum is formulated as follows:

If the point 𝒙 = 𝒂 is an extremum point of the function 𝒚 = 𝒇(𝒙); then
𝒅𝒚
𝒇′ (𝒂) = | =𝟎
𝒅𝒙 𝒙=𝒂

Note: The necessary condition does not guarantee the existence of an extremum. A classic
illustration here is the cubic function 𝑓(𝑥) = 𝑥 3 . Despite the fact that the derivative of the
function at the point 𝑥 = 0 is zero i.e., 𝑓 ′ (0) = 0, this point is not an extremum at 𝑥 = 𝑎.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Conditions for Maxima and Minima:

1) 𝑓(𝑥) is maximum at x⁡ = ⁡a if 𝑓 ′ (𝑎) = 0 and 𝑓 ′′ (𝑎) is ‘- ve’ [i.e., 𝑓 ′ (𝑎) changes sign
from ‘+ ve’ to ‘- ve’].

2) 𝑓(𝑥) is minimum at x⁡ = ⁡a if 𝑓 ′ (𝑎) = 0 and 𝑓 ′′ (𝑎) is ‘+ ve’ [i.e., 𝑓 ′ (𝑎) changes sign
from ‘- ve’ to ‘+ ve’].

Procedure for finding maxima and minima:

1. Consider the given function as 𝑓(𝑥).

2. Find 𝑓 ′ (𝑥) and equate it to zero. Solve 𝑓 ′ (𝑥) = 0 and let its roots be a,b,c,….

3. Find 𝑓 ′′ (𝑥) and then calculate the value of 𝑓 ′′ (𝑥) at x⁡ = ⁡a, b, c, …⁡

If 𝑓 ′′ (𝑎) < 0, 𝑓(𝑥) is maximum at x⁡ = ⁡a.

If 𝑓 ′′ (𝑎) > 0, 𝑓(𝑥) is minimum at x⁡ = ⁡a.

4. Sometimes 𝑓 ′′ (𝑥) maybe difficult to find out or 𝑓 ′′ (𝑥) = 0 at x⁡ = ⁡a. In such cases,
a) If 𝑓 ′ (𝑥) changes its sign from ‘+ ve’ to ‘- ve’ as x passes through a, then 𝑓(𝑥)
is maximum at x⁡ = ⁡a.
b) If 𝑓 ′ (𝑥) changes its sign from ‘- ve’ to ‘+ ve’ as x passes through a, then 𝑓(𝑥)
is minimum at x⁡ = ⁡a.
c) If 𝑓 ′ (𝑥) does not change its sign while passing through x⁡ = ⁡a, then 𝑓(𝑥) is
neither maximum nor minimum at x⁡ = ⁡a.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

PROBLEMS:

1. Find the maximum and minimum values of 3𝑥 4 − 2𝑥 3 − 6𝑥 2 + 6𝑥 + 1 in the interval

(0,2).

2. Find the maximum value of the function (x − 1)(x − 2)2 .

3. Find the maxima and minima for the following functions:

a) x 3 − 3𝑥 + 2;
b) 2x 3 − 3𝑥 2 + 6;
c) −3x 2 + 4𝑥 + 7;
d) 5x 3 + 2𝑥 2 − 3𝑥.

4. Show that sin 𝑥⁡ (1 + cos 𝑥) is a maximum when 𝑥 = π/3.

loge 𝑥
5. What is the maximum value of the function ?
𝑥

𝑎
6. The function 𝑓(𝑥) defined by 𝑓(𝑥) = 𝑥 + 𝑏𝑥; 𝑓(2) = 1, has an extremum at 𝑥 = 2.
Determine the value of a and b. Is this point (2,1), a point of maximum or minimum
on the graph of 𝑓(𝑥)?

7. Discuss the maxima and minima for the function 𝑦 = 𝑥𝑒 𝑥 .

8. Show that of all rectangles of given area, the square has the least parameter.

9. Show that the height of closed cylinder of given volume and least surface area is equal
to its diameter.

sin 3𝑥
10. What is the value of 𝑝 for which the function 𝑓(𝑥) = 𝑝 sin 𝑥 + has an extremum
3
𝜋
at 𝑥 = 3 .

11. It is given that at 𝑥 = 2, the function 𝑥 3 − 12𝑥 2 + 𝑘𝑥 − 8 attains its maximum value
on the interval [0,3]. Find the value of 𝑘.

12. A ball is thrown in the air. Its height at any time t is given by:
ℎ = 3 + 14t − t 2
What is its maximum height.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

Linear approximation
Consider a function 𝒚 = 𝒇(𝒙) is differentiable at a point 𝒙 = 𝒂. Then, the tangent line
to the graph of 𝒇(𝒙) at 𝒙 = 𝒂 is given by the equation:

𝒚 = 𝒇(𝒂) + 𝒇′ (𝒂)(𝒙 − 𝒂)

In general, for a differentiable function 𝒚 = 𝒇(𝒙), the equation of tangent line to 𝒇(𝒙) at 𝒙 =
𝒂 can be used to approximate 𝒇(𝒙) for 𝒙 near 𝒂. Therefore, we can write:

𝒇(𝒙) ≈ 𝒇(𝒂) + 𝒇′ (𝒂)(𝒙 − 𝒂) for 𝒙 near 𝒂.

We call the linear function,

𝑳(𝒙) = 𝒇(𝒂) + 𝒇′ (𝒂)(𝒙 − 𝒂)

The linear approximation, or tangent line approximation of the function 𝒇(𝒙) at 𝒙 = 𝒂. This
function L is known as the linearization of 𝒇(𝒙) at 𝒙 = 𝒂.

1
Example: Find the linear approximation of 𝑓(𝑥) = 𝑥 at 𝑥 = 2 and use the approximation to
1
estimate 2.1.
Solution: Since, we are looking for the linear approximation at 𝑥 = 2, then using the
linear approximation formula, we have:
𝐿(𝑥) = 𝑓(2) + 𝑓 ′ (2)(𝑥 − 2)
We need to find 𝑓(2) .and 𝑓 ′ (2)
1 1 1 1
𝑓(𝑥) = ⇒ 𝑓(2) = ⁡⁡;⁡⁡⁡⁡𝑓 , (𝑥) = − 2 ⇒ 𝑓 , (2) = −
𝑥 2 𝑥 4
Therefore, the linear approximation is given by:
𝟏 𝟏
𝑳(𝒙) = − (𝒙 − 𝟐)
𝟐 𝟒
𝟏
Using the linear approximation, we can estimate 𝟐.𝟏 by writing:
𝟏 𝟏 𝟏
= 𝒇(𝟐. 𝟏) ≈ 𝟐 − 𝟒 (𝟐. 𝟏 − 𝟐) ≈ 𝟎. 𝟒𝟕𝟓.
𝟐.𝟏

The actual value of 𝒇(𝟐. 𝟏) is given by:

𝟏
𝒇(𝟐. 𝟏) = 𝟐.𝟏 = 𝟎. 𝟒𝟕𝟔𝟏𝟗.
Therefore, the tangent line gives us a fairly good approximation of 𝒇(𝟐. 𝟏).

Note: However, for the value of 𝒙 far from 2, the equation of tangent line does not give us
good approximation. For example, if 𝒙 = 𝟏𝟎, the y-value of the corresponding point on the
tangent line is:
𝟏 𝟏 𝟏
𝐲 = − (𝟏𝟎 − 𝟐) = − 𝟐 = −𝟏. 𝟓
𝟐 𝟒 𝟐
𝟏
where as, the value of the function at 𝒙 = 𝟏𝟎 is 𝒇(𝟏𝟎) = 𝟏𝟎 = 𝟎. 𝟏.
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

PROBLEMS:

1. Find the linear approximation of f(x)=√x at x=9 and use the approximation to estimate
√9.1.
3
2. Find the linear approximation of f(x)= √𝑥 at x=8 and use the approximation to estimate
3
√8.1 to five decimal places.
3. Find the linear approximation of f(x)=(1 + 𝑥)𝑛 at x=0. Use the approximation to
estimate (1.01)3 .

INDETERMINATE FORMS

L’Hospital Rule : If f(x) and g(x) are two functions of x which can be expanded
by Taylor’s series in the neighborhood of x=a and f(a)=g(a)=0,then
𝒇(𝒙) 𝒇′ (𝒙)
𝐥𝐢𝐦 = 𝐥𝐢𝐦 ′
𝒙→𝒂 𝒈(𝒙) 𝒙→𝒂 𝒈 (𝒙)

Indeterminate Forms: There are seven types of indeterminate forms

given as follows:
0
•
0
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

∞
•
∞

• 0.∞
• ∞ − ⁡∞
• 1∞
• 00
• ∞0
where the limits for these can be calculated using L’Hospital Rule.

0
NOTE: To apply the L’Hospital rule, we need the given function in or
0
∞
form.
∞

Four standard limits :

𝑺𝒊𝒏⁡𝒙
(i) 𝐥𝐢𝐦⁡ ( )=𝟏
𝒙→𝟎 𝒙
𝒙
(ii) 𝐥𝐢𝐦⁡ ( )=𝟏
𝒙→𝟎 𝑺𝒊𝒏⁡𝒙
𝒕𝒂𝒏⁡𝒙
(iii) 𝐥𝐢𝐦⁡ ( )=𝟏
𝒙→𝟎 𝒙
𝒙
(iv) 𝐥𝐢𝐦⁡ ( )=𝟏
𝒙→𝟎 𝒕𝒂𝒏⁡𝒙

PROBLEMS:
𝑥−⁡𝑥 𝑥
1. Find lim⁡ ( ) Ans 2
𝑥→1 𝑥−1−𝑙𝑜𝑔⁡𝑥

𝑎 𝑥
2. Find lim⁡ ( − Cot⁡ ) Ans 0
𝑥
𝑥→0 𝑎
𝜋𝑥
𝑥 tan 2𝑎
3. Find lim⁡ (2 −⁡ )
𝑥→𝑎 𝑎

𝑒 2𝑥 −⁡(1+𝑥)2
4. Find lim⁡ ( ) Ans 1
𝑥→0 𝑥⁡log⁡(1+𝑥)

𝑆𝑖𝑛−1 𝑥−𝑥
5. Find lim⁡ ( ) Ans 1/6
𝑥→0 𝑥3
 Alliance School of Applied Mathematics
Semester I
Calculus and Linear Algebra

6. Find lim⁡ log tan 𝑥 tan 2𝑥

𝑥→0
1
(1+𝑥)𝑥 −𝑒 𝑒
7. Prove that lim⁡ ( ) = -⁡2
𝑥→0 𝑥

8. Show that lim⁡ log 𝑥 Sin 𝑥=1

𝑥→0

9. Evaluate lim⁡ tan 𝑥. log 𝑥⁡ Ans 0

𝑥→0
1 ⁡2
10.Evaluate lim⁡ - 𝐶𝑜𝑡 2 x Ans ⁡
𝑥→0 𝑥2 3

Note: For solutions, refer to the class notes.

Reference Books:

1. B.S. Grewal, Higher Engineering Mathematics, Latest edition, Khanna Publishers.

2. Peter V. O’Neil, Engineering Mathematics CENGAGE Learning India Pvt Ltd

.Publishers.

3. B.V. Ramana, Higher Engineering Mathematics, Latest Edition, Tata Mc. Graw Hill
Publications
4. Erwin Kreyszig, Advanced Engineering Mathematics, Latest edition, Wiley
Publications.