SOUTHWESTERN UNIVERSITY, NIGERIA
MAT 211 – MATHEMATICAL METHODS I
Lecture Note by DANIEL Deborah O.
COURSE OUTLINE
Real-Valued Function of a Real Variable,
Review of Differentiation and Integration and Their Applications,
Mean Value Theorem,
Taylor Series,
Real-Value Functions of Two or Three Variable,
Chain Rule,
Extrema,
Lagrange’s Multiplier,
Differentials and Linear Approximations,
Evaluation of Line Integral.
,Real Valued Function
Peter Dirichlet a German Mathematician (1829) conceived a function as a variable,
called the dependent variable having its value fixed or determine in some definite
manner by the value assigned to the independent variable or to several independent
variables . The value of both 𝑦 and 𝑥 are real. The statement 𝑦 = 𝑓(𝑥)
is read as 𝑦 is a function of 𝑥. Again, indicates the inter
dependence between the variable and . The function 𝑓(𝑥) is usually given
as an explicit formula such as for all real.
In algebraic expression, a real variable may take any value in a certain range. If
the lowest value of is 𝑎 and the highest value of 𝑥 is 𝑏 and may take any value
between 𝑎 and 𝑏, then is said to be a continuous variable in the range [𝑎, 𝑏] and
takes all values such that 𝑎 ≤ 𝑥 ≤ 𝑏 . Since the end points are included among the
values of which form this range, the interval is called a closed interval. The interval
defined by the inequality 𝑎 < 𝑥 < 𝑏 is called an open interval and is denoted
by(𝑎, 𝑏).
Review of Differentiation
Function of a Function.
If is a function of and that itself is a function of , then the derivative of with
respect to is
This is also called the chain rule of differentiation.
Example: Find the derivative of each of the following.
Solution
Let
,
, Derivative of a Product.
If where and are functions of , then the derivative of with respect to
is
Example: Find the derivative of each of the following.
Solution
Let
Let
Let
MAT 211 – MATHEMATICAL METHODS I
Lecture Note by DANIEL Deborah O.
COURSE OUTLINE
Real-Valued Function of a Real Variable,
Review of Differentiation and Integration and Their Applications,
Mean Value Theorem,
Taylor Series,
Real-Value Functions of Two or Three Variable,
Chain Rule,
Extrema,
Lagrange’s Multiplier,
Differentials and Linear Approximations,
Evaluation of Line Integral.
,Real Valued Function
Peter Dirichlet a German Mathematician (1829) conceived a function as a variable,
called the dependent variable having its value fixed or determine in some definite
manner by the value assigned to the independent variable or to several independent
variables . The value of both 𝑦 and 𝑥 are real. The statement 𝑦 = 𝑓(𝑥)
is read as 𝑦 is a function of 𝑥. Again, indicates the inter
dependence between the variable and . The function 𝑓(𝑥) is usually given
as an explicit formula such as for all real.
In algebraic expression, a real variable may take any value in a certain range. If
the lowest value of is 𝑎 and the highest value of 𝑥 is 𝑏 and may take any value
between 𝑎 and 𝑏, then is said to be a continuous variable in the range [𝑎, 𝑏] and
takes all values such that 𝑎 ≤ 𝑥 ≤ 𝑏 . Since the end points are included among the
values of which form this range, the interval is called a closed interval. The interval
defined by the inequality 𝑎 < 𝑥 < 𝑏 is called an open interval and is denoted
by(𝑎, 𝑏).
Review of Differentiation
Function of a Function.
If is a function of and that itself is a function of , then the derivative of with
respect to is
This is also called the chain rule of differentiation.
Example: Find the derivative of each of the following.
Solution
Let
,
, Derivative of a Product.
If where and are functions of , then the derivative of with respect to
is
Example: Find the derivative of each of the following.
Solution
Let
Let
Let
