Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
Building Competence. Crossing Borders.
Mathematics 2 for Business Schools
Section 1: Fundamentals of Differential Calculus
Spring Semester 2017
After finishing this section you should be able to …
• derive the difference quotient and the differential quotient of a function (repetition).
• explain the concept of the derivative of functions (repetition).
• derive and correctly apply the rule for the derivative of constant functions (repetition).
• derive and correctly apply the rule for the derivative of power functions (repetition).
• correctly apply the constant factor rule and the sum rule (repetition).
• correctly apply the product rule, the quotient rule, and the chain rule (new).
• find the derivative of exponential functions and logarithmic functions (new).
Learning objectives
2Spring semester 2017 Section 1: Fundamentals of differential calculus
Difference quotient – Definition
3
The slope 𝑚𝑠 of the secant through the
points
𝑃 𝑥0, 𝑓(𝑥0) and
𝑄 𝑥0 + Δ𝑥, 𝑓(𝑥0 + Δ𝑥)
of 𝑓, i.e., the average rate of change
of 𝑓 on the interval 𝑥0; 𝑥0 + Δ𝑥 is
called the difference quotient
𝑚𝑠 =Δ𝑓
Δ𝑥=𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0
Δ𝑥
at 𝑥0 (between 𝑃 and 𝑄).
Spring semester 2017 Section 1: Fundamentals of differential calculus
𝑥0 𝑥0 + ∆𝑥
𝑓(𝑥0)
𝑓(𝑥0 + ∆𝑥)
𝑃
𝑄
Δ𝑓
Δ𝑥
Local rate of change of a function
4
𝑚𝑡 = limΔ𝑥⟶0
𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0Δ𝑥
Finally, the secant becomes the tangent and the
slope of the secant becomes the slope of the
tangent.
Spring semester 2017 Section 1: Fundamentals of differential calculus
The average rate of change of a function between 𝑃 and 𝑄 becomes the local rate of
change in 𝑃 if 𝑄 is moved towards 𝑃, i.e. if ∆𝑥 tends to 0.
Differential quotient – Definition
5
The function 𝑓 is called differentiable in 𝑥0 if the
limit of the difference quotient
𝑚𝑡 = 𝑓′ 𝑥0 = limΔ𝑥⟶0
𝑓 𝑥0+Δ𝑥 −𝑓 𝑥0
Δ𝑥
exists in 𝑥0.
Our notation for this limit is 𝑓′ 𝑥0 and we call it
differential quotient,
derivative,
slope of the tangent or
local rate of change of 𝑓
in 𝑥0.
Spring semester 2017 Section 1: Fundamentals of differential calculus
𝑥0
𝑓(𝑥0)
Spring semester 2017 Section 1: Fundamentals of differential calculus
There are several notations used for the derivative.
The most widely used notation for the derivative of 𝑓 is 𝑓′. This notation was introduced
by Newton.
Since the derivative is the same as the differential quotient, the Leibniz notation is also
used quite often. Here, the derivative of 𝑓 is written as 𝑑𝑓
𝑑𝑥.
Both notations mean the same, namely the derivative of 𝑓. Therefore 𝑓′ =𝑑𝑓
𝑑𝑥.
The calculator can numerically find the derivative at a specific point 𝑥0. This is done by
using the following keys:
SHIFT 𝑑/𝑑𝑥,
enter the function,
enter the 𝑥-value where to calculate 𝑓′.
d-notation and calculator
6
Differentiation rules – Review
7Spring semester 2017 Section 1: Fundamentals of differential calculus
𝑓(𝑥) 𝑓′(𝑥) Remarks
1. 𝑐 0 𝑐 ∈ ℝ, any real constant
2. 𝑥 1
3. 𝑥𝑟 𝑟 ∙ 𝑥𝑟−1
4. 𝑐 ∙ 𝑔(𝑥) 𝑐 ∙ 𝑔′(𝑥) 𝑐 ∈ ℝ, factor rule
5. 𝑢 𝑥 + 𝑣(𝑥) 𝑢′ 𝑥 + 𝑣′(𝑥) sum rule
These rules can be expressed in a shorter way by omitting the argument x, e.g. the sum rule can be written as
𝑢 + 𝑣 ′ = 𝑢′ + 𝑣′ instead of 𝑢 𝑥 + 𝑣 𝑥 ′ = 𝑢′ 𝑥 + 𝑣′(𝑥)
Exercise – Repetition
8
Find the first and second derivatives of the following functions:
a) 𝑔 𝑥 = 5
b) 𝑝 𝑥 = 𝑥2
c) 𝑓 𝑡 = 3𝑥4
d) 𝑓 𝑡 = 5𝑡3
e) ℎ 𝑢 = 3𝑢4 −1
2𝑢2 + 3
f) 𝑓 𝑥 =3𝑥2 +
1
𝑥2−
1
𝑥
Spring semester 2017 Section 1: Fundamentals of differential calculus
Product rule
9
If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then 𝑓 𝑥 = 𝑢(𝑥) ∙ 𝑣(𝑥) is also
differentiable and its derivative is given by:
𝑓(𝑥)′ = 𝑢′ 𝑥 ⋅ 𝑣(𝑥) + 𝑢(𝑥) ⋅ 𝑣′(𝑥)
Spring semester 2017 Section 1: Fundamentals of differential calculus
Short notation: 𝑢 ∙ 𝑣 ′ = 𝑢′𝑣 + 𝑢𝑣′
Example: 𝑓 𝑥 = 𝑥 + 1 𝑥 − 1
𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1
𝑓′ 𝑥 = 1 ∙ 𝑥 − 1 + 𝑥 + 1 ∙ 1= 𝑥 − 1 + 𝑥 + 1 = 2𝑥
(One can reach the same result by first expanding the functional term and then differentiating the expression in its expanded form.)
Quotient rule
10
If 𝑢(𝑥) and 𝑣(𝑥) are differentiable functions, then the function 𝑓 𝑥 =𝑢(𝑥)
𝑣(𝑥)is
also differentiable and its derivative is given by:
𝑓(𝑥)′ =𝑢′(𝑥) ⋅ 𝑣 𝑥 − 𝑢 𝑥 ⋅ 𝑣′(𝑥)
𝑣(𝑥)2
Spring semester 2017 Section 1: Fundamentals of differential calculus
Short notation :𝑢
𝑣
′=
𝑢′𝑣−𝑢𝑣′
𝑣2
Example: 𝑓 𝑥 =𝑥+1
𝑥−1
𝑢 𝑥 = 𝑥 + 1 and 𝑢′ 𝑥 = 1𝑣 𝑥 = 𝑥 − 1 and 𝑣′ 𝑥 = 1
𝑓′ 𝑥 =1∙ 𝑥−1 − 𝑥+1 ∙1
𝑥−1 2 =𝑥−1−𝑥−1
𝑥−1 2 =−2
𝑥−1 2
Note theminus sign!
Chain rule
11
Example:
𝑓 𝑥 = 3𝑥2 + 6𝑥 79
Outer function: 𝑢 𝑣
Inner function: 𝑣(𝑥)
Is the composition of the two functions 𝑢 ∘ 𝑣 exists and if 𝑢 is
differentiable at 𝑣(𝑥) then derivative of the composition is given by:
𝑢 ∘ 𝑣 ′ 𝑥 = 𝑢 𝑣 𝑥 ′ = 𝑢′ 𝑣 𝑥 ⋅ 𝑣′(𝑥)
𝑓′ 𝑥 = 79 ⋅ 3𝑥2 + 6𝑥 78 ⋅ 6𝑥 + 6 = ⋯
Outer derivative : 𝑢′ 𝑣 Inner derivative: 𝑣′(𝑥)
Spring semester 2017 Section 1: Fundamentals of differential calculus
Short notation : 𝑢 ∘ 𝑢 ′ = 𝑢′(𝑣) ∙ 𝑣′ In other words: outer derivative multiplied by inner derivative
Examples – Product rule, quotient rule, and chain rule
12
Find the derivatives of
a) 𝑓 𝑥 = 2𝑥2 − 1 3𝑥 + 1
b) 𝑔 𝑥 =2𝑥2−1
3𝑥+1
c) ℎ 𝑥 =32𝑥2 − 1
Spring semester 2017 Section 1: Fundamentals of differential calculus
Derivatives of exponential and logarithmic functions
13
For 𝑎 ∈ ℝ+\ 1 the functions 𝑎𝑥 and log𝑎𝑥 are differentiable and
their derivatives are:
a) 𝑒𝑥 ′ = 𝑒𝑥
b) 𝑎𝑥 ′ = 𝑎𝑥 ⋅ ln 𝑎
c) ln 𝑥 ′ =1
𝑥
d) log𝑎 𝑥′ =
1
𝑥⋅
1
ln 𝑎
Spring semester 2017 Section 1: Fundamentals of differential calculus
with 𝑒 ≔ 2.71828… Euler number
Examples
14Spring semester 2017 Section 1: Fundamentals of differential calculus
Find the derivatives of
a) 𝑓 𝑥 = 𝑒−𝑥2
2
b1) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the product rule)
b2) 𝑔 𝑥 = 𝑥 ∙ 𝑒−𝑥 (solve using the quotient rule)
c) ℎ 𝑥 = log𝑎 2 𝑥
Rules of differentiation: Overview (Formulary)
15Spring semester 2017 Section 1: Fundamentals of differential calculus