Proof (a) Prove (Theorem 3.3) that d / d x [ x n ] = n x n − 1 for the case in which r is a rational number. (Hint: Write y = x p / q in the form y ″ = x n and differentiate implicitly. Assume that p and u are integers, where q > 0.) (b) Prove part (a) for the case in which r is an irrational number. (Hint: Let y = x r where r is a real number, and use logarithmic differentiation .)
Proof (a) Prove (Theorem 3.3) that d / d x [ x n ] = n x n − 1 for the case in which r is a rational number. (Hint: Write y = x p / q in the form y ″ = x n and differentiate implicitly. Assume that p and u are integers, where q > 0.) (b) Prove part (a) for the case in which r is an irrational number. (Hint: Let y = x r where r is a real number, and use logarithmic differentiation .)
Solution Summary: The author explains how to prove that n is an irrational number.
(a) Prove (Theorem 3.3) that
d
/
d
x
[
x
n
]
=
n
x
n
−
1
for the case in which r is a rational number. (Hint: Write
y
=
x
p
/
q
in the form
y
″
=
x
n
and differentiate implicitly. Assume that p and u are integers, where q > 0.)
(b) Prove part (a) for the case in which r is an irrational number. (Hint: Let
y
=
x
r
where r is a real number, and use logarithmic differentiation.)
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
Chapter 3 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY