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.
Problem 11 (a) A tank is discharging water through an orifice at a depth of T
meter below the surface of the water whose area is A m². The
following are the values of a for the corresponding values of A:
A 1.257 1.390
x 1.50 1.65
1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650
1.80 1.95 2.10 2.25 2.40 2.55 2.70
2.85
Using the formula
-3.0
(0.018)T =
dx.
calculate T, the time in seconds for the level of the water to drop
from 3.0 m to 1.5 m above the orifice.
(b) The velocity of a train which starts from rest is given by the fol-
lowing table, the time being reckoned in minutes from the start
and the speed in km/hour:
| † (minutes) |2|4 6 8 10 12
14 16 18 20
v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0
Estimate approximately the total distance ran in 20 minutes.
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