(a) Show tm/n = ( n√t)m for all m, n ∈ N.(b) Show log(tx) = x log t, for all t > 0 and x ∈ R.(c) Show tx is differentiable on R and find the derivative.Finding the right definition for x! is harder than defining tx, but the strategyis essentially the same. We are seeking a formula of the form n! = g(n) where g yields a meaningful formula when n is replaced by x. What might such a function g(x) = x! look like when graphed over R? For x ≥ 0 it must grow extremely rapidly to keep up with n!, but how about on x < 0? Using a functional equation for x! we can create a reasonable artist’s rendering of the function we are looking for.
(a) Show tm/n = ( n√t)m for all m, n ∈ N.(b) Show log(tx) = x log t, for all t > 0 and x ∈ R.(c) Show tx is differentiable on R and find the derivative.Finding the right definition for x! is harder than defining tx, but the strategyis essentially the same. We are seeking a formula of the form n! = g(n) where g yields a meaningful formula when n is replaced by x. What might such a function g(x) = x! look like when graphed over R? For x ≥ 0 it must grow extremely rapidly to keep up with n!, but how about on x < 0? Using a functional equation for x! we can create a reasonable artist’s rendering of the function we are looking for.
(a) Show tm/n = ( n√t)m for all m, n ∈ N.(b) Show log(tx) = x log t, for all t > 0 and x ∈ R.(c) Show tx is differentiable on R and find the derivative.Finding the right definition for x! is harder than defining tx, but the strategyis essentially the same. We are seeking a formula of the form n! = g(n) where g yields a meaningful formula when n is replaced by x. What might such a function g(x) = x! look like when graphed over R? For x ≥ 0 it must grow extremely rapidly to keep up with n!, but how about on x < 0? Using a functional equation for x! we can create a reasonable artist’s rendering of the function we are looking for.
(a) Show tm/n = ( n√t)m for all m, n ∈ N. (b) Show log(tx) = x log t, for all t > 0 and x ∈ R. (c) Show tx is differentiable on R and find the derivative. Finding the right definition for x! is harder than defining tx, but the strategy is essentially the same. We are seeking a formula of the form n! = g(n) where g yields a meaningful formula when n is replaced by x. What might such a function g(x) = x! look like when graphed over R? For x ≥ 0 it must grow extremely rapidly to keep up with n!, but how about on x < 0? Using a functional equation for x! we can create a reasonable artist’s rendering of the function we are looking for.
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
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.
