In some situations, we would like to differentiate a function more than once. We call this higher-order differentiation, and we can define it in the following way: f(®(x) = f(x) flne1)(x) = (f\")'(x) . Caleulate f Chowyour working ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi- cient is c,. For example, the polynomial above is of order 5, since it has highest non-zero coefficient c5 2. Show that f+1)(x)= 0 im) Now let )=si Caleulate iv) Fix some a e R. With f (x) = sin(x) as before, we define a new function g: N R such that g(n) = f(")(a). Write an alternative definition of g(n) that does not involve differentiation.
In some situations, we would like to differentiate a function more than once. We call this higher-order differentiation, and we can define it in the following way: f(®(x) = f(x) flne1)(x) = (f\")'(x) . Caleulate f Chowyour working ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi- cient is c,. For example, the polynomial above is of order 5, since it has highest non-zero coefficient c5 2. Show that f+1)(x)= 0 im) Now let )=si Caleulate iv) Fix some a e R. With f (x) = sin(x) as before, we define a new function g: N R such that g(n) = f(")(a). Write an alternative definition of g(n) that does not involve differentiation.
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter10: Radical Functions And Equations
Section: Chapter Questions
Problem 15CT
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![b) In some situations, we would like to differentiate a function more than once. We call
this higher-order differentiation, and we can define it in the following way:
f(0(x) = f (x)
fla+1) (x) = (f") (x)
Calculate fa Chowyour working,
ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi-
cient is c,. For example, the polynomial above is of order 5, since it has highest
non-zero coefficient c5 = 2. Show that f("+1)(x) = 0
iii) Now let ) = si Caleulate
our working
iv) Fix some a e R. With f(x) = sin(x) as before, we define a new function g: N→R
such that g(n) = f(n)(a). Write an alternative definition of g(n) that does not
involve differentiation.
c) Below is an alternative but equivalent definition of the definite integral of f : R → R
between a and b, given x = a+ k() for 0 <k<n:
n-1
b-a
I„(f,a,b) =
f(xk)
k=0
f (x)dx = lim I,„f,a,b)
i) For f(x) = x² , calculate I5(f,-2, 8). Show your working.
ii) We now define
h -a
Inf,a,b)= |
k=1
For f(x) = x² , calculate J5(f,-2,8) . Show your working.
A Coloula
iv) We call a function f: R → R monotone increasing if x > y implies f(x) > f (y).
Show that, if f is monotone increasing, then, for any a,be R, a < b , and n e N
In(f, a,b)< Jn(f , a, b)
v) Without formally proving it, explain why:
lim J„(f , a, b) =
f (x)dx = lim I,(f,a,b)
100](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621668a2-d7f9-4b5a-8e75-1c56081c0500%2Fbf77af36-04fe-4d60-a1db-972f9b14c70e%2Fo6yybym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:b) In some situations, we would like to differentiate a function more than once. We call
this higher-order differentiation, and we can define it in the following way:
f(0(x) = f (x)
fla+1) (x) = (f") (x)
Calculate fa Chowyour working,
ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi-
cient is c,. For example, the polynomial above is of order 5, since it has highest
non-zero coefficient c5 = 2. Show that f("+1)(x) = 0
iii) Now let ) = si Caleulate
our working
iv) Fix some a e R. With f(x) = sin(x) as before, we define a new function g: N→R
such that g(n) = f(n)(a). Write an alternative definition of g(n) that does not
involve differentiation.
c) Below is an alternative but equivalent definition of the definite integral of f : R → R
between a and b, given x = a+ k() for 0 <k<n:
n-1
b-a
I„(f,a,b) =
f(xk)
k=0
f (x)dx = lim I,„f,a,b)
i) For f(x) = x² , calculate I5(f,-2, 8). Show your working.
ii) We now define
h -a
Inf,a,b)= |
k=1
For f(x) = x² , calculate J5(f,-2,8) . Show your working.
A Coloula
iv) We call a function f: R → R monotone increasing if x > y implies f(x) > f (y).
Show that, if f is monotone increasing, then, for any a,be R, a < b , and n e N
In(f, a,b)< Jn(f , a, b)
v) Without formally proving it, explain why:
lim J„(f , a, b) =
f (x)dx = lim I,(f,a,b)
100
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