Numerical differentiation: Consider a smooth function f(r). The central difference approxima- tion to the first derivative for small h> 0 is Def(1) = _ƒ (z + h) − f (x − h) 2h while f'(x) = Def(x) + Ch² for some constant C that depends on f". Discarding the error term we have f'(2) Def(x). Similarly we define the forward difference approximation to the first derivative for small h as Dif(2)= and the backward difference approximation 1. f(z)-f(z-h) Dof(2)= with truncation error of O(h). There is a general numerical procedure to determine the order of accuracy of the approximation (i.e. the exponent p in the error formula ChP): We consider a problem with known solution. In our case we consider a given function f(r) and values h = h₁, h₂, h with hi+1
Numerical differentiation: Consider a smooth function f(r). The central difference approxima- tion to the first derivative for small h> 0 is Def(1) = _ƒ (z + h) − f (x − h) 2h while f'(x) = Def(x) + Ch² for some constant C that depends on f". Discarding the error term we have f'(2) Def(x). Similarly we define the forward difference approximation to the first derivative for small h as Dif(2)= and the backward difference approximation 1. f(z)-f(z-h) Dof(2)= with truncation error of O(h). There is a general numerical procedure to determine the order of accuracy of the approximation (i.e. the exponent p in the error formula ChP): We consider a problem with known solution. In our case we consider a given function f(r) and values h = h₁, h₂, h with hi+1
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Please do the following questions with Python Coding given as answers. The question I need question 2 answered. It follows from question 1. The quesetion is in the image.
![Numerical differentiation: Consider a smooth function f(r). The central difference approxima-
tion to the first derivative for small h> 0 is
Def(r):
while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term
we have f'(r) Def(x).
Similarly we define the forward difference approximation to the first derivative for small h as
Djf(x)
and the backward difference approximation
1.
2.
f(r+h)-f(r-h)
2h
Dof(x)=
with truncation error of O(h). There is a general numerical procedure to determine the order of
accuracy of the approximation (i.e. the exponent p in the error formula Chp): We consider a problem
with known solution. In our case we consider a given function f(r) and values h = h₁, h₂,... hn
with hi+1 <h. We compute the quantity Df(z) and the exact value f'(x) and then we compute
the error E₁ = f'(x) - Df(r)] for each h,. This error must be E; = Ch. Taking the values E;
and E+1 we can approximate the order p as follows: We compute the fraction
E₁/E₁+1 = (hi/hi+1)P
and then using logarithms and solving for p we get the approximation
log(E₁/E+1)
log(h₂/hi+1)
p=
f(x+h)-f(x)
(x + h) -
f(x)-f(x-h)
h
Consider the function f(x)=sin(z). Approximate the derivative f'(x) with central,
forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005,
0.0001 and plot the graph of the error f'(x) - Df() for the corresponding values of h.
Compute the values of p using the above method and make a table with the values.
Comment on the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Faf03cfd0-25b8-4af2-bbf8-08c0974d4aa8%2F4dz8q8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Numerical differentiation: Consider a smooth function f(r). The central difference approxima-
tion to the first derivative for small h> 0 is
Def(r):
while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term
we have f'(r) Def(x).
Similarly we define the forward difference approximation to the first derivative for small h as
Djf(x)
and the backward difference approximation
1.
2.
f(r+h)-f(r-h)
2h
Dof(x)=
with truncation error of O(h). There is a general numerical procedure to determine the order of
accuracy of the approximation (i.e. the exponent p in the error formula Chp): We consider a problem
with known solution. In our case we consider a given function f(r) and values h = h₁, h₂,... hn
with hi+1 <h. We compute the quantity Df(z) and the exact value f'(x) and then we compute
the error E₁ = f'(x) - Df(r)] for each h,. This error must be E; = Ch. Taking the values E;
and E+1 we can approximate the order p as follows: We compute the fraction
E₁/E₁+1 = (hi/hi+1)P
and then using logarithms and solving for p we get the approximation
log(E₁/E+1)
log(h₂/hi+1)
p=
f(x+h)-f(x)
(x + h) -
f(x)-f(x-h)
h
Consider the function f(x)=sin(z). Approximate the derivative f'(x) with central,
forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005,
0.0001 and plot the graph of the error f'(x) - Df() for the corresponding values of h.
Compute the values of p using the above method and make a table with the values.
Comment on the results.
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