You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
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    filename: derivative_MPL_first_order_taylor_approximation.m
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    the code of the file until where you have to start completion: %% Differential and Marginal Product
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% 
% In economics papers, we often see these symbols: $\Delta$, $d$, $\partial$
% 
% $\Delta{y}$ and $\Delta{x}$ are changes along the function graph: given some 
% $x_0$: 
%% 
% * $\Delta{y} = f(x_0 + \Delta{x}) - f(x_0)$
%% 
% $\text{d}y$ and $\text{d}x$ are *di
    
function graph: given some 
% $x_0$: 
%% 
% * $\Delta{y} = f(x_0 + \Delta{x}) - f(x_0)$
%% 
% $\text{d}y$ and $\text{d}x$ are *differentials*, which are, at each point 
% $(x, f(x))$, the changes in y for the tangent line given a change in $x$: 
%% 
% * $\text{d}{x} = \Delta{x}$
% * $\text{d}y = f'(x_0) \cdot \text{d}x$
%% 
% we have seen that the tangent line to $f(x)$ at $x_0$ approximates the function 
% $f(x)$ around $x_0$ (and is identical at $x_0$), so approximately, for small 
% $\Delta{x}$:
%% 
% * $\Delta{y} \approx \text{d} y$
%% MPL for Cobb-Douglas
% With this Cobb-Douglas production function:
% 
% $$F(K,L) = K^{\alpha} \cdot L^{1-\alpha}$$
% 
% As derived earlier, the derivative with respect to labor is (MPL=marginal 
% product of labor):
% 
% $$\text{MPL}(K,L) = (1-\alpha) \cdot K^{\alpha} \cdot L^{-\alpha}$$
%% Interpreting MPL
% In the above problem, suppose $K_0=1$ and $L_0=1$, and $\alpha=0.5$. Without 
% a calculator, we can calculate what output and marginal product of labor is:
% 
% $$F(K_0=1, L_0=1) = 1$$
% 
% $$\text{MPL}(K_0=1, L_0=1) = 0.5$$
% 
% This means the total output with one unit of worker and one unit of capital 
% is $1$.
% 
% Becareful about interpreting the $\text{MPL}$ term (we are treating it as 
% a function of continuous $L$, some define MPL in terms of discrete increases 
% in $L$), it is a derivative, which as we have discussed is the slope of the 
% tangent line to the production function line with fixed $K$ and $L$ along the 
% $\text{x-axis}$. Which means if you increase labor by a infinitestimally small 
% amount when existing $K=1$ and $L=1$, the *slope* of output increase will be 
% $0.5$. The actual output increase is that infinitestimally small increase in 
% labor multiplied by $0.5$. It is perhaps difficult to conceptualize what it 
% means to multiply something infinitely small by another number. To make the 
% idea more conconcret, we will think using $\text{MPL}$ to approximate the increase 
% in output given a small increase in labor.
%% Exact Output Calculated with Matlab
% Continuing with the two numbers we can calculate without a calculator:
% 
% $$F(K_0=1, L_0=1) = 1$$
% 
% $$\text{MPL}(K_0=1, L_0=1) = 0.5$$
% 
% Suppose we are interested in the increase in output when labor increases from 
% $L_0=1$ to $L_1=1.03$, what is the new output? What is the increase in output? 
% (You can think of this as increasing the number of workers by 3 percentage points.)
% 
% _*Exact* Solution_*:* We can directly calculate this, very hard by hand, but 
% using matlab:

% Define parameters, fixed K0
alpha = 0.5;
K0 = 1;
% Define equation with L is unknown
syms L
f(L) = K0^(alpha)*L^(1-alpha);
% two different L levels
L0 = 1;
L1 = 1.03;
% Fill the L0 and L1 values into the symbolic function
YL0 = subs(f, L0)
YL1 = subs(f, L1)
% Take difference
increaseOutput = YL1 - YL0
% Turn symbolic answer to double (easier to read), increase in output
increaseOutput = double(increaseOutput)
% new level of output
newLevelOutput = double(YL1)
%% *Approximate Output Increase with Derivative (MPL)*
% Remember as we have seen, the slope of the tangent line at $L_0$ is similar 
% to the slope of the line between $L_0+h$ and $L_0$, from the definition of derivative, 
% for $h$ small, the following should be true:
%% 
% * $F_{L}'(K_0, L_0) \approx \frac{F(K_0, L_0+h)-F(K_0, L_0)}{h}$
%% 
% Just move the $h$ from the right to the left, the *increase in output* is 
% _approximately_:
%% 
% * $F(K_0, L_0+h)-F(K_0, L_0) \approx F_{L}'(K_0, L_0) \cdot h$
%% 
% Furthermore, the *level of output* is _approximately_:
%% 
% * $F(K_0, L_0+h) \approx F(K_0, L_0) + F_{L}'(K_0, L_0) \cdot h$
%% 
% In our case above, we can now approximate output levels using the two numbers 
% we calculated by hand, with $K_0 =1$ and $L_0=1$:
%% 
% * $F(K_0, L_0+h) \approx F(K_0, L_0) + F_{L}'(K_0, L_0) \cdot h = 1 + 0.5 
% \cdot h$
%% 
% Now with $\mathbf{1 + 0.5 \cdot h}$, that is something we can use very easily, 
% back to 1st grade math. We calculated previously that if $h=0.03$, the exact 
% new level of output is $1.0149$:

newLevelOutput
%% 
% What is our approximated increase that we can calculate by hand? It is $1.015$

approximatedLevelOutput = 1 + 0.5 * 0.03
%% 
% These are almost identical.
%% First Order Taylor Polynomial Approximation
% What we have just done is called *First Order Taylor Polynomial Approximation*, 
% which can be written more generally as:
%% 
% * $F(X_0+h) \approx F(X_0) + F'(X_0) \cdot h$
%% 
% Often you see this written as below, these are equivalent:
%% 
% * $f(x)  \approx f(a) + f'(a) \cdot (x-a)$
%% 
% This is just another way to write down the differential formula described 
% at the beginning
%% 
% * $F( X_0 + \Delta{x}) - F(X_0) = \Delta{Y} \approx  \text{d}{Y} =  F'(X_0) 
% \cdot \text{d}{X}$
%% 
% When solving economics problems, we often end