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Questions and Answers
What is the primary purpose of the fsolve function in the provided code?
What is the primary purpose of the fsolve function in the provided code?
- To calculate resistance values
- To optimize resistor design
- To find voltage values that satisfy two simultaneous equations (correct)
- To simulate diode behavior
The diode variable is calculated using the equation I0*(np.exp((v-v)/VT)-1.0).
The diode variable is calculated using the equation I0*(np.exp((v-v)/VT)-1.0).
False (B)
What does the variable VT represent in the diode code?
What does the variable VT represent in the diode code?
Diode threshold voltage
The maximum search method described is known as the _____ Search.
The maximum search method described is known as the _____ Search.
Match the variables with their functions in the diode circuit:
Match the variables with their functions in the diode circuit:
What is the purpose of the Newton's method in the provided code?
What is the purpose of the Newton's method in the provided code?
The relaxation method can converge to a solution regardless of the initial guess.
The relaxation method can converge to a solution regardless of the initial guess.
What are the parameters required by the 'newtmethod' function?
What are the parameters required by the 'newtmethod' function?
The relaxation method rewrites the equation in the form _____ .
The relaxation method rewrites the equation in the form _____ .
Match the methods with their characteristics:
Match the methods with their characteristics:
In the provided code, what value is used for tolerance in the Newton's method second call?
In the provided code, what value is used for tolerance in the Newton's method second call?
The output of the Newton's method showed that it estimates the value of pi accurately.
The output of the Newton's method showed that it estimates the value of pi accurately.
Name one disadvantage of the relaxation method.
Name one disadvantage of the relaxation method.
What does the variable R represent in the context of the Lagrange point calculation?
What does the variable R represent in the context of the Lagrange point calculation?
The gravitational force between the Earth and the Moon varies with the square of the distance between them.
The gravitational force between the Earth and the Moon varies with the square of the distance between them.
What is the primary purpose of the function lagrange_calc(r) in the code?
What is the primary purpose of the function lagrange_calc(r) in the code?
The gravitational constant G is equal to ________.
The gravitational constant G is equal to ________.
Match the following variables with their meanings:
Match the following variables with their meanings:
Which library is NOT imported in the Lagrange point code?
Which library is NOT imported in the Lagrange point code?
The variable OMEGA refers to the mass of the Moon.
The variable OMEGA refers to the mass of the Moon.
What is the expected output when the fsolve function is called in the code?
What is the expected output when the fsolve function is called in the code?
What is the primary advantage of Newton's Method?
What is the primary advantage of Newton's Method?
Newton's Method can only find a single root for a given function.
Newton's Method can only find a single root for a given function.
What is the purpose of the factor 'r' in the modified Newton's Method?
What is the purpose of the factor 'r' in the modified Newton's Method?
In the function funcpi(x), the term np.cos(x) contributes to the ______ of the function.
In the function funcpi(x), the term np.cos(x) contributes to the ______ of the function.
Match the following terms with their descriptions:
Match the following terms with their descriptions:
Which numerical method requires prior knowledge of the function's derivative?
Which numerical method requires prior knowledge of the function's derivative?
In Newton's Method, if the derivative at a guess is small, the method can converge quickly.
In Newton's Method, if the derivative at a guess is small, the method can converge quickly.
What is the purpose of the relax function in the provided code?
What is the purpose of the relax function in the provided code?
What is the output of the user-defined Newton's Method according to the code?
What is the output of the user-defined Newton's Method according to the code?
The function dfuncpi(x) is designed to calculate the ______ of the function funcpi(x).
The function dfuncpi(x) is designed to calculate the ______ of the function funcpi(x).
The maximum number of iterations defined in the code is 10000.
The maximum number of iterations defined in the code is 10000.
What does the REPORT_MOD variable represent in the code?
What does the REPORT_MOD variable represent in the code?
Which library is used in Python for plotting in the example code?
Which library is used in Python for plotting in the example code?
In the binary search method, after finding $f(x_3)$, set _____ to $x_3$ if $f(x_3) < 0$.
In the binary search method, after finding $f(x_3)$, set _____ to $x_3$ if $f(x_3) < 0$.
What message is printed if a starting value does not converge?
What message is printed if a starting value does not converge?
The numpy.linalg.solve function is designed to find roots of equations.
The numpy.linalg.solve function is designed to find roots of equations.
In the context of the Earth's gravitational forces, what does the example aim to find between the Earth and the Moon?
In the context of the Earth's gravitational forces, what does the example aim to find between the Earth and the Moon?
What is the purpose of the effincandescent_bulb function?
What is the purpose of the effincandescent_bulb function?
The constant BOLTZ represents the speed of light in a vacuum.
The constant BOLTZ represents the speed of light in a vacuum.
What does the variable COEFF3 represent in the calculations?
What does the variable COEFF3 represent in the calculations?
The integration is performed using the ______ function imported from the scipy.integrate module.
The integration is performed using the ______ function imported from the scipy.integrate module.
Match the following constants with their values:
Match the following constants with their values:
Which method is used to find the temperature that gives the peak efficiency?
Which method is used to find the temperature that gives the peak efficiency?
The variable temps is created to hold a range of efficiencies.
The variable temps is created to hold a range of efficiencies.
At what temperature is the initial efficiency for the incandescent bulb calculated?
At what temperature is the initial efficiency for the incandescent bulb calculated?
The Planck's constant is denoted as ______ in the code.
The Planck's constant is denoted as ______ in the code.
What output does the code provide regarding actual filament temperatures?
What output does the code provide regarding actual filament temperatures?
The gravitational force between the Earth and the Moon increases with the square of the distance between them.
The gravitational force between the Earth and the Moon increases with the square of the distance between them.
What is the gravitational constant G used in the calculations?
What is the gravitational constant G used in the calculations?
The function lagrange_calc(r) aims to find the _____ where the forces balance.
The function lagrange_calc(r) aims to find the _____ where the forces balance.
Which numerical method is primarily used to find the root in the Lagrange point code?
Which numerical method is primarily used to find the root in the Lagrange point code?
The mass of the moon, represented as 'MM', is 5.974e24 kg.
The mass of the moon, represented as 'MM', is 5.974e24 kg.
What is the purpose of the constant OMEGA in the context of the Lagrange point calculations?
What is the purpose of the constant OMEGA in the context of the Lagrange point calculations?
What is one of the main advantages of Newton's Method?
What is one of the main advantages of Newton's Method?
Newton's Method can be used without knowing the derivative of the function.
Newton's Method can be used without knowing the derivative of the function.
What factor is suggested to limit the jump in Newton's method when the derivative is small?
What factor is suggested to limit the jump in Newton's method when the derivative is small?
Match the following variables with their roles in the Newton's Method code:
Match the following variables with their roles in the Newton's Method code:
Which of the following is a potential disadvantage of Newton's Method?
Which of the following is a potential disadvantage of Newton's Method?
The function 'dfuncpi(x)' is designed to calculate the derivative of 'funcpi(x)'.
The function 'dfuncpi(x)' is designed to calculate the derivative of 'funcpi(x)'.
What language is used for the programming example in the document?
What language is used for the programming example in the document?
In Newton's Method, if the derivative at a guess is small, the method can converge too ______.
In Newton's Method, if the derivative at a guess is small, the method can converge too ______.
What does the 'newton' function from the scipy.optimize library do?
What does the 'newton' function from the scipy.optimize library do?
In the golden search algorithm, what condition prompts a sliding up of T1 and T2?
In the golden search algorithm, what condition prompts a sliding up of T1 and T2?
The variable x3 in the golden search represents the rightmost bound of the search interval.
The variable x3 in the golden search represents the rightmost bound of the search interval.
What is the primary purpose of the function 'goldsearch'?
What is the primary purpose of the function 'goldsearch'?
The golden search will stop if the values of x1 and x4 are closer than ______.
The golden search will stop if the values of x1 and x4 are closer than ______.
Match the variables with their roles in the golden search algorithm:
Match the variables with their roles in the golden search algorithm:
Which line in the 'goldsearch' function indicates that a sliding down of T3 happens?
Which line in the 'goldsearch' function indicates that a sliding down of T3 happens?
The golden search method always starts with fixed values for x1, x2, x3, and x4.
The golden search method always starts with fixed values for x1, x2, x3, and x4.
What condition is checked first in the golden search function to determine if the search should continue?
What condition is checked first in the golden search function to determine if the search should continue?
What is the output of the function func(x) when evaluated at $x = 2$?
What is the output of the function func(x) when evaluated at $x = 2$?
The maximum value of the function func(x) occurs at $x = 0$.
The maximum value of the function func(x) occurs at $x = 0$.
Which method is used to find the maximum point of the function func(x)?
Which method is used to find the maximum point of the function func(x)?
The peak temperature for maximizing visible light is calculated using the equation 𝐼 = 2𝜋𝐴ℎ𝑐 4 𝑒 :;!< − 1 at a wavelength of _____ Kelvin.
The peak temperature for maximizing visible light is calculated using the equation 𝐼 = 2𝜋𝐴ℎ𝑐 4 𝑒 :;!< − 1 at a wavelength of _____ Kelvin.
Match the following temperatures with their outputs in context of light bulb efficiency:
Match the following temperatures with their outputs in context of light bulb efficiency:
In the equation for visible light, what does the variable $𝜆$ represent?
In the equation for visible light, what does the variable $𝜆$ represent?
Ignoring constants means that the final result is independent of the specific values used in the calculations.
Ignoring constants means that the final result is independent of the specific values used in the calculations.
The function that evaluates the maximum efficiency of an incandescent bulb describes its behavior in terms of ______.
The function that evaluates the maximum efficiency of an incandescent bulb describes its behavior in terms of ______.
What is the primary purpose of the effincandescent_bulb function?
What is the primary purpose of the effincandescent_bulb function?
The variable BOLTZ represents Planck’s Constant.
The variable BOLTZ represents Planck’s Constant.
What range of temperatures is suggested for actual filament temperatures in the code?
What range of temperatures is suggested for actual filament temperatures in the code?
Which of the following constants is used to calculate COEFF3?
Which of the following constants is used to calculate COEFF3?
The efficiency at 300 K is considered very efficient.
The efficiency at 300 K is considered very efficient.
The variable C represents the speed of light in __________.
The variable C represents the speed of light in __________.
What value is initially used for temperature in the peak efficiency search?
What value is initially used for temperature in the peak efficiency search?
What is the maximum efficiency of an incandescent bulb at its peak temperature?
What is the maximum efficiency of an incandescent bulb at its peak temperature?
The typical operating temperature range for an incandescent bulb is from 2000 K to 3300 K.
The typical operating temperature range for an incandescent bulb is from 2000 K to 3300 K.
What is the goal of optimization in mathematical functions?
What is the goal of optimization in mathematical functions?
The six-hump function evaluates at a given ______________ coordinate.
The six-hump function evaluates at a given ______________ coordinate.
Match the following temperatures with their corresponding efficiencies:
Match the following temperatures with their corresponding efficiencies:
Which algorithm is prone to getting trapped in local minima during optimization?
Which algorithm is prone to getting trapped in local minima during optimization?
The efficiency of an incandescent bulb increases with temperature.
The efficiency of an incandescent bulb increases with temperature.
What library is used in Python for optimization functions?
What library is used in Python for optimization functions?
Flashcards
Finding Roots of Nonlinear Equations
Finding Roots of Nonlinear Equations
Determining the values of x where a function f(x) equals zero. Numerical methods are often used because precise solutions may be difficult.
Numerical Methods
Numerical Methods
Approximating solutions to equations and problems that do not have precise solutions using calculations.
fsolve
fsolve
A Python function that finds roots or zeros of functions using numerical methods.
Newton's Method
Newton's Method
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𝑓′(𝑥)
𝑓′(𝑥)
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Convergence Rate
Convergence Rate
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Multiple Roots
Multiple Roots
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Derivative
Derivative
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Iteration
Iteration
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Approximating Roots
Approximating Roots
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Convergence
Convergence
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Tolerance (Tol)
Tolerance (Tol)
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Relaxation Method
Relaxation Method
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Iterative process
Iterative process
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Root finding
Root finding
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Root Finding (fsolve)
Root Finding (fsolve)
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Binary Search
Binary Search
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Lagrange Point
Lagrange Point
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Initial Guess (x)
Initial Guess (x)
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Function (f(x))
Function (f(x))
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Lagrange Point Calculation
Lagrange Point Calculation
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Gravitational Constant (G)
Gravitational Constant (G)
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Mass of Earth (ME)
Mass of Earth (ME)
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Mass of Moon (MM)
Mass of Moon (MM)
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Distance to Moon (R)
Distance to Moon (R)
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Orbital Angular Velocity (Ω)
Orbital Angular Velocity (Ω)
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fsolve Function
fsolve Function
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Diode Circuit Analysis
Diode Circuit Analysis
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Node Voltage Analysis
Node Voltage Analysis
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Simultaneous Equations
Simultaneous Equations
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Golden Search Algorithm
Golden Search Algorithm
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Lightbulb Efficiency
Lightbulb Efficiency
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Integration Limits
Integration Limits
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Boltzmann Constant
Boltzmann Constant
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Planck's Constant
Planck's Constant
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Golden Search
Golden Search
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Temperature (Lightbulb)
Temperature (Lightbulb)
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Wavelength Range
Wavelength Range
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Optimization Algorithm
Optimization Algorithm
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Efficiency Calculation
Efficiency Calculation
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Integration Function
Integration Function
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Derivative (f'(x))
Derivative (f'(x))
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Limit ∆x
Limit ∆x
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f(T2) > f(T3) Case
f(T2) > f(T3) Case
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f(T2) < f(T3) Case
f(T2) < f(T3) Case
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Interval (x1, x2, x3, x4)
Interval (x1, x2, x3, x4)
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x1, x2, x3, x4 (New)
x1, x2, x3, x4 (New)
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Golden Search Algorithm (In Python Code)
Golden Search Algorithm (In Python Code)
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Optimization Goal
Optimization Goal
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Local Minima
Local Minima
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scipy.optimize
scipy.optimize
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What is the Six-Hump Problem?
What is the Six-Hump Problem?
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Why is optimization difficult?
Why is optimization difficult?
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What does 'coords' represent in the Six-Hump Problem?
What does 'coords' represent in the Six-Hump Problem?
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What is 'sixhump(coords)'?
What is 'sixhump(coords)'?
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What is the purpose of 'numpy'?
What is the purpose of 'numpy'?
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What is the function of goldsearch?
What is the function of goldsearch?
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What's the purpose of effincandescent_bulb?
What's the purpose of effincandescent_bulb?
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What do COEFF1, COEFF2, and COEFF3 represent?
What do COEFF1, COEFF2, and COEFF3 represent?
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What is the expfun function used for?
What is the expfun function used for?
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What is quad used for?
What is quad used for?
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What are the lowerlimit and upperlimit?
What are the lowerlimit and upperlimit?
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What is the purpose of the for loop in the final code snippet?
What is the purpose of the for loop in the final code snippet?
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How does temperature affect lightbulb efficiency?
How does temperature affect lightbulb efficiency?
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Earth and Moon Gravitational Force
Earth and Moon Gravitational Force
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Centrifugal Force
Centrifugal Force
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Initial Guess
Initial Guess
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Lagrange Point Code Setup
Lagrange Point Code Setup
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Study Notes
Course Title and Instructor
- Python for Rapid Engineering Solutions
- Steve Millman
Objectives
- Finding roots of nonlinear equations
- Examining numerical methods
- Using fsolve to find roots
- Examining examples
- Finding minima and maxima of functions (new)
- (Potential) Solving for voltages in a diode circuit. (new)
Solving Nonlinear Equations
- Many equations require numerical methods.
Newton's Method
- Formula: xnew = x - f(x)/f'(x)
- Advantage: Converges quickly
- Disadvantage: Needs the derivative f'(x)
- Disadvantage: May have multiple roots
- Disadvantage: May jump too far when f'(x) is small
- Possible Solution: Use a factor (r) to limit Δx in the formula xnew = x - r*f(x)/f'(x)
Relaxation Method
- Goal: Rearrange the equation to the form x = f(x)
- Steps:
- Start with a guess for x
- Plug the guess into f(x) to get a new value of x
- Repeat steps 1 and 2 until Δx < ε (a tolerance value
Relaxation Method Example
- Equation: x² + 2x + 1 = 0 (roots are -1)
- Rearranged: x = -(x² + 1)/2
- Disadvantage: May have multiple roots
- Disadvantage: Doesn't always converge
fsolve
- Function from scipy.optimize to find roots of an equation. It can also be used for finding minima and maxima, but the nature of the function is a crucial factor.
Lagrange Point
- Finding the point between two objects (like Earth and Moon) where their gravitational forces cancel.
Lagrange Point Formula (Earth/Moon)
- GMEarthmsat /r² = GMMoonmsat/(R-r)² - msat ω²r = 0
Lagrange Point Code
- Includes constants representing gravitational constant, Earth mass, Moon mass, distance, and angular rate of the Moon.
- Uses
fsolvefunction fromscipy.optimizeto find the solution. - Includes plotting to visualize the function and the root.
Binary Search
- Goal: Finding an x1 where f(x1) < 0. Then finding an x2 where f(x2) > 0.
- Steps:
- Set x3 = (x1 + x2) / 2.
- If f(x3) < 0, set x1 = x3. Otherwise, set x2 = x3
- Repeat until the difference between x1 and x2 is smaller than a tolerance value.
Golden Search
- A method for finding the maximum of a function (opposite of finding a minimum)
- Uses 4 points that are close together, trying to zoom in on the maximum value by squeezing the 4 values together, and by comparing the output of function at those points.
- Useful for optimization problems where a function has a maximum or minimum.
Optimization
- Goal: Finding the minimum or maximum of a function
- Complexity: Optimization problems can be challenging and complex. There are many powerful optimizers in Python to deal with complex situations
Six-Hump Problem
- A challenging problem found in the optimization area used as a demonstration of the optimization procedures
- Has multiple local minima to be calculated by optimization methods
Residuals and Nonlinear Fits
- Goal: Fitting parameterized curves to data
- Method: Using a least squares (a minimization algorithm) to fine-tune the parameters to minimize the difference between predicted and actual values
leastsq
- Function from
scipy.optimizefor finding the best parameters in an optimization problem, by minimizing the sum of squares. - Used for nonlinear curve fits or when data doesn't exactly match a formula.
Light Bulb Optimization
- Goal: Finding the temperature at which the energy output (or efficiency) of a light bulb is maximized
- Method: Using integration and the Golden Search approach to find that temperature.
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Description
This quiz explores key concepts in diode circuit analysis, focusing on numerical methods such as Newton's method and the relaxation method. Participants will match variables with their functions, understand the purpose of specific functions like fsolve, and discuss the characteristics of various solution techniques.
