An industrial process requires a large motor to drive a heavy load. The efficiency of the motor is

a function of the current $($I$)$ in Amp can be expressed as:

$(I)=-0\mathrm{.}003I-0\mathrm{.}3e^{(-0\mathrm{.}2I)}+0\mathrm{.}788$

We would like to find the current that maximizes the efficiency of the motor.

To solve this problem, we first convert the optimization problem to a root finding problem. The

derivative of the efficiency function with respect to I is:

$\frac{d}{dI}=-0\mathrm{.}003+0\mathrm{.}06e^{(-0\mathrm{.}2I)}$

To find the root of the derivative function, we can apply any appropriate root finding methods,

such as the False Position method or the built$-$in Brent's Method of PYTHON. You would need

to complete the following tasks:

$1.$Plot the efficiency curve of the motor as a function of the current I. Add labels on both axes.

$2.$Plot the derivative of efficiency with respect to current. Add labels on both axes.

$3.$ Select an appropriate bracketing interval. Apply the False Position method code to solve this

optimization problem, with an accuracy of $5$ significant figures. In addition, insert a counter to

count the number of iterations. Print the solution with correct significant figures and number

of iterations using an informative statement.

$4.$Apply the PYTHON built$-$in brentq function to solve the optimization problem. Print the

solution.

$5.$ The False Position algorithm can be improved. One such improved version is called Illinois

algorithm. The Illinois algorithm is a bracketed root finding method. This algorithm modifies

the updating rule and formula of the False Position method. Starting with a valid bracketing

interval $($can be the same interval as the False Position$),$ the Illinois algorithm performs the

following:

$1)$ First guess

xr $=$

$($no ea since no previous iteration$)$

$2)$ Check

ME$2315$ Computer Aided Analysis EXAM $1$ Coding Part

If sign of f$($xl$))$ is the same as sign of f$($xr$),$ replace lower bound xl by xr

$,$ and the next

guess is xr $=$

Otherwise, replace upper bound, xu by xr

$,$ and the next guess is xr $=$

$3)$ Calculate ea $=|$

Current Guess $$Previous Guess

Current Guess $|\backslash$times $100\%.$ If ea $$ es

then stop. Otherwise,

repeat by going to step $2.$

In the above algorithm, note that there are two different updating formulas based on the signs

of F$($xl$),$ f$($xu$),$ and f$($xr$).$

Please use the same starting bracketing interval as in the False Position method. Insert a counter

to count the number of iterations. Print the solution with correct significant figures and number

of iterations using an informative statement.