Answered

Rewrite the tasks and questions so that they are easier to read. Fix any grammar or spelling errors. Remove phrases that are not part of the tasks or questions. Do not remove or change LaTeX formatting. Do not change or remove [tex] [/tex] tags. Do not translate the questions or any parts of the questions. If the questions are nonsense, rewrite them so that they make sense.

---

1. Use the function to calculate [tex]$y(-2)$[/tex] and [tex]$y(4)$[/tex].
2. Use the function to make a plot of the function [tex]$y(x)$[/tex] for [tex]$-4 \leq x \leq 8$[/tex].

Write a user-defined MATLAB function for the following math function:
[tex]\[ r(\theta) = 3 \sin (3 \cos (0.5 \theta)) \][/tex]

The input to the function is [tex]$\theta$[/tex] (in radians) and the output is [tex]$r$[/tex]. Write the function such that [tex]$\theta$[/tex] can be a vector.

(a) Use the function to calculate [tex]$r(\pi / 6)$[/tex] and [tex]$r(5 \pi / 6)$[/tex].
(b) Use the function to plot (polar plot) [tex]$r(\theta)$[/tex] for [tex]$0 \leq \theta \leq 4 \pi$[/tex].

In the U.S., fuel efficiency of cars is specified in miles per gallon (mpg). In Europe, it is often expressed in liters per 100 km. Write a MATLAB user-defined function that converts fuel efficiency from mpg to liters per 100 km. For the function name and arguments, use [tex] \text{Lkm} = \text{mpgToLpkm}(\text{mpg}) [/tex]. The input argument mpg is the efficiency in [tex]$mi / gl$[/tex], and the output argument Lkm is the efficiency in liters per 100 km (rounded to the nearest hundredth).



Answer :

GinnyAnswer
Certainly! Let's work through each part of this problem step-by-step.

### Part 1: User-defined MATLAB Function for `y(x)`

We'll define the function `y(x)` and make sure it handles vector input for `x`.

1. Define the Function `y`:

```matlab
function y = y_function(x)
y = -2 x.^5 + 3 x.^4 - x.^3 + 6 x.^2 - 2 x + 4;
end
```

2. Calculate `y(-2)` and `y(4)`:

To calculate the values, we call the function with the specific inputs:

```matlab
% Calculate y(-2)
y_neg2 = y_function(-2);

% Calculate y(4)
y_4 = y_function(4);

% Display the results
disp(['y(-2) = ', num2str(y_neg2)]);
disp(['y(4) = ', num2str(y_4)]);
```

3. Plot the Function for `-4 <= x <= 8`:

To plot the function over a specified interval, use:

```matlab
% Create the x values
x = linspace(-4, 8, 1000);

% Calculate the corresponding y values
y = y_function(x);

% Plot the function
figure;
plot(x, y);
xlabel('x');
ylabel('y(x)');
title('Plot of y(x) for -4 <= x <= 8');
grid on;
```

### Part 2: User-defined MATLAB Function for `r(\theta)`

We'll define the function `r(\theta)` for given angle `\theta`.

1. Define the Function `r`:

```matlab
function r = r_function(theta)
r = 3 sin(3 cos(0.5 theta));
end
```

2. Calculate `r(pi/6)` and `r(5pi/6)`:

To calculate these specific values:

```matlab
% Calculate r(pi/6)
r_pi_6 = r_function(pi / 6);

% Calculate r(5 pi / 6)
r_5pi_6 = r_function(5 pi / 6);

% Display the results
disp(['r(pi/6) = ', num2str(r_pi_6)]);
disp(['r(5pi/6) = ', num2str(r_5pi_6)]);
```

3. Polar Plot `r(\theta)` for `0 <= \theta <= 4\pi`:

To create a polar plot over the interval:

```matlab
% Create the theta values
theta = linspace(0, 4 pi, 1000);

% Calculate the corresponding r values
r = r_function(theta);

% Polar plot
figure;
polarplot(theta, r);
title('Polar Plot of r(\theta) for 0 <= \theta <= 4\pi');
```

### Part 3: Fuel Efficiency Conversion Function

We'll now write a function to convert fuel efficiency from miles per gallon (mpg) to liters per 100 kilometers (L/100 km).

1. Define the Conversion Function:

```matlab
function Lkm = mpgToLpkm(mpg)
% Conversion factor: 1 mile/gallon to 1 kilometer/liter
miles_per_km = 1 / 1.60934; % miles per kilometer
liters_per_gallon = 3.78541; % liters per gallon

% Conversion from mpg to L/100 km
Lkm = liters_per_gallon / (miles_per_km mpg) 100;

% Round to the nearest hundredth
Lkm = round(Lkm, 2);
end
```

To use this function:

```matlab
% Convert 25 miles/gallon to L/100 km
Lkm_25mpg = mpgToLpkm(25);

% Display the result
disp(['25 mpg is equivalent to ', num2str(Lkm_25mpg), ' liters per 100 km']);
```

By following these detailed instructions, you should be able to define the necessary MATLAB functions and perform the calculations and visualizations as required.