README.md

# 05_curve_fitting
## Problem 2
Coefficient of Determination
- Part A: 0.9801
- Part B: 0.9112
- Part C: 0.9462
- Part D: 0.9219

Best Fit Lines
### Part A
- y = 0.375+0.98644x+0.84564/x

![Part A Best Fit Line](./figures/figure1.png)

### Part B
- y = 22.47-1.36x+0.28x^2

![Part B Best Fit Line](./figures/figure2.png)

### Part C
- y = 4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x)

![Part C Best Fit Line](./figures/figure3.png)

### Part D
- y = 0.99sin(t)+0.5sin(3t)

![Part D Best Fit Line](./figures/figure4.png)

## Problem 3
### Part A
```matlab
darts = dlmread('compiled_data.csv',',',1,0);
x_darts =darts(:,2).*cosd(darts(:,3));
y_darts =darts(:,2).*sind(darts(:,3));

ur = darts(:,1);
i = 1;

for r = 0:32
    i_interest = find(ur==r);
    mx_ur(i)= mean(x_darts(i_interest));
    my_ur(i)= mean(y_darts(i_interest));
    i = i +1;
end
    accuracy = mx_ur + my_ur;
    val = 0; %value trying to be closest to
    abs_accuracy = abs(accuracy-val); %takes absolute value of accuracy vector
    [~, index] = min(abs_accuracy)
    closest_value = accuracy(index)
```

The most accurate dart thrower was person 31, with a combined x and y average of 0.0044 cm away from zero.

### Part B
```matlab
darts = dlmread('compiled_data.csv',',',1,0);
x_darts =darts(:,2).*cosd(darts(:,3));
y_darts =darts(:,2).*sind(darts(:,3));

ur = darts(:,1);
i = 1;

for r = 0:32
    i_interest = find(ur==r);
    stdx_ur(i)= std(x_darts(i_interest));
    stdy_ur(i)= std(y_darts(i_interest));
    i = i +1;
end
    precision = stdx_ur + stdy_ur;
    val = 0; %value trying to be closest to
    abs_accuracy = abs(precision-val); %takes absolute value of accuracy vector
    [~, index] = min(abs_accuracy)
    closest_value = precision(index)
```

The most precise dart thrower was person 32, with a combined x and y standard deviation of 3.407.

## Problem 4
### Part A
```matlab
function [mean_buckle_load,std_buckle_load]=buckle_monte_carlo(E,r_mean,r_std,L_mean,L_std)
    r= normrnd(r_mean,r_std,[100 1]);
    L= normrnd(L_mean,L_std,[100 1]);
    p_cr = (pi.^3.*E.*r.^4)./(16.*L.^2);
    mean_buckle_load = mean(p_cr);
    std_buckle_load = std(p_cr);
end
```
Output:
- Mean_buckle_load = 160.81 N
- Std_buckle_load = 70.21 N

### Part B
```matlab
N=100;
r_mean=0.01;
r_std=.001;
p_cr = 160.81; %N - from part A
r=normrnd(r_mean,r_std,[N,1]);
L = ((pi^3*E.*r.^4)./(16*p_cr)).^0.5;
L_mean = mean(L)
```
Output:
- Length (L) = 4.955 m

## Problem 5
### Part A
```matlab
cd_out_linear = sphere_drag(300,'linear') = 0.1750
cd_out_spline = sphere_drag(300,'spline') = 0.1809
cd_out_pchip = sphere_drag(300,'pchip') = 0.1807
```

### Part B
![Drag Force vs. Velocity - 3 Interpolation Methods](./figures/figure5.png)

## Problem 6
| Method | Value | Error |
| --- | --- | --- |
| Analytical | 8.375 | 0% |
| 1 Gauss Point | 8.229 | 1.74% |
| 2 Gauss Point | 8.375 | 0% |
| 3 Gauss Point | 8.375 | 0% |