# Question 1:
Solve the following equations

## a. $$3x - 3 = x + 5$$

<details> <summary> Example: </summary>

$$2x - 4 = x + 6$$

We are tasked with solving the equation:
$$
2x - 4 = x + 6
$$
We will break it down into 4 steps.

### Step 1: Move all terms involving $x$ to one side
To isolate the terms with $x$, subtract $x$ from both sides:
$$
2x - x - 4 = 6
$$
This simplifies to:
$$
x - 4 = 6
$$

### Step 2: Move the constant terms to the other side
Now, add $4$ to both sides to move the constants to the right side:
$$
x = 6 + 4
$$
This simplifies to:
$$
x = 10
$$

### Step 3: Verify the solution
Substitute $x = 10$ back into the original equation to check:
$$
2(10) - 4 = 10 + 6
$$
$$
20 - 4 = 16 \quad \text{and} \quad 10 + 6 = 16
$$
Both sides are equal, so the solution is correct.

Thus, the solution to the equation is:
$$
x = 10
$$

</details>

## b. $$5 - x = x - 7$$

<details> <summary> Example: </summary>

$$4 - x = x - 6$$

We are tasked with solving the equation:
$$
4 - x = x - 6
$$
We will break it down into 4 steps.

### Step 1: Move all terms involving $x$ to one side
To isolate the terms with $x$, add $x$ to both sides:
$$
4 = 2x - 6
$$

### Step 2: Move the constant terms to the other side
Now, add $6$ to both sides to move the constants to the right side:
$$
4 + 6 = 2x
$$
This simplifies to:
$$
10 = 2x
$$

### Step 3: Solve for $x$
Divide both sides by $2$ to solve for $x$:
$$
x = \frac{10}{2} = 5
$$

### Step 4: Verify the solution
Substitute $x = 5$ back into the original equation to check:
$$
4 - 5 = 5 - 6
$$
$$
-1 = -1
$$
Both sides are equal, so the solution is correct.

Thus, the solution to the equation is:
$$
x = 5
$$

</details>

## c. $$x^2 + 15 = 7x + 5$$

<details> <summary> Example: </summary>

$$x^2 +20= 9 x +2$$

We are tasked with solving the equation:
$$
x^2 + 20 = 9x + 2
$$
We will break it down into 4 steps.

### Step 1: Move all terms to one side
To set the equation equal to $0$, subtract $9x + 2$ from both sides:
$$
x^2 + 20 - 9x - 2 = 0
$$
This simplifies to:
$$
x^2 - 9x + 18 = 0
$$

### Step 2: Factor the quadratic equation
We now factor the quadratic equation $x^2 - 9x + 18 = 0$. Find two numbers that multiply to $18$ and add to $-9$. These numbers are $-6$ and $-3$:
$$
(x - 6)(x - 3) = 0
$$

### Step 3: Solve for $x$
Set each factor equal to $0$ and solve for $x$:
$$
x - 6 = 0 \quad \text{or} \quad x - 3 = 0
$$
Thus:
$$
x = 6 \quad \text{or} \quad x = 3
$$

### Step 4: Verify the solutions
Substitute $x = 6$ into the original equation:
$$
(6)^2 + 20 = 9(6) + 2
$$
$$
36 + 20 = 54 + 2 \quad \text{which simplifies to} \quad 56 = 56
$$
Now substitute $x = 3$:
$$
(3)^2 + 20 = 9(3) + 2
$$
$$
9 + 20 = 27 + 2 \quad \text{which simplifies to} \quad 29 = 29
$$
Both solutions are correct.

Thus, the solutions to the equation are:
$$
x = 6 \quad \text{or} \quad x = 3
$$

</details>

## d. $$10 - x^2 = -20 + x$$

<details> <summary> Example: </summary>

$$25-x^2= 3 x -3$$

We are tasked with solving the equation:
$$
25 - x^2 = 3x - 3
$$
We will break it down into 4 steps.

### Step 1: Move all terms to one side
To set the equation equal to $0$, subtract $3x - 3$ from both sides:
$$
25 - x^2 - 3x + 3 = 0
$$
This simplifies to:
$$
-x^2 - 3x + 28 = 0
$$
Multiply through by $-1$ to make the quadratic term positive:
$$
x^2 + 3x - 28 = 0
$$

### Step 2: Factor the quadratic equation
We now factor the quadratic equation $x^2 + 3x - 28 = 0$. Find two numbers that multiply to $-28$ and add to $3$. These numbers are $7$ and $-4$:
$$
(x + 7)(x - 4) = 0
$$

### Step 3: Solve for $x$
Set each factor equal to $0$ and solve for $x$:
$$
x + 7 = 0 \quad \text{or} \quad x - 4 = 0
$$
Thus:
$$
x = -7 \quad \text{or} \quad x = 4
$$

### Step 4: Verify the solutions
Substitute $x = -7$ into the original equation:
$$
25 - (-7)^2 = 3(-7) - 3
$$
$$
25 - 49 = -21 - 3 \quad \text{which simplifies to} \quad -24 = -24
$$
Now substitute $x = 4$:
$$
25 - (4)^2 = 3(4) - 3
$$
$$
25 - 16 = 12 - 3 \quad \text{which simplifies to} \quad 9 = 9
$$
Both solutions are correct.

Thus, the solutions to the equation are:
$$
x = -7 \quad \text{or} \quad x = 4
$$

</details>

# Question 2:
Find the following derivatives

## a. For the function $$f(x) = \ln(2x - 4)$$ find $$f'(x) = $$

<details> <summary> Example: </summary>

For the function $$f(x) = \ln(3x - 5)$$ find $$f'(x) = $$

We are tasked with finding the derivative of the function:
$$
f(x) = \ln(3x - 5)
$$
To do this, we will use the chain rule.

### Chain Rule and Logarithmic Differentiation
The derivative of $\ln(u(x))$ is:
$$
f'(x) = \frac{1}{u(x)} \cdot u'(x)
$$
Where $u(x) = 3x - 5$.

### Step 1: Differentiate $u(x)$
First, we find the derivative of $u(x)$:
$$
u'(x) = 3
$$

### Derivative Table

| | Argument of $\ln$ | Derivative of Argument |
| --- | --- | --- |
| **Original** | $3x - 5$ | $3$ |

### Step 2: Apply the chain rule
Substitute $u(x) = 3x - 5$ and $u'(x) = 3$ into the chain rule:
$$
f'(x) = \frac{1}{3x - 5} \cdot 3
$$

### Step 3: Simplify
The derivative simplifies to:
$$
f'(x) = \frac{3}{3x - 5}
$$

Thus, the derivative of the function is:
$$
f'(x) = \frac{3}{3x - 5}
$$

</details>

## b. For the function $$g(x) = x^2 e^{2x}$$ find: $$g'(x) = $$

<details> <summary> Example: </summary>

$$g(x) = x^3 e^{4x}$$ find: $$g'(x) = $$

We are tasked with finding the derivative of the function:
$$
g(x) = x^3 e^{4x}
$$
To do this, we will use the product rule and the chain rule.

### Product Rule
The product rule states that for two functions $u(x)$ and $v(x)$, the derivative of their product is:
$$
g'(x) = u'(x)v(x) + u(x)v'(x)
$$
Let:
- $u(x) = x^3$
- $v(x) = e^{4x}$

### Step 1: Differentiate $u(x)$ and $v(x)$
- $u'(x) = 3x^2$
- $v'(x)$ requires the chain rule. The derivative of $e^{4x}$ is:
$$
v'(x) = 4e^{4x}
$$

### Derivative Table

| | First Function ($u(x)$) | Second Function ($v(x)$) |
| --- | --- | --- |
| **Original** | $x^3$ | $e^{4x}$ |
| **Derivative** | $3x^2$ | $4e^{4x}$ |

### Step 2: Apply the product rule
Substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the product rule:
$$
g'(x) = (3x^2)(e^{4x}) + (x^3)(4e^{4x})
$$

### Step 3: Simplify
Factor out $e^{4x}$:
$$
g'(x) = e^{4x}(3x^2 + 4x^3)
$$
This simplifies to:
$$
g'(x) = e^{4x}x^2(3 + 4x)
$$

Thus, the derivative of the function is:
$$
g'(x) = e^{4x}x^2(3 + 4x)
$$

</details>

## c. For the function $$h(x) = \frac{x - 9}{x}$$ find: $$h'(x) = $$

<details> <summary> Example: </summary>

For the function $$h(x) = \frac{x - 5}{x}$$ find: $$h'(x) = $$

We are tasked with finding the derivative of the function:
$$
h(x) = \frac{x - 5}{x}
$$
To do this, we will use the quotient rule.

### Quotient Rule
The quotient rule states that for two functions $u(x)$ and $v(x)$, the derivative of their quotient is:
$$
h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
$$
Let:
- $u(x) = x - 5$
- $v(x) = x$

### Step 1: Differentiate $u(x)$ and $v(x)$
- $u'(x) = 1$
- $v'(x) = 1$

### Derivative Table

| | Numerator (First Function) | Denominator (Second Function) |
| --- | --- | --- |
| **Original** | $x - 5$ | $x$ |
| **Derivative** | $1$ | $1$ |

### Step 2: Apply the quotient rule
Substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the quotient rule:
$$
h'(x) = \frac{(1)(x) - (x - 5)(1)}{x^2}
$$

### Step 3: Simplify
Expand the numerator:
$$
h'(x) = \frac{x - (x - 5)}{x^2}
$$
Simplify further:
$$
h'(x) = \frac{x - x + 5}{x^2} = \frac{5}{x^2}
$$

Thus, the derivative of the function is:
$$
h'(x) = \frac{5}{x^2}
$$

</details>

# Question 3:
Lorenz curves also can be used to provide a relative measure of the distribution of the total assets of a country. Using data in a report by the U.S. Congressional Joint Economic Committee, an economist produced the following Lorenz curves for the distribution of total assets in the United States in 1963 and 1983:

$$f(x) = x^{10} \text{ Lorenz curve for 1963}$$

$$g(x) = x^{12} \text{ Lorenz curve for 1983}$$

Find the Gini index of asset concentration for each Lorenz curve, and interpret the result.

<details> <summary> Example: </summary>

$$f(x) = x^{11} \text{ Lorenz curve for 1963}$$

$$g(x) = x^{13} \text{ Lorenz curve for 1983}$$

We are tasked with finding the Gini index for the Lorenz curves representing the distribution of total assets in the United States in 1963 and 1983. The Gini index is a measure of income or wealth inequality, and it is calculated using the formula:

$$
G = 1 - 2 \int_0^1 L(x) \, dx
$$

Where $L(x)$ is the Lorenz curve.

### Lorenz curves given:
- 1963: $f(x) = x^{11}$
- 1983: $g(x) = x^{13}$

We will compute the Gini index for each year.

### Step 1: Find the antiderivative of $f(x) = x^{11}$ (1963)

The antiderivative of $f(x) = x^{11}$ is:
$$
\int_0^1 x^{11} \, dx = \frac{x^{12}}{12} \Bigg|_0^1
$$
Evaluating this at the bounds:
$$
\frac{1^{12}}{12} - \frac{0^{12}}{12} = \frac{1}{12}
$$

### Step 2: Compute the Gini index for 1963

Substitute the result into the Gini index formula:
$$
G_{1963} = 1 - 2 \times \frac{1}{12} = 1 - \frac{2}{12} = 1 - \frac{1}{6} = \frac{5}{6}
$$

Thus, the Gini index for 1963 is:
$$
G_{1963} = \frac{5}{6} \approx 0.833
$$

### Step 3: Find the antiderivative of $g(x) = x^{13}$ (1983)

The antiderivative of $g(x) = x^{13}$ is:
$$
\int_0^1 x^{13} \, dx = \frac{x^{14}}{14} \Bigg|_0^1
$$
Evaluating this at the bounds:
$$
\frac{1^{14}}{14} - \frac{0^{14}}{14} = \frac{1}{14}
$$

### Step 4: Compute the Gini index for 1983

Substitute the result into the Gini index formula:
$$
G_{1983} = 1 - 2 \times \frac{1}{14} = 1 - \frac{2}{14} = 1 - \frac{1}{7} = \frac{6}{7}
$$

Thus, the Gini index for 1983 is:
$$
G_{1983} = \frac{6}{7} \approx 0.857
$$

### Interpretation of the Results:
- The Gini index for 1963 is approximately $0.833$, which indicates a high level of asset concentration (inequality) in 1963.
- The Gini index for 1983 is approximately $0.857$, which is even higher, indicating that asset inequality increased between 1963 and 1983.

</details>

# Question 4:
Using data from the U.S. Census Bureau, an economist produced the following Lorenz curves for the distribution of U.S. income in 1962 and 1972:

$$f(x) = \frac{3}{10}x + \frac{7}{10}x^2 \text{ Lorenz curve for 1962}$$

$$g(x) = \frac{1}{2}x + \frac{1}{2}x^2 \text{ Lorenz curve for 1972}$$

Find the Gini index of income concentration for each Lorenz curve, and interpret the result.

<details> <summary> Example: </summary>

$$f(x) = \frac{2}{10}x + \frac{8}{10}x^2 \text{ Lorenz curve for 1962}$$

$$g(x) = \frac{1}{3}x + \frac{2}{3}x^2 \text{ Lorenz curve for 1972}$$

We are tasked with finding the Gini index for the Lorenz curves representing the distribution of U.S. income in 1962 and 1972. The Gini index is a measure of income or wealth inequality, and it is calculated using the formula:

$$
G = 1 - 2 \int_0^1 L(x) \, dx
$$

Where $L(x)$ is the Lorenz curve.

### Lorenz curves given:
- 1962: $f(x) = \frac{2}{10}x + \frac{8}{10}x^2$
- 1972: $g(x) = \frac{1}{3}x + \frac{2}{3}x^2$

We will compute the Gini index for each year.

### Step 1: Find the antiderivative of $f(x) = \frac{2}{10}x + \frac{8}{10}x^2$ (1962)

The antiderivative of $f(x)$ is:
$$
\int_0^1 \left( \frac{2}{10}x + \frac{8}{10}x^2 \right) dx = \frac{2}{10} \cdot \frac{x^2}{2} + \frac{8}{10} \cdot \frac{x^3}{3} \Bigg|_0^1
$$
Simplify the expression:
$$
= \frac{2}{20}x^2 + \frac{8}{30}x^3 \Bigg|_0^1
$$
Evaluate at the bounds:
$$
= \left( \frac{2}{20}(1)^2 + \frac{8}{30}(1)^3 \right) - \left( \frac{2}{20}(0)^2 + \frac{8}{30}(0)^3 \right)
$$
$$
= \frac{2}{20} + \frac{8}{30} = \frac{1}{10} + \frac{4}{15} = \frac{3}{30} + \frac{8}{30} = \frac{11}{30}
$$

### Step 2: Compute the Gini index for 1962

Substitute the result into the Gini index formula:
$$
G_{1962} = 1 - 2 \times \frac{11}{30} = 1 - \frac{22}{30} = \frac{8}{30} = \frac{4}{15} \approx 0.267
$$

### Step 3: Find the antiderivative of $g(x) = \frac{1}{3}x + \frac{2}{3}x^2$ (1972)

The antiderivative of $g(x)$ is:
$$
\int_0^1 \left( \frac{1}{3}x + \frac{2}{3}x^2 \right) dx = \frac{1}{3} \cdot \frac{x^2}{2} + \frac{2}{3} \cdot \frac{x^3}{3} \Bigg|_0^1
$$
Simplify the expression:
$$
= \frac{1}{6}x^2 + \frac{2}{9}x^3 \Bigg|_0^1
$$
Evaluate at the bounds:
$$
= \left( \frac{1}{6}(1)^2 + \frac{2}{9}(1)^3 \right) - \left( \frac{1}{6}(0)^2 + \frac{2}{9}(0)^3 \right)
$$
$$
= \frac{1}{6} + \frac{2}{9} = \frac{3}{18} + \frac{4}{18} = \frac{7}{18}
$$

### Step 4: Compute the Gini index for 1972

Substitute the result into the Gini index formula:
$$
G_{1972} = 1 - 2 \times \frac{7}{18} = 1 - \frac{14}{18} = \frac{4}{18} = \frac{2}{9} \approx 0.222
$$

### Interpretation of the Results:
- The Gini index for 1962 is approximately $0.267$, indicating a moderate level of income inequality in 1962.
- The Gini index for 1972 is approximately $0.222$, indicating a slight decrease in income inequality between 1962 and 1972.

</details>