Let's tackle each part of the question step by step:

1. Compute $R_4$ for $f(x) = 4 - x$ over the interval $[0, 4]$

The right endpoint approximation ($R_n$) uses the right endpoints of the subintervals to approximate the area under a curve. We are asked to compute $R_4$, which means we will divide the interval $[0, 4]$ into $4$ subintervals.

Step 1: Determine the width of each subinterval.

The formula for the width ($\Delta x$) of each subinterval is:

$\Delta x = \frac{b - a}{n}$

Here, $a = 0$, $b = 4$, and $n = 4$. So,

$\Delta x = \frac{4 - 0}{4} = 1$

Step 2: Determine the right endpoints.

The right endpoints for the 4 subintervals are:

$x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4$

Step 3: Evaluate the function at the right endpoints.

We are using the function $f(x) = 4 - x$. Evaluate the function at the right endpoints:

$f(1) = 4 - 1 = 3$ $f(2) = 4 - 2 = 2$ $f(3) = 4 - 3 = 1$ $f(4) = 4 - 4 = 0$

Step 4: Compute $R_4$.

The formula for the right endpoint approximation is:

$R_4 = \Delta x \left( f(x_1) + f(x_2) + f(x_3) + f(x_4) \right)$

Substituting the values:

$R_4 = 1 \left( 3 + 2 + 1 + 0 \right) = 1 \times 6 = 6$

Thus, $R_4 = 6$.

2. Determine the value of $\lim_{n \to \infty} R_n$.

As $n \to \infty$, the right endpoint approximation converges to the exact value of the definite integral of the function over the interval. So, we need to compute the definite integral of $f(x) = 4 - x$ over the interval $[0, 4]$.

The definite integral is:

$\int_0^4 (4 - x) \, dx$

Step 1: Find the antiderivative of $4 - x$.

The antiderivative of $4 - x$ is:

$\int (4 - x) \, dx = 4x - \frac{x^2}{2}$

Step 2: Evaluate the definite integral.

Now, evaluate the antiderivative at the endpoints $4$ and $0$:

$\left[ 4x - \frac{x^2}{2} \right]_0^4 = \left( 4(4) - \frac{4^2}{2} \right) - \left( 4(0) - \frac{0^2}{2} \right)$

Simplifying:

$= \left( 16 - \frac{16}{2} \right) - 0 = 16 - 8 = 8$

Thus, the value of $\lim_{n \to \infty} R_n$ is $8$.

Final Answers:

1. $R_4 = 6$
2. $\lim_{n \to \infty} R_n = 8$

Would you like any further explanation on these steps?

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Relative Questions:

1. How is the left endpoint approximation different from the right endpoint approximation?
2. How does the midpoint approximation compare to the right endpoint approximation?
3. Can the trapezoidal rule provide a more accurate estimate for $f(x) = 4 - x$?
4. What is the formula for the error in a right endpoint approximation?
5. How would $R_4$ change if the function were $f(x) = x^2$?

Tip: To improve approximation accuracy, increasing the number of subintervals $n$ gives a better estimate of the integral.