Let Rn be the right endpoint approximation to the area under the graph of f with N intervals, 1.) Compute R4 for f=4-x over the interval [0,4]. 2.) determine the value of limit n approaches infinity of Rn

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.

Let Rn be the right endpoint approximation to the area under the graph of f with N intervals, 1.) Compute R4 for f=4-x over the interval [0,4]. 2.) Determine the value of limit n approaches infinity of Rn.

Learn how to compute the right endpoint approximation R4 for the function f(x) = 4 - x over the interval [0,4]. This problem covers key calculus concepts such as Riemann sums, definite integrals, and limits. The solution includes a step-by-step explanation of calculating R4 and determining the limit as n approaches infinity. Suitable for college-level calculus students.

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