Chapter 4: Q. 57 (page 400)

For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.
$\ln e$

Short Answer

Expert verified

Using left and right sums with four rectangles is $0.8999 \leq \ln e \leq 1.149$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\ln e$

Step 2. Use Riemann sums for left sum definition.

Consider the function,

$f (x) = \frac{1}{t}$

The left-sum defined for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} f (x_{k - 1}) △ x$ .

Where, $△ x = \frac{b - a}{n}, x_{k} = a + k △ x$

Now $△ x$ ,

$= \frac{2.8 - 0}{14} = \frac{28}{140} = 0.2$

Step 3. Find the left sum.

Consider the given question,

$x_{k} = 0 + k (0.2) = 0.2 k$

In the left sum, $x_{k - 1}$ is the leftmost point in the interval $[x_{k - 1}, x_{k}]$ .

So, $x_{k - 1} = (k - 1) 0.2 = 0.2 k - 0.2$

The left sum is given below,

$\sum_{k = 0}^{2.8} \ln (0.2 k - 0.2) (0.2) \approx 0.8999$

Step 5. Find the right sum.

The right-sum defined for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} f (x) △ x$ .

The right sum is given below,

$= \sum_{k = 0}^{2.8} \ln (0.2 k - 0.2) (0.2) \approx 1.149$

Therefore, the value of $\ln e$ is $0.8999 \leq \ln e \leq 1.149$ .

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For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.
$\ln e$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use Riemann sums for left sum definition.

Step 3. Find the left sum.

Step 5. Find the right sum.

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For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.lne

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use Riemann sums for left sum definition.

Step 3. Find the left sum.

Step 5. Find the right sum.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.
$\ln e$