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Chapter 4: Q. 42 (page 340)

In Exercises 39鈥�44, write out the sigma notation for the Riemann sum described in such a way that the only letter which appears in the general term of the sum is k. Don鈥檛 calculate the value of the sum; just write it down in sigma notation.
$f (x) = \ln (x), [a, b] = [2, 5], l e f t s u m, n = 100$

Short Answer

Expert verified

The left sum is $= 0.03 {鈭�}_{k = 1}^{100} \ln (2 + 0.03 (k - 1))$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \ln (x), [a, b] = [2, 5], l e f t s u m, n = 100$

02

Step 2. Finding ∆x and xk

$鈭� x = \frac{b - a}{n} = \frac{5 - 2}{100} = \frac{3}{100} = 0.03$

$x_{k} = a + k 鈭� x = 2 + 0.03 k x_{k - 1} = a + (k - 1) 鈭� x = 2 + 0.03 (k - 1)$

03

Step 3. Finding Reimann Sum

We have $f (x) = \ln (x)$ then $f (x_{k - 1}) = \ln (x_{k - 1})$

Now, ${鈭�}_{k = 1}^{100} f (x_{k - 1}) 鈭� x = {鈭�}_{k = 1}^{100} \ln (2 + 0.03 (k - 1)) (0.03) = 0.03 {鈭�}_{k = 1}^{100} \ln (2 + 0.03 (k - 1))$

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Most popular questions from this chapter

Explain why it would be difficult to write the sum $\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{8} + \frac{1}{11} + \frac{1}{12} + \frac{1}{13}$ in sigma notation.

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{鈭� 3}^{2} 鈥� \frac{2 x}{x^{2} + 5} d x$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{鈭� 1}^{1} 鈥� \frac{2^{x}}{4^{x}} d x$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{0}^{1} 鈥� \frac{x}{1 + x^{2}} d x$

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

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