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Chapter 4: Q 16. (page 339)

Explain why the sum $\sum_{k = 1}^{100} f (2 + 0.1 (k - 1)) (0.1)$ can't be a left sum for f on $[a, b] = [2, 5]$

Short Answer

Expert verified

Part of the sum denotes $0.25$ and the other part indicates $0.5$

Step by step solution

01

Step 1. Given information

Sum is $\sum_{k = 1}^{100} f (2 + 0.1 (k - 1)) (0.1)$

02

Step 2. Explanation

Consider the sum $\sum_{k = 1}^{100} f (2 + 0.1 (k - 1)) (0.1)$

$\sum_{k = 1}^{100} f (2 + 0.1 (k - 1)) (0.1) = f (2 + 0.1 (1 - 1)) (0.1) + f (2 + 0.1 (2 - 1)) (0.1) + f (2 + 0.1 (3 - 1)) (0.1) + . . .$

Part of it denotes 0.25 and other part denotes 0.5

So, the sum is incorrect.

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Most popular questions from this chapter

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n \to \infty} \sum_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}$

Fill in each of the blanks:

(a) $\int x^{6} d x = + C .$

(b) is an antiderivative of role="math" localid="1648619282178" $x^{6} .$

(c) The derivative of is $x^{6} .$

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

$\int_{0}^{π} (\sin^{2} x + \cos^{2} x) d x$

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [−π, π].

(b) The average value of f(x) = cos x on [0, 2π].

(c) The area of the region between the graphs of f(x) = $\sqrt{4 - x^{2}} and g (x) = - \sqrt{4 - x^{2}} on [- 2, 2] .$

Explain why the formula for the integral of $x^{k}$ does not

apply when $k = - 1 .$ What is the integral of $x^{- 1} ?$

See all solutions

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