Q. 53 Use Definition 4.8 to prove that... [FREE SOLUTION]

Chapter 4: Q. 53 (page 341)

Use Definition 4.8 to prove that if a function f is positive and concave up on $[a, b]$ , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Short Answer

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It is proved that if a function f is positive and concave up on $[a, b]$ , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Step by step solution

Step 1. Given Information

We are given thatif a function f is positive and concave up on $[a, b]$ , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Step 2. Proving the statement

The right sum defined for n rectangles on $[a, b]$ is ${鈭�}_{k = 1}^{n} f (x_{k}) 螖 x$ .

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

The average of left sum and right sum is,

$\frac{{鈭�}_{k = 1}^{n} f (x_{k - 1}) 螖 x + {鈭�}_{k = 1}^{n} f (x_{k}) 螖 x}{2} = {鈭�}_{k = 1}^{n} \frac{f (x_{k - 1}) + f (x_{k})}{2} 螖 x$

The function is concave up and is positive so the average of the left sum and the right sum will be over-approximation.

The trapezoid sum for n rectangles on $[a, b]$ is ${鈭�}_{k = 1}^{n} \frac{f (x_{k - 1}) + f (x_{k})}{2} 螖 x$ .

Hence, the trapezoid sum approximation will also be an over approximation.

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91影视

Use Definition 4.8 to prove that if a function f is positive and concave up on $[a, b]$ , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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91影视

Use Definition 4.8 to prove that if a function f is positive and concave up on [a,b], then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Probability and Statistics

Geometry

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use Definition 4.8 to prove that if a function f is positive and concave up on $[a, b]$ , then the trapezoid sum with n trapezoids is an always an over-approximation for the actual area.