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Chapter 4: Q. 70 (page 401)

Prove in your own words the last part of Theorem 4.37: If we define $\ln (x) = {鈭�}_{1}^{鈭�} \frac{1}{t} d t$ for $x > 0$ , then $\ln (x)$ is one-to-one on $(0, 鈭�)$ .

Short Answer

Expert verified

We have proved the theorem.

Step by step solution

Step 1. Given Information.

The objective is to prove the last part of Theorem 4.37.

Step 2. The proof.

The following graph shows that the signed area under the graph of $f = \frac{1}{t}$ and x-axis is,

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Most popular questions from this chapter

Prove part (b) of theorem 4.4 in the case when n is even: if n is a positive even integer, then ${鈭�}_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

$f (x) = \sin (x), [a, b] = [0, 蟺]$ , n = 3 with

a) Trapezoid sim b) Upper sum

Fill in each of the blanks:

(a) $鈭� x^{6} d x = + C .$

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

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{0}^{4} ((2 x - 3)^{2} + 5) d x$

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ , and ${鈭�}_{3}^{6} g (x) d x = 3$ , then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{- 2}^{- 2} x (f (x) + 3)^{2} d x$

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Prove in your own words the last part of Theorem 4.37: If we define lnx=鈭�1鈭�1tdtfor x>0, then lnxis one-to-one on 0,鈭�.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The proof.

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Prove in your own words the last part of Theorem 4.37: If we define $\ln (x) = {鈭�}_{1}^{鈭�} \frac{1}{t} d t$ for $x > 0$ , then $\ln (x)$ is one-to-one on $(0, 鈭�)$ .