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Chapter 10: Problem 805

If \(y=\left(x^{2}+7 x+2\right)\left(e^{x}-\log x\right)\) and \((d y / d x)\) \(=\left(x^{2}+A x+B\right)\left|e^{x}-\\{1 / x\\}\right|+\left(e^{x}-\log x\right)(C x+D)\) then \(\mathrm{A}+\mathrm{B}-\mathrm{C}-\mathrm{D}=\) (a) 0 (b) 7 (c) 2 (d) 9

Short Answer

Expert verified
The value of \(A + B - C - D\) is 0. The correct answer is (a) 0.

Step by step solution

01

Identify the functions u and v

Let's define \(u(x) = x^2 + 7x + 2\) and \(v(x) = e^x - \log x\). Then, \(y(x) = u(x)v(x)\).
02

Calculate the derivatives of u and v

We find the derivatives of u and v with respect to x: \(\frac{du}{dx} = 2x + 7\) \(\frac{dv}{dx} = e^x - \frac{1}{x}\)
03

Apply the product rule

Now, let's apply the product rule to find the derivative of y with respect to x: \(\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx} = (x^2 + 7x + 2) (e^x - \frac{1}{x}) + (e^x - \log x)(2x + 7)\)
04

Compare the obtained expression with the given one

Now we have the expression for \(\frac{dy}{dx}\) in the form: \((x^2 + 7x + 2) \left(e^x - \frac{1}{x}\right) + (e^x - \log x)(2x + 7)\) Comparing it to the given expression: \((x^2 + Ax + B) \left| e^x - \frac{1}{x}\right| + (e^x - \log x)(Cx + D)\) We can identify the constants A, B, C, and D as follows: \(A = 7\) \(B = 2\) \(C = 2\) \(D = 7\)
05

Calculate the sum A + B - C - D

Now, let's calculate the sum \(A + B - C - D\): \(A + B - C - D = (7) + (2) - (2) - (7) = 0\) So, the value of \(A + B - C - D = 0\). The correct answer is (a) 0.

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Most popular questions from this chapter

\(\mathrm{f}(\mathrm{x})=|[\mathrm{x}] \mathrm{x}|,-1 \leq \mathrm{x} \leq 2\) then (a) continuous at \(\mathrm{x}=0\) (b) discontinuous at \(\mathrm{x}=0\) (c) diffemtiable at \(\mathrm{x}=0\) (d) continuous at \(\mathrm{x}=2\)

If \(x=\left[\left(e^{2 y}+1\right) /\left(e^{2 y}-1\right)\right]\) then \((d y / d x)=\) (a) \(\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (b) \(\left[1 /\left(\mathrm{x}^{2}-1\right)\right]\) (c) \(\left[1 /\left(1-x^{2}\right)\right]\) (d) \(\left[1 /\left(x^{2}-1\right)\right]\)

In which interval \(\mathrm{f}(\mathrm{x})=\tan ^{-1}(\sin \mathrm{x}+\cos \mathrm{x})\) is strictly increasing? (a) \([\\{(-\pi) / 2\\}, 0]\) (b) \([0,(\pi / 4)]\) (c) \([\\{(-\pi) / 2\\},(\pi / 2)]\) (d) \((-\pi, \pi)\)

If \(\mathrm{x}=\mathrm{a}\left(1-\cos ^{3} \theta\right), \mathrm{y}=\mathrm{a} \sin ^{3} \theta\) and \(\left|\left(\mathrm{d}^{2} \mathrm{y}\right) /\left(\mathrm{dx}^{2}\right)\right|(\pi / 6)\) \(=(\mathrm{A} / \mathrm{a})\) then \(\mathrm{A}=\) (a) \((27 / 32)\) (b) \((32 / 27)\) (c) - (32 / 27) (d) \((27 / 32)\)

If \(\mathrm{x}=\sqrt\left[\left(1-\mathrm{t}^{2}\right) /\left(1+\mathrm{t}^{2}\right)\right] \text { and } \mathrm{y}=\left[\left\\{\sqrt{\left(1+\mathrm{t}^{2}\right)-\sqrt{ \left.\left(1-\mathrm{t}^{2}\right)\right\\}} /}\right.\right.\) (a) \(-1\) (b) (c) - 2 (d) 2
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