Chapter 4: Q2P (page 236)

If $s = \int_{u}^{v} \frac{1 - e^{t}}{t} dt,$ find $\frac{\partial s}{\partial v} and \frac{\partial s}{\partial u}$ and also their limits as $u a n d v$ tend to zero.

Short Answer

Expert verified

The values of $\frac{\partial s}{\partial v} is \frac{1 - e^{v}}{v}, \frac{\partial s}{\partial u} is - (\frac{1 - e^{u}}{u}), \lim_{v \to 0} \frac{1 - e^{v}}{v} is - 1 and \lim_{u \to 0} \frac{e^{u} - 1}{u} is 1 .$

Step by step solution

Given Information

Given the value of $s = \int_{u}^{v} \frac{1 - e^{t}}{t} d t$ .

Finding ∂s∂vand∂s∂u

$If I = \int_{u}^{v} f (t) dt then, \frac{\partial I}{\partial v} = f (v) and \frac{\partial I}{\partial u} = - f (u) .$

So, the partial derivatives

$\frac{\partial s}{\partial v} = f (v) = \frac{1 - e^{v}}{v}$

$\frac{\partial s}{\partial u} = - f (u) = - (\frac{1 - e^{u}}{u})$

As the limits $u a n d v$ tends to zero, then

$\underset{v \to 0}{l i m} \frac{1 - e^{v}}{v} = - 1 \underset{u \to 0}{l i m} \frac{e^{u} - 1}{u} = 1$

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If $s = \int_{u}^{v} \frac{1 - e^{t}}{t} dt,$ find $\frac{\partial s}{\partial v} and \frac{\partial s}{\partial u}$ and also their limits as $u a n d v$ tend to zero.

Short Answer

Step by step solution

Given Information

Finding ∂s∂vand∂s∂u

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Finding ∂s∂vand∂s∂u

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If $s = \int_{u}^{v} \frac{1 - e^{t}}{t} dt,$ find $\frac{\partial s}{\partial v} and \frac{\partial s}{\partial u}$ and also their limits as $u a n d v$ tend to zero.