Chapter 4: Q2P (page 236)

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

Short Answer

Expert verified

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

Step by step solution

Given Information

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

Finding ∂s∂vand∂s∂u

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

So, the partial derivatives

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

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

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

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

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91影视

If $s = {鈭�}_{u}^{v} \frac{1 - e^{t}}{t} dt,$ find $\frac{鈭� s}{鈭� v} and \frac{鈭� s}{鈭� u}$ 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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91影视

If s=鈭�uv1-ettdt,find鈭�s鈭�vand鈭�s鈭�u and also their limits as uandvtend to zero.

Short Answer

Step by step solution

Given Information

Finding ∂s∂vand∂s∂u

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Geometrical and Physical Optics

Circular Motion and Gravitation

Modern Physics

Energy Physics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

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