Q. 23 In Exercises 21鈥�26, find the d... [FREE SOLUTION]

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Chapter 12: Q. 23 (page 976)

In Exercises 21鈥�26, find the discriminant of the given function.
$f (s, t) = s^{3} e^{t / 2}$

Short Answer

Expert verified

The answer is $- \frac{3}{4} s^{4} e^{t}$ .

Step by step solution

Step 1. Given Information.

The function is $f (s, t) = s^{3} e^{t / 2}$ .

Step 2. Explanation.

The discriminant is calculated by formula,

$d e t (H_{g}) = \frac{{鈭�}^{2} g}{鈭� s^{2}} 路 \frac{{鈭�}^{2} g}{鈭� t^{2}} - {(\frac{{鈭�}^{2} g}{鈭� s 鈭� t})}^{2}$ .

Calculate $\frac{鈭� g}{鈭� s}$ , $\frac{鈭� g}{鈭� t}$ , $\frac{{鈭�}^{2} g}{鈭� s^{2}}$ , $\frac{{鈭�}^{2} g}{鈭� t^{2}}$ and $\frac{{鈭�}^{2} g}{鈭� s 鈭� t}$ .

$\frac{鈭� g}{鈭� s} = 3 s^{2} e^{\frac{t}{2}}$ , $\frac{{鈭�}^{2} g}{鈭� s^{2}} = 6 s e^{\frac{t}{2}}$ .

$\frac{鈭� g}{鈭� t} = \frac{1}{2} s^{3} e^{\frac{t}{2}}$ , $\frac{{鈭�}^{2} g}{鈭� t^{2}} = \frac{1}{4} s^{3} e^{\frac{t}{2}}$ .

role="math" localid="1649922485848" $\frac{{鈭�}^{2} g}{鈭� s 鈭� t} = \frac{3}{2} s^{2} e^{\frac{t}{2}}$ .

Step 3. Calculation.

Calculate $d e t (H_{g}) = \frac{{鈭�}^{2} g}{鈭� s^{2}} 路 \frac{{鈭�}^{2} g}{鈭� t^{2}} - {(\frac{{鈭�}^{2} g}{鈭� s 鈭� t})}^{2}$ .

role="math" localid="1649922348949" $d e t (H_{g}) = \frac{{鈭�}^{2} g}{鈭� s^{2}} 路 \frac{{鈭�}^{2} g}{鈭� t^{2}} - {(\frac{{鈭�}^{2} g}{鈭� s 鈭� t})}^{2} = (6 s e^{\frac{t}{2}}) (\frac{1}{4} s^{3} e^{\frac{t}{2}}) - {(\frac{3}{2} s^{2} e^{\frac{t}{2}})}^{2} = \frac{6}{4} s^{4} e^{t} - \frac{9}{4} s^{4} e^{t} = - \frac{3}{4} s^{4} e^{t}$

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91影视

In Exercises 21鈥�26, find the discriminant of the given function.
$f (s, t) = s^{3} e^{t / 2}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

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91影视

In Exercises 21鈥�26, find the discriminant of the given function.f(s,t)=s3et/2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 21鈥�26, find the discriminant of the given function.
$f (s, t) = s^{3} e^{t / 2}$