Chapter 1: Q26P (page 42)

Find the values of several derivatives of $e^{- 1 / t^{2}}$ at t = 0. Hint:Calculate a few derivatives (as functions of t); then make the substitution $x = 1 / t^{2}$ , and use the result of Problem 24(f) or 25.

Short Answer

Expert verified

The solution is $\underset{x \to \infty}{l i m} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = 0$ . t

Step by step solution

Given information

A function $e^{- 1 / t^{2}}$ is given.

Definition of Maclaurin Series.

The definition of the Maclaurin series used in this solution is given as:

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . + \frac{x^{n}}{n!} + \frac{x^{n + 1} n}{(n + 1)} + . . .$

Perform differentiation several times and make appropriate substitutions

Assume $f (t) = - \frac{1}{t^{2}}$ .

This implies $e^{- 1 / t^{2}} = e^{f (t)}$ .

$\frac{d}{d t} (e^{f (t)}) = f' (t) e^{f (t)} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = \frac{d}{d t} f' (t) e^{f (t)} = (f' (t)) e^{f (t)} + f " (t) e^{f (t)} \frac{d^{3}}{d t^{3}} (e^{f (t)}) = \frac{d}{d t} ((f' (t)) e^{f (t)} + f " (t) e^{f (t)}) = 2 f' (t) f " (t) e^{f (t)} + {(f' (t))}^{3} e^{f (t)} + f''' (t) e^{f (t)} + f' (t) f'' (t) e^{f (t)}$

Note the values of derivatives of all orders.

Derivatives of all the orders are calculated.

$f (t) = - \frac{1}{t^{2}} f' (t) = \frac{2}{t^{3}} f " (t) = - \frac{6}{t^{4}} f''' (t) = \frac{24}{t^{5}}$

Substitute the values of derivatives in the equation.

Substitute the derivates back in the equation obtained.

$\frac{d}{d t} (e^{f (t)}) = \frac{2}{t^{3}} e^{- 1 / t^{2}} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = \frac{4}{t^{6}} e^{- 1 / t^{2}} - \frac{6}{t^{4}} e^{- 1 / t^{2}} \frac{d^{3}}{d t^{3}} (e^{f (t)}) = 2 f' (t) f " (t) e^{f (t)} + {(f' (t))}^{3} e^{f (t)} + f''' (t) e^{f (t)} + f' (t) f'' (t) e^{f (t)} \frac{d^{3}}{d t^{3}} (e^{f (t)}) = \frac{24 e^{(^{- 1 / t^{2}})} t^{4} - 36 e^{(^{- 1 / t^{2}})} t^{2} + 8 e^{(^{- 1 / t^{2}})}}{t^{9}}$

It is known that $\underset{x \to \infty}{l i m} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = 0$ .

Continue evaluation.

$\underset{x \to \infty}{l i m} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = \underset{x \to \infty}{l i m} 4 x^{3} e^{x} - \underset{x \to \infty}{l i m} 6 x^{2} e^{- x} \underset{x \to \infty}{l i m} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = 0 - 0 = 0$

Thus, the solution is $\underset{x \to \infty}{l i m} \frac{d^{2}}{d t^{2}} (e^{f (t)}) = 0$ .

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Find the values of several derivatives of $e^{- 1 / t^{2}}$ at t = 0. Hint:Calculate a few derivatives (as functions of t); then make the substitution $x = 1 / t^{2}$ , and use the result of Problem 24(f) or 25.

Short Answer

Step by step solution

Given information

Definition of Maclaurin Series.

Perform differentiation several times and make appropriate substitutions

Note the values of derivatives of all orders.

Substitute the values of derivatives in the equation.

One App. One Place for Learning.

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Find the values of several derivatives ofe-1/t2at t = 0. Hint:Calculate a few derivatives (as functions of t); then make the substitutionx=1/t2, and use the result of Problem 24(f) or 25.

Short Answer

Step by step solution

Given information

Definition of Maclaurin Series.

Perform differentiation several times and make appropriate substitutions

Note the values of derivatives of all orders.

Substitute the values of derivatives in the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Electric Charge Field and Potential

Classical Mechanics

Work Energy and Power

Thermodynamics

Nuclear Physics

Study anywhere. Anytime. Across all devices.

Find the values of several derivatives of $e^{- 1 / t^{2}}$ at t = 0. Hint:Calculate a few derivatives (as functions of t); then make the substitution $x = 1 / t^{2}$ , and use the result of Problem 24(f) or 25.