Q. 44 Use the Second Fundamental Theor... [FREE SOLUTION]

Chapter 4: Q. 44 (page 400)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t$

Short Answer

Expert verified

Ans: $\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t = e^{x} f (e^{x}) g (e^{x}) + e^{2 x} f^{鈥�} (e^{x}) g (e^{x}) + e^{2 x} f (e^{x}) g^{鈥�} (e^{x})$

Step by step solution

Step 1. Given information.

given expression,

$\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t$

Step 2. The objective is to find the above derivative using the Second Fundamental Theorem of Calculus.

Recollect that if $f$ is continuous on $[a, b]$ , then

$\frac{d}{d x} {鈭�}_{a}^{u (x)} 鈥� f (t) d t = f (u (x)) 鈰� u^{鈥�} (x)$

Therefore,

role="math" $\begin{aligned} \frac{d}{d x} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t = f (e^{x}) g (e^{x}) 鈰� {(e^{x})}^{鈥�} \\ = f (e^{x}) g (e^{x}) 鈰� e^{x} \\ = e^{x} f (e^{x}) g (e^{x}) \end{aligned}$

Step 3. Note that the function exfexgex is continuous on [1,b] and differentiable on (1,b)

So, by the Second Fundamental Theorem of calculus,

$\begin{aligned} \frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t = \frac{d}{d x} [\frac{d}{d x} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t] \\ = \frac{d}{d x} [e^{x} f (e^{x}) g (e^{x})] \\ = \frac{d}{d x} (e^{x}) 鈰� f (e^{x}) g (e^{x}) + e^{x} 鈰� \frac{d}{d x} (f (e^{x})) 鈰� g (e^{x}) + e^{x} f (e^{x}) 鈰� \frac{d}{d x} (g (e^{x})) \\ = e^{x} f (e^{x}) g (e^{x}) + e^{x} 鈰� f^{鈥�} (e^{x}) 鈰� e^{x} 鈰� g (e^{x}) + e^{x} f (e^{x}) 鈰� g^{鈥�} (e^{x}) 鈰� e^{x} \\ = e^{x} f (e^{x}) g (e^{x}) + e^{2 x} f^{鈥�} (e^{x}) g (e^{x}) + e^{2 x} f (e^{x}) g^{鈥�} (e^{x}) \end{aligned}$

Therefore, $\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t = e^{x} f (e^{x}) g (e^{x}) + e^{2 x} f^{鈥�} (e^{x}) g (e^{x}) + e^{2 x} f (e^{x}) g^{鈥�} (e^{x})$

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91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to find the above derivative using the Second Fundamental Theorem of Calculus.

Step 3. Note that the function exfexgex is continuous on [1,b] and differentiable on (1,b)

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91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below. d2dx2鈭�1ex鈥�f(t)g(t)dt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to find the above derivative using the Second Fundamental Theorem of Calculus.

Step 3. Note that the function exfexgex is continuous on [1,b] and differentiable on (1,b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Probability and Statistics

Mechanics Maths

Calculus

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Study anywhere. Anytime. Across all devices.

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{1}^{e^{x}} 鈥� f (t) g (t) d t$