Q. 34 Use (a) the h鈫�0 definition of ... [FREE SOLUTION]

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Chapter 2: Q. 34 (page 184)

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = 2 e^{x}$

Short Answer

Expert verified

(a) $f^{'} (c) = 2$

(b) $f^{'} (c) = 2$

Step by step solution

Part (a) Step 1. Given information.

Given function is $f (x) = 2 e^{x}$

We have to find $f^{'} (c)$

Part (a) Step 2. Find the f'(c)

We have to find the derivative of the function using h鈫�0 definition,

Therefore,

$\begin{aligned} lim_{h 鈫� 0} 鈥� \frac{f (0 + h) 鈭� f (0)}{h} = lim_{h 鈫� 0} 鈥� \frac{2 e^{(0 + h)} 鈭� 2 e^{0}}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{e^{h} 鈭� 2}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{2 (1 + \frac{h}{1!} + \frac{h^{2}}{2!} + 鈰�) 鈭� 2}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{\frac{2 h}{1!} + \frac{2 h^{2}}{2!} + 鈰�}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{h (\frac{2}{1!} + \frac{2 h}{2!} + 鈰�)}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{2}{1!} + \frac{2 h}{2!} + 鈰� \end{aligned} = 2$

Part (b) Step 1. Find f'(c)

Find the derivate of the function using x鈫�0 definition
Therefore,

$\begin{aligned} lim_{x 鈫� 0} 鈥� \frac{f (x) 鈭� f (0)}{x 鈭� 0} = lim_{x 鈫� 0} 鈥� \frac{2 e^{x} 鈭� 2 e^{0}}{x} \\ = lim_{x 鈫� 0} 鈥� \frac{2 e^{x} 鈭� 2}{x} \\ = lim_{x 鈫� 0} 鈥� \frac{2 (1 + \frac{x}{1!} + \frac{x^{2}}{2!} + 鈰�) 鈭� 2}{x} \\ = lim_{x 鈫� 0} 鈥� \frac{x (1 + \frac{x}{2!} + 鈰�)}{x} \\ = lim_{x 鈫� 0} 鈥� (1 + \frac{x}{2!} + 鈰�) \\ = 2 \end{aligned}$

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91影视

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = 2 e^{x}$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the f'(c)

Part (b) Step 1. Find f'(c)

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Calculus

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91影视

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.f(x)=2ex

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the f'(c)

Part (b) Step 1. Find f'(c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = 2 e^{x}$