Chapter 2: Q. 34 (page 184)

Use (a) the h→0 definition of the derivative and then (b) the z→c 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→0 definition,

Therefore,

$\begin{aligned} lim_{h \to 0} \frac{f (0 + h) - f (0)}{h} = lim_{h \to 0} \frac{2 e^{(0 + h)} - 2 e^{0}}{h} \\ = lim_{h \to 0} \frac{e^{h} - 2}{h} \\ = lim_{h \to 0} \frac{2 (1 + \frac{h}{1!} + \frac{h^{2}}{2!} + \dots) - 2}{h} \\ = lim_{h \to 0} \frac{\frac{2 h}{1!} + \frac{2 h^{2}}{2!} + \dots}{h} \\ = lim_{h \to 0} \frac{h (\frac{2}{1!} + \frac{2 h}{2!} + \dots)}{h} \\ = lim_{h \to 0} \frac{2}{1!} + \frac{2 h}{2!} + \dots \end{aligned} = 2$

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

Find the derivate of the function using x→0 definition
Therefore,

$\begin{aligned} lim_{x \to 0} \frac{f (x) - f (0)}{x - 0} = lim_{x \to 0} \frac{2 e^{x} - 2 e^{0}}{x} \\ = lim_{x \to 0} \frac{2 e^{x} - 2}{x} \\ = lim_{x \to 0} \frac{2 (1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \dots) - 2}{x} \\ = lim_{x \to 0} \frac{x (1 + \frac{x}{2!} + \dots)}{x} \\ = lim_{x \to 0} (1 + \frac{x}{2!} + \dots) \\ = 2 \end{aligned}$

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Use (a) the h→0 definition of the derivative and then (b) the z→c 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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Use (a) the h→0 definition of the derivative and then (b) the z→c 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

Statistics

Probability and Statistics

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Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

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