Chapter 2: Q. 33 (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.
$f (x) = e^{x}, x = 0$

Short Answer

Expert verified

(a) $f^{'} (c) = 1$

(b) $f^{'} (c) = 1$

Step by step solution

Part (a) Step 1. Given information.

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

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,

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

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

Find the derivate of the function using x→0 definition

Therefore,

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

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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.
$f (x) = e^{x}, x = 0$

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)=ex,x=0

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

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

Discrete Mathematics

Mechanics 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.
$f (x) = e^{x}, x = 0$