Chapter 2: Q. 37 (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.
$f (x) = \tan x, x = 0$

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 $f (x) = \tan x$

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,

$\begin{aligned} lim_{h \to 0} \frac{f (0 + h) - f (0)}{h} = lim_{h \to 0} \frac{\tan (0 + h) - \tan 0}{h} \\ = lim_{h \to 0} \frac{\tan (h) - 0}{h} \\ = lim_{h \to 0} \frac{\sin (h)}{h \cos (h)} \\ = lim_{h \to 0} \frac{\sin (h)}{h} \cdot lim_{h \to 0} \frac{1}{\cos (h)} \end{aligned} = 1$

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

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

$\begin{aligned} lim_{x \to 0} \frac{f (x) - f (0)}{x - 0} = lim_{x \to 0} \frac{\tan x - \tan 0}{x} \\ = lim_{x \to 0} \frac{\tan x}{h} \\ = lim_{x \to 0} \frac{\sin x}{x \cos x} \\ = lim_{x \to 0} \frac{\sin x}{x} \cdot lim_{x \to 0} \frac{1}{\cos x} \\ = 1 \end{aligned}$

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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.
$f (x) = \tan x, x = 0$

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)=tan⁡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)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Applied Mathematics

Geometry

Pure Maths

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.
$f (x) = \tan x, x = 0$