Question

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case. $$ \lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}+h\right)-1}{h} $$

Name: Problem 47, Chapter 2: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2023-04-28T23:28:16-08:00
Duration: 1 min 18 s
Channel: Ma. Theresa Alin
Description: Each limit represents the derivative of some function f at some number a. State such an f and a in each case. limh →0 (tan((π)/(4)+h)-1)/(h)

   Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$$
\lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}+h\right)-1}{h}
$$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 2, Problem 47

Instant Answer

Solved by Expert Ma. Theresa Alin

04/28/2023

Step 1

Step 1: We are given the limit $$ \lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}+h\right)-1}{h} $$ We can see that this limit is in the form of the definition of the derivative of a function at a point, which is $$ f'(a) = \lim_{h \rightarrow 0} Show more…

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Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case. $$ \lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}+h\right)-1}{h} $$

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Key Concepts

Derivative

The derivative of a function at a given point is defined as the limit of the difference quotient as the increment approaches zero. It represents the instantaneous rate of change of the function at that point and geometrically corresponds to the slope of the tangent line to the function’s graph.

Limit Definition of the Derivative

The limit definition of the derivative is given by the expression lim??0 [f(a + h) - f(a)]/h. This formulation is fundamental because it precisely captures how a function changes at a specific point and forms the basis for differential calculus.

Trigonometric Function Derivatives

Trigonometric functions, such as the tangent function, have specific rules for differentiation. Understanding how to differentiate trigonometric functions is crucial when the limit definition incorporates these functions, as it enables one to identify the function f and the specific point a where the derivative is being taken.

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