Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. a. arctan 1
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.153a
153. The linearization of 2ˣ
a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.
Verified step by step guidance1
Identify the function to be linearized: \(f(x) = 2^{x}\), and the point of linearization: \(x = 0\).
Recall that the linearization of a function \(f(x)\) at \(x = a\) is given by the formula: \(L(x) = f(a) + f'(a)(x - a)\).
Calculate \(f(0)\) by substituting \(x = 0\) into the function: \(f(0) = 2^{0}\).
Find the derivative of the function: \(f'(x) = 2^{x} \ln(2)\), then evaluate it at \(x = 0\): \(f'(0) = 2^{0} \ln(2)\).
Write the linearization formula using the values found: \(L(x) = f(0) + f'(0)(x - 0)\), then round the coefficients \(f(0)\) and \(f'(0)\) to two decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linearization of a Function
Linearization approximates a function near a point using the tangent line at that point. It is given by L(x) = f(a) + f'(a)(x - a), where a is the point of approximation. This simplifies complex functions to linear ones for easier calculations near a.
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Linearization
Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x is f'(x) = a^x ln(a). This rule helps find the slope of the tangent line at any point, which is essential for constructing the linearization.
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Evaluating Functions and Derivatives at a Point
To find the linearization at x = 0, you must compute both f(0) and f'(0). These values provide the y-intercept and slope of the tangent line, respectively, which are then used in the linear approximation formula.
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Related Practice
Textbook Question
In Exercises 1–4, solve for t.
1. a. e^(-0.3t) = 27
Textbook Question
75. a. Find the open intervals on which the function is increasing and decreasing.
g(x) = x(ln x)²
Textbook Question
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
Textbook Question
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
139. a. y=arctan(tan x)