Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.75a

Chapter 7, Problem 7.3.75a

Shallow-water velocity equation
a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.

Verified step by step guidance

Recall that the linear approximation (or linearization) of a function \( f(x) \) at a point \( a \) is given by the formula: \[ L(x) = f(a) + f'(a)(x - a) \]

Identify the function and the point of approximation: here, \( f(x) = \tanh x \) and \( a = 0 \).

Calculate \( f(a) = \tanh 0 \). Since \( \tanh 0 = 0 \), this simplifies the linear approximation formula.

Find the derivative of \( f(x) = \tanh x \). Recall that \( \frac{d}{dx} \tanh x = \operatorname{sech}^2 x \).

Evaluate the derivative at \( a = 0 \): \( f'(0) = \operatorname{sech}^2 0 \). Since \( \operatorname{sech} 0 = 1 \), this gives \( f'(0) = 1 \). Substitute these values into the linear approximation formula to get \( L(x) = 0 + 1 \cdot (x - 0) = x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Approximation (Linearization)

Linear approximation involves approximating a function near a point using the tangent line at that point. It is given by L(x) = f(a) + f'(a)(x - a), which simplifies calculations by replacing complex functions with linear ones close to a. This method is useful for estimating function values near the point a.

Derivative of tanh(x)

The derivative of tanh(x) is sech²(x), which measures the rate of change of tanh(x) at any point x. Evaluating this derivative at a specific point, such as x = 0, is essential for finding the slope of the tangent line used in linear approximation.

Properties of tanh(x) at x = 0

At x = 0, tanh(0) = 0, and its derivative f'(0) = sech²(0) = 1. These values simplify the linear approximation formula to L(x) = 0 + 1*(x - 0) = x, confirming that near zero, tanh(x) behaves approximately like the identity function.

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Related Practice

Textbook Question

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.

f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

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Textbook Question

Zero net area Consider the function f(x) = (1 − x)/x

a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

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Textbook Question

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.

a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

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Textbook Question

A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.

a. Show that a satisfies the equation cosh 50/a − 1 = 10/a.

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Textbook Question

Wave velocity Use Exercise 73 to do the following calculations.

a. Find the velocity of a wave where λ = 50 m and d = 20 m.

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Textbook Question

Terminal velocity Refer to Exercises 95 and 96.

a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).

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