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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.2.44
Chapter 3, Problem 3.2.44

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.


g(x) = { 2x − x³ − 1, x ≥ 0
x − (1 / (x + 1)), x < 0

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First, ensure the function is continuous at x = 0. Evaluate the left-hand limit and the right-hand limit of g(x) as x approaches 0, and check if they are equal to g(0).
Calculate the right-hand limit: For x ≥ 0, g(x) = 2x - x³ - 1. Evaluate the limit as x approaches 0 from the right: lim (x -> 0+) (2x - x³ - 1).
Calculate the left-hand limit: For x < 0, g(x) = x - (1 / (x + 1)). Evaluate the limit as x approaches 0 from the left: lim (x -> 0-) (x - (1 / (x + 1))).
Check if the left-hand limit and the right-hand limit are equal. If they are, the function is continuous at x = 0.
To determine differentiability, calculate the derivative of each piece of the function and evaluate the left-hand and right-hand derivatives at x = 0. If these derivatives are equal, the function is differentiable at x = 0.

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Key Concepts

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

Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each applying to a specific interval of the domain. Understanding how these sub-functions interact at their boundaries is crucial, especially when determining continuity and differentiability at points where the function's definition changes.
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Continuity at a Point

For a function to be differentiable at a point, it must first be continuous there. Continuity at a point x = c means that the left-hand limit, right-hand limit, and the function's value at c are all equal. This ensures there are no jumps or breaks in the graph of the function at that point.
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Differentiability and Derivatives

A function is differentiable at a point if it has a defined derivative there, meaning the function's graph has a tangent line at that point. This requires the left-hand and right-hand derivatives to be equal. Differentiability implies continuity, but a continuous function is not necessarily differentiable if it has a sharp corner or cusp at the point.
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Related Practice
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Find all points on the curve y = tan x, −π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent lines together, labeling each with its equation.

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In Exercises 1–12, find the first and second derivatives.


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Derivative of multiples Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer.

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Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.


f(x) = { 2x − 1, x ≥ 0

x² + 2x + 7, x < 0

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