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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 3.2.41
Chapter 3, Problem 3.2.41

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

Verified step by step guidance
1
First, check the continuity of the function at x = 0. Evaluate the left-hand limit and the right-hand limit of f(x) as x approaches 0. For x ≥ 0, f(x) = 2x - 1, so the right-hand limit as x approaches 0 is f(0) = 2(0) - 1 = -1. For x < 0, f(x) = x² + 2x + 7, so the left-hand limit as x approaches 0 is f(0) = 0² + 2(0) + 7 = 7.
Since the left-hand limit (7) and the right-hand limit (-1) are not equal, the function is not continuous at x = 0. A function must be continuous at a point to be differentiable there.
Even though the function is not continuous at x = 0, let's check the derivatives from both sides for completeness. For x ≥ 0, the derivative of f(x) = 2x - 1 is f'(x) = 2.
For x < 0, the derivative of f(x) = x² + 2x + 7 is f'(x) = 2x + 2. Evaluate this derivative as x approaches 0 from the left, which gives f'(0) = 2(0) + 2 = 2.
Since the function is not continuous at x = 0, it cannot be differentiable at that point, even though the derivatives from both sides are equal. Differentiability requires continuity at the point in question.

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct formulas: one for x greater than or equal to 0 and another for x less than 0. Understanding how to evaluate and analyze these functions at specific points is crucial for determining properties like continuity and differentiability.
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Piecewise Functions

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which means the function must be continuous and have a consistent slope from both sides. To check differentiability at x = 0, we need to ensure that the left-hand derivative (from x < 0) and the right-hand derivative (from x ≥ 0) are equal. If they differ, the function is not differentiable at that point.
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Continuity

Continuity at a point means that the function's value at that point matches the limit of the function as it approaches that point from both sides. For f(x) to be differentiable at x = 0, it must first be continuous there. This involves checking that the values of the two pieces of the function agree at x = 0, ensuring no jumps or breaks in the graph.
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Intro to Continuity
Related Practice
Textbook Question

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.


y = (1 / x²), (−1, 1)

Textbook Question

Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = cos u, u = −x/3

Textbook Question

Find the derivatives of the functions in Exercises 1–42.


𝔂 = x⁻¹/² sec (2x)²

Textbook Question

Second Derivatives


Find y'' in Exercises 59–64.


y = x(2x + 1)⁴

Textbook Question

Derivatives


In Exercises 1–18, find dy/dx.


y = (sec x + tan x)(sec x − tan x)

Textbook Question

Find the derivatives of the functions in Exercises 1–42.


𝔂 = (x + 1)² (x² + 2x)