Textbook Question
Consider the function
f(x) = { x² cos(2/x), x ≠ 0
0, x = 0
b. Determine f' for x ≠ 0.
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Ch. 3 - Derivatives
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.2.60b
b. Show that
f(x) = { x² sin(1/x), x ≠ 0
0, x = 0
is differentiable at x = 0 and find f′(0).
Verified step by step guidance1
First, understand the definition of differentiability at a point. A function f(x) is differentiable at x = a if the limit of the difference quotient exists: lim(x -> a) [f(x) - f(a)] / (x - a).
For the given function f(x), we need to check differentiability at x = 0. The function is defined as f(x) = x² sin(1/x) for x ≠ 0 and f(x) = 0 for x = 0.
Calculate the difference quotient for f(x) at x = 0: [f(x) - f(0)] / (x - 0) = [x² sin(1/x) - 0] / x = x sin(1/x).
Evaluate the limit of x sin(1/x) as x approaches 0. Use the fact that |sin(1/x)| ≤ 1, so |x sin(1/x)| ≤ |x|. As x approaches 0, |x| approaches 0, thus the limit of x sin(1/x) is 0.
Since the limit exists and is equal to 0, f(x) is differentiable at x = 0. Therefore, f'(0) = 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiability
A function is differentiable at a point if it has a defined derivative at that point. This means the function must be continuous at the point, and the limit of the difference quotient must exist as the point is approached. For f(x) to be differentiable at x = 0, we need to check these conditions specifically at x = 0.
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Limit Definition of Derivative
The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero: f'(a) = lim (h -> 0) [(f(a+h) - f(a))/h]. For f(x) at x = 0, this involves evaluating the limit of [x² sin(1/x)]/x as x approaches 0, which simplifies to finding the behavior of x sin(1/x) as x approaches 0.
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Squeeze Theorem
The Squeeze Theorem is used to find the limit of a function trapped between two other functions that have the same limit at a point. If -1 ≤ sin(1/x) ≤ 1, then -x ≤ x sin(1/x) ≤ x. As x approaches 0, both -x and x approach 0, allowing us to conclude that x sin(1/x) also approaches 0, which is crucial for proving differentiability at x = 0.
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Related Practice
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Find y⁽⁴⁾ = d⁴y/dx⁴ if:
b. y = 9 cos x
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______
S = πr √ r² + h².
b. How is dS/dt related to dh/dt if r is constant?
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Generalizing the Product Rule The Derivative Product Rule gives the formula
d/dx (uv) = u (dv/dx) + (du/dx) v
for the derivative of the product uv of two differentiable functions of x.
b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?
Textbook Question
Tolerance
b. About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?