Textbook Question
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.
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 3.2.44
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
Verified step by step guidance1
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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05:36
Piecewise Functions
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.
Recommended video:
05:34Intro to Continuity
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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05:53Finding Differentials
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
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
Textbook Question
The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is
A(q) = (km / q) + cm + (hq / 2),
where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).
Find dA/dq and d²A/dq².
Textbook Question
Slopes and Tangent Lines
In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 2 tan² x - sec² x