Textbook Question
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
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 3.3.66
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.
Verified step by step guidance1
Identify the function A(q) given in the problem: A(q) = (km / q) + cm + (hq / 2). This is the function for which we need to find the first and second derivatives with respect to q.
To find the first derivative, dA/dq, apply the power rule and the constant rule. The term (km / q) can be rewritten as km * q^(-1). The derivative of cm with respect to q is 0 since it is a constant. The derivative of (hq / 2) is h/2 since it is a linear term in q.
Calculate the derivative of the first term: d/dq [km * q^(-1)] = -km * q^(-2).
Combine the derivatives to find dA/dq: dA/dq = -km * q^(-2) + 0 + h/2.
To find the second derivative, d²A/dq², differentiate dA/dq with respect to q. The derivative of -km * q^(-2) is 2km * q^(-3), and the derivative of h/2 is 0 since it is a constant. Thus, d²A/dq² = 2km * q^(-3).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input changes. In this context, finding dA/dq involves applying differentiation rules to determine how the average weekly cost A(q) changes with respect to the order quantity q. This helps in understanding the rate of change of cost with varying order sizes.
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05:44
Derivatives
Second Derivative
The second derivative, denoted as d²A/dq², provides information about the curvature or concavity of the function A(q). It indicates whether the function is concave up or down, which is crucial for identifying local minima or maxima. In inventory management, this helps determine the optimal order quantity that minimizes costs.
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06:02The Second Derivative Test: Finding Local Extrema
Optimization
Optimization involves finding the best solution from a set of possible choices. In this problem, it refers to determining the order quantity q that minimizes the average weekly cost A(q). By analyzing the first and second derivatives, one can identify critical points and assess their nature to find the optimal order size that balances ordering and holding costs.
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10:13Intro to Applied Optimization: Maximizing Area
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 (x² + 2x)
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