교과서 질문
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
3장, 문제 3.6.29a
Consider the following cost functions.
a. Find the average cost and marginal cost functions.
C(x) = 1000+0.1x, 0≤x≤5000, a=2000
검증된 단계별 안내1
To find the average cost function, divide the total cost function C(x) by the number of units x. The average cost function A(x) is given by A(x) = C(x) / x.
Substitute the given cost function C(x) = 1000 + 0.1x into the average cost formula: A(x) = (1000 + 0.1x) / x.
Simplify the expression for A(x) to get A(x) = 1000/x + 0.1.
To find the marginal cost function, take the derivative of the total cost function C(x) with respect to x. The marginal cost function MC(x) is given by MC(x) = dC(x)/dx.
Differentiate C(x) = 1000 + 0.1x with respect to x. Since the derivative of a constant is 0 and the derivative of 0.1x is 0.1, the marginal cost function MC(x) = 0.1.
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주요 개념
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Average Cost Function
The average cost function, denoted as AC(x), represents the total cost C(x) divided by the quantity produced x. It provides insight into the cost per unit of production, helping businesses determine pricing strategies. For the given cost function C(x) = 1000 + 0.1x, the average cost can be calculated as AC(x) = C(x)/x, which simplifies to (1000/x) + 0.1.
추천 영상:
가이드 코스
06:37
Average Value of a Function
Marginal Cost Function
The marginal cost function, denoted as MC(x), measures the additional cost incurred by producing one more unit of output. It is derived from the derivative of the total cost function C(x) with respect to x. For the cost function C(x) = 1000 + 0.1x, the marginal cost is found by calculating MC(x) = dC/dx, which results in a constant value of 0.1, indicating that each additional unit costs 0.1.
추천 영상:
가이드 코스
06:21Properties of Functions
Cost Function
A cost function describes the relationship between the quantity of output produced and the total cost incurred in production. It typically includes fixed costs, which do not change with output, and variable costs, which do. In the provided function C(x) = 1000 + 0.1x, the fixed cost is 1000, while the variable cost is represented by the term 0.1x, indicating that costs increase linearly with production.
추천 영상:
가이드 코스
06:21Properties of Functions
관련 실천
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Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
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y = 2^x x
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a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
g(s) = 4s³ - 8s² +4s / 4s
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a. Use implicit differentiation to find dy/dx.
(x+y)^2/3=y; (4, 4)
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Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>
a. At which points is the slope of the curve negative?
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