Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 3x2 - 4x; P(1, -1)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 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
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 8 - 2x2; P(0, 8)
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1
Textbook Question
Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.
a. Show that the stones reach their high points at the same time.
Textbook Question
31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)
s(t)= −16t²+100t
Textbook Question
The following table gives the distance f(t) fallen by a smoke jumper seconds after she opens her chute. <IMAGE>
a. Use the forward difference quotient with ℎ = 0.5 to estimate the velocity of the smoke jumper at t=2 seconds.