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Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. Predict the future population from the present value and the population growth rate. Use derivatives to calculate marginal cost and revenue in a business situation.
In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.
A linear cost function expresses cost as a linear function of the number of items. In other words, Here, C is the total cost, and x is the number of items. In this context, the slope m is called the marginal cost and b is called the fixed cost.
17 sie 2024 · Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. Predict the future population from the present value and the population growth rate. Use derivatives to calculate marginal cost and revenue in a business situation.
Find the average rate of change of the \(x\)-coordinate of the car with respect to time. Using the formula, we get \[ \text{Rate} = \dfrac{\Delta x}{\Delta t} = \dfrac{14 - 2}{6 - 0} = 2 \text{ m/s}.\ _\square\]
25 kwi 2021 · Let’s do an example where we calculate the marginal cost, revenue, and profit of producing a specific number of goods. Example. A smart phone manufacturer knows that the cost of producing ???x??? phones is given by ???C(x)=6x^2+34x+2,500??? and that the demand function for their phones is ???p=60x???.
We want to find the increase in total cost when increasing production from 5000 items to 5001 items. This is equivalent to finding the average rate of change on the interval \([5000, 5001]\). The total cost at 5000 items is: \(TC(5000) = 2000 + 50\sqrt {5000} \approx 5535.53\)
