A company purchases a new machine for which the rate of...

A company purchases a new machine for which the rate of depreciation can be modeled by $\frac{d V}{d t}=10,000(t-6), \quad 0 \leq t \leq 5$ where $V$ is the value of the machine after $t$ years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years.

A company purchases a new machine for which the rate of depreciation can be modeled by
$\frac{d V}{d t}=10,000(t-6), \quad 0 \leq t \leq 5$
where $V$ is the value of the machine after $t$ years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years.

Key Concepts

Differential Equations

Differential equations relate a function to its rate of change, allowing one to model how a quantity evolves over time. In many applications, including financial depreciation, the derivative represents an instantaneous rate of change, and solving the differential equation helps determine the cumulative change over a given period.

Net Change Theorem

The net change theorem states that the definite integral of a rate of change over a time interval gives the total change in the quantity over that interval. This concept is crucial when using integration to determine the accumulated effect of continuously changing systems, as seen in computing total depreciation or other cumulative processes.

Definite Integral

A definite integral computes the accumulation of a quantity over an interval by summing the values of a function, often representing rates, across that interval. It is the mathematical tool that transforms a continuously varying rate of change into a total change, providing a precise measure of overall loss, gain, or other accumulations over time.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing that integration can be reversed by differentiation. This theorem is key in evaluating definite integrals, as it allows one to compute the total change in a quantity by finding an antiderivative of the rate function and calculating the difference at the endpoints of the interval.

Transcript

00:03 Here we have a company that purchases a new machine for which the rate of depreciation can be modeled by this equation.

00:12 Dvdt equals 10 ,000 times t minus 6 from 0 to 5.

00:16 V is the value of the machine, maybe in dollars, after t years.

00:22 What we want to do is set up and evaluate a definite interval that gives the total loss of value of the machine for the first three years.

00:29 Now, if you notice, dvdt from 0 to 5, if you plug in for values of t, it's going to give you a negative number.

00:38 That makes sense.

00:39 That means that the value is decreasing.

00:42 So if we integrate from 0 to 3, we will get the total amount of value units, dollars, that we have lost.

00:53 And that's what it's asking for.

00:54 It's not asking for the value of the machine to figure out the actual value of the machine.

00:59 We would need to know what the initial value was of the machine.

01:03 So all we need to do is set it up.

01:05 So we'll say the loss of value, loss of value, and that loss is already interpreting the sign.

01:14 That's going to end up being negative.

01:16 The loss of value, okay? i'm not going to put equals here because i want to be careful.

01:22 If i put equals, then my negative number, you know, a double negative means positive.

01:29 So i'm just going to set this up here as the integral of 1 ,000 or 10 ,000 it is, t minus 6d, and over the first three years, that's from 0 to 3.

01:43 Now, if this is a calculator question, you could just plug and chug and go for it.

01:48 I'm going to assume that it's not.

01:50 So to do this, i'm going to go ahead and pull the 10 ,000 out front just to make it easier.

01:57 And we'll put a colon up here, sorry, loss of value colon, t minus 6 with respect to t.

02:03 So now it's just a simple little integral...

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