Question

Let F(t)=0.01e^−0.2tbe the rate at which a chemical is produced during a reaction in milligrams per minute where t is in minutes. - Determine the antiderivative of F(t). - Draw the graph of F(t) and estimate the area between the graph and the t-axis from t=0 to t=5 minutes. - What are the units of ∫ _0^5 F(t)dt ? - Suppose at the start of the reaction, there are 2 milligrams of the chemical. Explain how to get the total amount of the chemical after ten minutes and compute it.

Answer 1

The antiderivative of F(t) = 0.01e^(-0.2t) is given by G(t) = -0.05e^(-0.2t) + C, where C is the constant of integration.

Step-by-step explanation:

The antiderivative of a function represents the reverse process of differentiation. For F(t) = 0.01e^(-0.2t), we find the antiderivative G(t) by applying the power rule for integration. The antiderivative is

G(t) = -0.05e^(-0.2t) + C, where C is the constant of integration. This constant is introduced because the derivative of a constant is zero, and when finding an antiderivative, we lose information about the constant term.

To draw the graph of F(t) and estimate the area between the graph and the t-axis from t=0 to t=5 minutes, we would plot the function and find the definite integral ∫_0^5 F(t)dt. The units of this integral represent the area under the curve, which in this context is milligrams multiplied by minutes, giving milligram-minutes as the units.

If at the start of the reaction there are 2 milligrams of the chemical, the total amount after ten minutes is found by evaluating G(t) at t=10 and subtracting the initial amount: G(10) - G(0) = [-0.05e^(-2) + C] - [-0.05e^(0) + C]. After simplifying, we get the final result, representing the total amount of the chemical after ten minutes.

Answer 2

The antiderivative of the function F(t) is -0.05e^{-0.2t} plus a constant. The units of the integrated function over time are in milligrams. To find the total chemical quantity after ten minutes, add the initial amount to the integral of the rate function from 0 to 10 minutes.

Step-by-step explanation:

Antiderivative of the Rate Function

The given rate function is F(t) = 0.01e^{-0.2t}. To find the antiderivative of F(t), we can use the basic rules of integration. The antiderivative of e^{ax} with respect to x is \frac{1}{a}e^{ax}. Therefore, the antiderivative of F(t) is -0.05e^{-0.2t} plus a constant of integration, C.

Graphing and Estimating Area

We would need to graph F(t) and look at the curve from t=0 to t=5 minutes. The exact area under the curve can be found by integrating F(t) from 0 to 5, which gives us the area as part of the solution to the previous part. This area can be estimated visually if the graph is drawn on a grid.

Units of the Area

The units of \int_0^5 F(t)\,dt are milligrams (mg), since the integration of a rate (mg per minute) over time (minutes) yields a quantity in milligrams.

Total Amount of the Chemical

To get the total amount of the chemical after ten minutes, we would add the initial amount to the integral of the rate function from 0 to 10.

So, the total amount is 2 mg + \int_0^{10} 0.01e^{-0.2t}\,dt. After calculating the integral, we obtain the total amount, which in this case is 2 + (0.01/-0.2)*(e^{-0.2*10}-1) mg.