The concentration, C, in ng/ml of a drug in the blood as a function of the time, t, in hours since the drug was administered is given by $C = 15 t e ^ { - 0.2 t }.$ The area under the concentration curve is a measure of the overall effect of the drug on the body, called the bioavailability. Find the bioavailability of the drug between t = 0 and t = 3.

The concentration C (in ng/mL) of a drug in the bloodstream t hours after ingestion is modeled by $C=\dfrac{600t}{t^3+125}.$ (A) Determine the concentration at 1 hr, 12 hr, 24, hr, and 48 hr. Round to 1 decimal place. (B) What appears to be the limiting concentration for large values of t?

The concentration, c, of a drug in the bloodstream t hours after the drug was taken orally is given by $c(t)=\frac{5 t}{t^{2}+7}$, where c is measured in milligrams per litre. a) Calculate the average rate of change in the drug’s concentration during the first 2 h since ingestion. b) Estimate the rate at which the concentration of the drug is changing after exactly 3 h. c) Graph c(t) on a graphing calculator. When is the concentration of the drug increasing the fastest in the bloodstream? Explain.

During a surge in the demand for electricity, the rate, r, at which energy is used can be approximated by $r=t e^{-a t}$, where t is the time in hours and a is a positive constant. Find the total energy, E, used in the first T hours. Give your answer as a function of a.

The concentration, C, in ng/rnl, of a drug in the blood as a function of the time, t, in hours since the drug was administered is given by $C=15 t e^{-0.2 t}$. The area under the concentration curve is a measure of the overall effect of the drug on the body, called the bioavailability. Find the bioavailability of the drug between t=0 and t=3.

The London Eye is a large Ferris wheel that has diameter 135 meters and revolves continuously. Passengers enter the cabins at the bottom of the wheel and complete one revolution in about 27 minutes. One minute into the ride a passenger is rising at 0.06 meters per second. How fast is the horizontal motion of the passenger at that moment?

Find the exact area. Under the first arch of f(x) = x sin x.

If t is in hours, the drug concentration curve for a drug is given by $C=17.2 t e^{-0.4 t}$. The minimum effective concentration is 10 ng/ml. If the second dose of the drug is to be administered when the first dose becomes ineffective, when should the second dose be given?

Derive the formula (called a reduction formula): $\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x$.

The curves in the accompanying figure model the birth rates and death rates (in millions of people per year) for a country over a 50-year period. What does the area A between the curves over the interval [1960,2010] represent? Justify your answer.

If time, t, is in hours and concentration, C, is in ng/ml, the drug concentration curve for a drug is given by $C=12.4 t e^{-0.2 t}$. If the minimum effective concentration is 10 ng/ml, during what time period is the drug effective?

After a foreign substance is introduced into the blood, the rate at which antibodies are made is given by

The total number of people infected with a virus often grows like a logistic curve. Suppose that time, t, is in weeks and that 10 people originally have the virus. In the early stages, the number of people infected is increasing exponentially with k = 1.78. In the long run, 5000 people are infected. (a) Find a logistic function to model the number of people infected. (b) Sketch a graph of your answer to part (a). (c) Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point?

Multiply.

$$( 5 \sqrt { 3 } + 6 ) ( 5 \sqrt { 3 } - 6 )$$