Calculus Problems

Question

 After a foreign substance is introduced into the blood, the rate at which antibodies are produced is given by    where A is thousands of antibodies and t is the time in minutes since introduction. How many antibodies have been produced during the first five minutes? How many antibodies have been produced between five and ten minutes?

Answer 1

A'(t) = ( t / t² + 2) This is the derivative of A'(t) . To get A(t) between 0 and 5 minutes, you need5 mi to integrate A'(t) over bounds of 0 and 5 minutes such that:

5

A(t) = ∫₀ ( t / t²+2) dt , let u= t²+2 , du = 2( t)^2-1 + 0 = 2(t)¹ = 2t, divide both sides by 2, then 1/2 du = t,

5 5 5

Substitute in the integral, A(t) = 1/2 ∫₀ du / u = 1/2 ln(u) ]₀ = 1/2 ln( t²+2) ]₀ =1/2 [ ln(5²+2)-ln(0²+2)] =

1/2 [ ln 27 - ln 2] = 1/2 ln(27/2) = 1.3013*1000 = 1301 antibodies

10 10

Between 5 and 10 minutes = A(t) = 1/2 A(t) = 1/2 ∫₅ du / u = 1/2 ln(u)]₅ = 1/2[ ln( 10²+2) - ln( 5²+2)] =

1/2[( ln(102)-ln(27)] = 1/2 ln( 102/27) = 0.664*1000= 664 antibodies

Note that the rate of Antibodies production decreases with time because the function is a decreasing function if you plot t /t²+2