Mathematical Methods · Unit 3 · Introduction to integration · Anti-differentiation
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A particle moves along a straight line such that its acceleration at time \(t\) seconds is given by \(a(t) = 6t - 8\) m/s². At \(t = 2\) seconds, the particle has velocity \(v = 5\) m/s and is located at position \(s = 12\) m from the origin. Determine the position function \(s(t)\) of the particle.
A pharmaceutical company is developing a time-release medication. The rate at which the drug enters the bloodstream is modelled by \( R(t) = 15t e^{-0.4t} \) mg/hour, where \( t \) is the time in hours after the medication is administered, \( t \geq 0 \). The concentration of the drug in the bloodstream is given by the accumulated amount of the drug absorbed up to time \( t \). (a) Determine an expression for \( C(t) \), the total amount of drug (in mg) absorbed into the bloodstream by time \( t \), given that no drug is present at \( t = 0 \). You must show algebraic working without technology for the integration step. [3 marks] (b) The medication is considered therapeutically effective when at least 18 mg has been absorbed. Determine the time (to the nearest minute) when the medication first becomes therapeutically effective. Technology may be used for this calculation. [2 marks]
A pharmaceutical company models the rate of drug concentration in a patient's bloodstream (in mg/L per hour) as shown in the graph below, where t is the time in hours after administration. If the initial concentration is c(0) = 0, which expression correctly models the concentration function c(t)?
A particle moves along a straight line such that its acceleration at time \( t \) seconds is given by \( a(t) = 6t - 8 \) m/s². At \( t = 2 \) seconds, the velocity is \( 5 \) m/s and the displacement is \( 12 \) m. Determine the displacement function \( s(t) \) for the particle.
A particle's velocity (in m s$^{-1}$) at time $t$ (in seconds) is given by $v(t) = 6t^2 - 4t + 3$. The particle starts at position $s = 5$ m when $t = 0$. Which expression represents the displacement function $s(t)$?