Determine 𝑓(𝑥) given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏.

A warehouse is being heated during winter. The rate of change of temperature in the warehouse is given by T'(t) = 4e^(0.2t), where T is the temperature in degrees Celsius and t represents hours after heating begins. (a) Determine the general temperature function T(t). [2 marks] (b) Given that the initial temperature is 8°C, determine the particular solution for T(t). [1 mark] (c) Determine the time required for the warehouse to reach a temperature of 88°C. Express your answer in the form a ln(b). [1 mark]

A medical researcher monitoring a patient's medication levels determines that the rate of change of drug concentration in the bloodstream is given by \(\frac{dc}{dt} = 4e^{-0.5t} + 2\), where \(c\) is the concentration (in mg/L) and \(t\) is the time in hours after administration. At \(t = 0\), the initial concentration is \(c(0) = 1\) mg/L. (a) Determine the general antiderivative of \(\frac{dc}{dt}\). [1] (b) Use the initial condition to find the constant of integration. [1] (c) Hence, determine the concentration \(c(t)\) as a function of \(t\). [1]

Given that $f'(x) = 6x^2 - 4x + 1$ and $f(1) = 3$, which of the following is $f(x)$?

A pharmaceutical research laboratory models the concentration of an experimental drug in the bloodstream. The rate of change of concentration \(C(t)\) (measured in micrograms per millilitre per hour) is given by \[C'(t) = 12t e^{-0.2t^2} - 3e^{-0.5t}\] where \(t\) is the time in hours after administration. (a) Determine the general expression for \(C(t)\). [2 marks] (b) When the drug is first administered at \(t = 0\), the initial concentration in the bloodstream is 5 micrograms per millilitre. Determine the particular solution for \(C(t)\). [1 mark] (c) Calculate the concentration of the drug in the bloodstream exactly 3 hours after administration. Express your answer correct to two decimal places. [2 marks]

A function \( f(x) \) has derivative \( f'(x) = 6x^2 - 8x + 3 \) and satisfies the condition \( f(2) = 11 \). The function \( f(x) \) is