Derivative of Functions PDF

Exercices 1 Drug Absorption Rate The concentration of a drug in the bloodstream over time is given by C (t) = 5t 2 − 3t + 2. Find the rate of change of the drug concentration every hour from begining to 5 hours, and the rate of change with respect to time (i.e.,...

Exercices 1 Drug Absorption Rate The concentration of a drug in the bloodstream over time is given by C (t) = 5t 2 − 3t + 2. Find the rate of change of the drug concentration every hour from begining to 5 hours, and the rate of change with respect to time (i.e., the derivative C ′ (t)). Interpret what the sign of C ′ (t) indicates about the absorption or elimination of the drug. 2 Pharmacokinetics: Drug Metabolism The metabolism of a drug follows M(t) = e −0.2t , where t is the time in hours. Compute the derivative M ′ (t). Explain what the result tells us about the rate at which the drug is being metabolized as time increases. 47/52 3 Logarithmic Function: pH Calculation The pH of a solution is given by pH = − log10 ([H + ]), where [H + ] is the concentration of hydrogen ions. Calculate the derivative of pH with respect to [H + ] and discuss what it implies about changes in pH as the hydrogen ion concentration increases. 4 Exponential Growth: Bacterial Culture The number of bacteria in a culture is modeled by N(t) = 200e 0.3t. Find the first derivative N ′ (t) to determine the rate of bacterial growth. Explain how the rate changes as time progresses. 5 Second Derivative: Drug Release Kinetics The release rate of a drug from a tablet can be modeled by R(t) = 4t 3 − 6t 2 + 2t. Compute the second derivative R ′′ (t) and interpret its significance regarding the acceleration or deceleration of drug release. 48/52 6 Chain Rule Application: Compound Concentration The concentration of a compound in the bloodstream depends on time and is given by C (x ) = (2x 2 + 5)3/2 , where x is the time in hours. Use the chain rule to find C ′ (x ). Discuss the implications for how rapidly the concentration is changing over time. 49/52 7 Polynomial Function: Drug Dosage A patient’s reaction to a drug dosage is modeled by R(d) = d 4 − 8d 3 + 20d, where d is the dosage, 0 < d < 10. Find Break-even Points (roots) and graph the function. Find the critical points by calculating the first derivative and setting it to zero. Determine where the drug’s effectiveness is increasing or decreasing. 50/52 8 Given that the concentration of a drug in the bloodstream varies with time according to the function: f (t) = sin(3t 2 + 2t) find f ′ (t) using the chain rule. 9 A medication’s efficacy follows a model where efficacy E over time t is represented as: 5   E (t) = cos t +1 Determine E ′ (t) using the chain rule. 10 The rate of change in the absorption of a drug can be modeled by: A(x ) = tan(4x + x 2 ) Find A′ (x ) with respect to x by applying the chain rule. 51/52 11 Suppose the bio-availability of a drug is given by: B(c) = sec(2c 2 − 3c) where c is the concentration of the drug. Calculate B ′ (c) using the chain rule. 12 The dissolution rate of a drug in the body is modeled by: D(t) = csc(3t 3 + 2t) where t is time in hours. Find D ′ (t) using the chain rule. 13 The decay of a drug’s concentration can be represented as: C (x ) = cot(6x − x 2 ) where x is time in minutes. Find C ′ (x ) by applying the chain rule. 52/52