Applied Calculus in Biology Quiz

1. What is the growth rate for a population P(t) = 1000/(1 + 9e^(-0.2t)) at t = 5?

a) 9000e^(-1)/(1 + 9e^(-1))² individuals/year
b) 1000e^(-0.2t) individuals/year
c) 0.2 individuals/year
d) 1000 individuals/year

2. For a logistic model dP/dt = 0.3P(1 - P/500), what is the growth rate at P = 250?

a) 37.5 individuals/hour
b) 75 individuals/hour
c) 18.75 individuals/hour
d) 150 individuals/hour

3. True or False: The carrying capacity in a logistic model is where dP/dt = 0.

a) True
b) False

4. What is the rate of drug concentration decay if C(t) = 10e^(-0.05t) mg/L at t = 2 hours?

a) -0.5e^(-0.1) mg/L/hour
b) -10e^(-0.05) mg/L/hour
c) -0.05 mg/L/hour
d) -10 mg/L/hour

5. In enzyme kinetics, v(S) = 5S/(2 + S) (rate in µM/s, S in µM), what is dv/dS at S = 3?

a) 10/25 µM/s per µM
b) 5/25 µM/s per µM
c) 15/25 µM/s per µM
d) 2/5 µM/s per µM

6. What is the total population from t = 0 to t = 10 for dP/dt = 0.1P, P(0) = 100?

a) 100e^(0.1*10) individuals
b) 100 individuals
c) 0.1*10 individuals
d) 1000 individuals

7. For a tumor growth model V(t) = 2t/(1 + t²) cm³, what is dV/dt at t = 1?

a) 1/2 cm³/day
b) 2 cm³/day
c) 1 cm³/day
d) 0 cm³/day

8. True or False: Calculus is used to model predator-prey dynamics.

a) True
b) False

9. What is the maximum growth rate for dP/dt = 0.4P(1 - P/200)?

a) 20 individuals/hour
b) 40 individuals/hour
c) 10 individuals/hour
d) 80 individuals/hour

10. For oxygen diffusion F(x) = 3e^(-0.1x) (µmol/cm²/s, x in cm), what is dF/dx at x = 0?

a) -0.3 µmol/cm³/s
b) 0.3 µmol/cm³/s
c) -3 µmol/cm³/s
d) 0 µmol/cm³/s