Question 1

Given the function $$f(x) = \frac{1}{x-3}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq 3$$

Question 2

The concentration of a drug in the bloodstream is modeled by $$C(t) = C_0$ e$$^{-kt}$$, where C_0$ and k$ are constants. After how many hours is 25% of the original concentration left?

Accepted Answer

$$t = \frac{\ln(0.25)}{k}$$

Question 3

If $$C(t) = 3 \left(1 - \frac{t}{4}\$$right)^2$$, what is the rate of change of the drug concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -0.75$$

Question 4

The rate of elimination of a drug from the bloodstream is described by $$E(t) = 3t^2$ - t. $$What is the total amount of drug eliminated from $$t = 0 to$$ $$t = 3$$ hours?

Accepted Answer

$$24$$

Question 5

Given the differential equation $$2\frac{dC}{dt} = 4C$$, what is the general solution for $$C(t)$$?

Accepted Answer

$$C(t) = Ce$$^{2t}$$

Question 6

A patient receives a drug dose every 8 hours. If the drug concentration halves every 8 hours, what percentage of the original dose remains after 24 hours?

Accepted Answer

$$12.5\%$$

Question 7

The function $$f(x) = x^2$ e$$^{-x}$$ models the concentration of a substance. At what value of x$ does f(x$$)$$ reach its maximum?

Accepted Answer

$$x = 2$$

Question 8

In a clinical trial, the probability that a patient has Disease A is $$P(A)=0.3$$, Disease B is $$P(B)=0.4$$, and both is $$P(A \cap B)=0.1. $$What is the probability that a patient has either Disease A or Disease B?

Accepted Answer

$$0.7$$

Question 9

Given the function $$f(x) = \ln(x)$$, what is the range of $$f(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 10

A random variable $$X$$ follows a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that $$X \u003e 130$$?

Accepted Answer

$$0.0228$$

Question 11

The equation $$\frac{d^2$y}{dx^2$} + 2\frac{dy}{dx} + y = 0$$ describes:

Accepted Answer

A second-order ordinary differential equation (ODE)

Question 12

Which probability distribution is most appropriate for modeling the number of times a patient experiences a rare side effect in a year?

Accepted Answer

Poisson distribution

Question 13

If $$y = ax^b$$, where $$a$$ and $$b$$ are constants, what type of relationship does this represent?

Accepted Answer

Power law relationship

Question 14

A patient is given a drug with concentration $$C(t) = 200e$$^{-0.5t}$$. After how many hours is 10% of the original concentration left?

Accepted Answer

$$t = \frac{\ln(0.1)}{-0.5}$$

Question 15

The coefficient of $$x in $$the expression $$3x^2 + 4x + 5$ is:

Accepted Answer

$$4$$

Question 16

In a study, the correlation coefficient between two variables is $$r = -0.85. $$What does this indicate?

Accepted Answer

A strong negative linear relationship

Question 17

If $$f(x) = x^3$ - 6x^2$ + 9x$$, what are the critical points?

Accepted Answer

$$x = 0, 3$$

Question 18

A patient’s blood pressure readings are normally distributed with mean $$120$$ mmHg and standard deviation $$8$$ mmHg. What is the probability that a reading is less than $$104$$ mmHg?

Accepted Answer

$$0.0228$$

Question 19

The function $$f(x) = \frac{1}{x^2$} is$$:

Accepted Answer

Decreasing for $$x \u003e 0$$

Question 20

In a clinical trial, the probability of a patient experiencing a specific side effect after taking a new medication cannot be:

Accepted Answer

Greater than one

Calculus and Differential Equations in Mathematical Modeling and Statistics

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