Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value a function approaches as the input approaches a certain value.

Question 2

Suppose the population of a bacteria culture grows according to the exponential model $$P(t) = P_0$ e$$^{kt}$$. If the initial population is 100 and after 3 hours the population is 200, what is the value of k$?

Accepted Answer

$$k = \frac{\ln(2)}{3}$$

Question 3

Which rule is used to differentiate the function $$f(x) = (3x^2$ + 2x)(e^x)$$?

Accepted Answer

Product Rule

Question 4

A medication is administered and its concentration in the bloodstream decays according to $$C(t) = C_0$ e$$^{-kt}$$. If the half-life of the medication is 4 hours, what is the value of k$?

Accepted Answer

$$k = \frac{\ln(2)}{4}$$

Question 5

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

4

Question 6

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It connects differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 7

A population grows logistically according to $$P(t) = \frac{K}{1 + Ae$$^{-kt}$$}. If$$ $$K is $$the carrying capacity, what happens to $$P(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$P(t)$$ approaches $$K$$

Question 8

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 9

Which of the following is the correct application of the Chain Rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 10

A patient’s body temperature cools according to Newton’s Law of Cooling: $$T(t) = T_{env} + (T_0$ - T_{env})e$$^{-kt}$$. If the initial temperature is 40$$^\circ$$C, the environment is 20$$^\circ$$C, and after 2 hours the temperature is 30$$^\circ$$C, which equation can be used to solve for k$?

Accepted Answer

$$30 = 20 + (40 - 20)e$$^{-2k}$$

Question 11

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$) dx.$$

Accepted Answer

8

Question 12

Which of the following is the correct setup for finding the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$?

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) dx$$

Question 13

If $$f(x) is $$continuous on $$[a, b]$$, which theorem guarantees that $$f(x)$$ attains both a maximum and a minimum on $$[a, b]$$?

Accepted Answer

Extreme Value Theorem

Question 14

Which of the following is the correct antiderivative of $$f(x) = 5e$$^{2x}$$?

Accepted Answer

$$\frac{5}{2}e$$^{2x}$$ + C$$

Question 15

A researcher measures the rate of change of a chemical concentration as $$\frac{dC}{dt} = -0.5C. $$What type of function best models the concentration $$C(t)$$ over time?

Accepted Answer

Exponential decay

Question 16

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$}{\sin x}$$?

Accepted Answer

$$f'(x) = \frac{2x \sin x - x^2$ \cos x}{\sin^2$ x}$$

Question 17

Which of the following integrals can be solved using integration by parts?

Accepted Answer

$$\int x e^x dx$$

Question 18

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at what $$x$$ value does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 19

Which of the following best describes the average value of a continuous function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) dx$$

MAT 251: Calculus for Life Sciences – Syllabus and Course Structure

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