Question 1

Suppose you are tasked with finding the area under the curve $$y = e$$^{-x^2}$$ from x = 0$ to x = 1$. Which calculus technique is most appropriate for approximating this area?

Accepted Answer

Use numerical integration methods such as Simpson's Rule or the Trapezoidal Rule.

Question 2

A pharmaceutical company needs to model the rate at which a drug is absorbed into the bloodstream. If the absorption rate is proportional to the amount of drug remaining, which type of differential equation best describes this scenario?

Accepted Answer

A separable first-order differential equation.

Question 3

Evaluate the improper integral $$\int_{1}^{\infty} \frac{1}{x^2$} \, dx. $$What is the result?

Accepted Answer

$$1$$

Question 4

A population of bacteria grows according to the logistic model $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}ight). If$$ $$P(0) = 100$$, $$r = 0.2$$, and $$K = 1000$$, what happens to $$P(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$P(t)$$ approaches $$1000$$

Question 5

Which of the following is the correct solution to the differential equation $$\frac{dy}{dx} = 3y$$ with initial condition $$y(0) = 2$$?

Accepted Answer

$$y(x) = 2e$$^{3x}$$

Question 6

A tank contains 100 liters of water with 10 grams of salt dissolved. Water with 0.5 grams of salt per liter flows in at 2 liters per minute, and the mixture is kept uniform and flows out at the same rate. What is the differential equation for the amount of salt $$S(t) in $$the tank?

Accepted Answer

$$\frac{dS}{dt} = 1 - \frac{2S}{100}$$

Question 7

Which of the following integrals can be solved using substitution $$u = x^2$?

Accepted Answer

$$\int x e$$^{x^2}$$ \, dx$$

Question 8

A radioactive isotope decays at a rate proportional to its current mass. If the half-life is 5 years, what is the decay constant $$k in $$the equation $$m(t) = m_0$ e$$^{-kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{5}$$

Question 9

Evaluate $$\int_{0}^{\pi} \sin x \, dx.$$

Accepted Answer

$$2$$

Question 10

Which of the following is a correct application of integration by parts to $$\int x \ln x \, dx$$?

Accepted Answer

Let $$u = \ln x$$, $$dv = x \, dx$$

Question 11

A medical researcher models the spread of a virus using the equation $$\frac{dN}{dt} = kN. If $$the number of cases doubles every 3 days, what is the value of $$k$$?

Accepted Answer

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

Question 12

Which of the following series converges?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$

Question 13

A chemical reaction is modeled by $$\frac{dC}{dt} = -kC$$, where $$C is $$concentration. If $$C(0) = 10$$ and $$k = 0.1$$, what is $$C(5)$$?

Accepted Answer

$$C(5) = 10 e$$^{-0.5}$$

Question 14

Find the volume of the solid obtained by rotating $$y = x^2$ about the x$-axis from x = 0$ to x = 1$.

Accepted Answer

$$\frac{\pi}{5}$$

Question 15

Which of the following is the correct Taylor series expansion for $$e^x$$ about $$x = 0$$?

Accepted Answer

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Question 16

A tank is being filled with water at a rate of $$5$$ liters per minute and leaks at a rate proportional to the amount of water present, $$W(t). $$Which equation models the water volume?

Accepted Answer

$$\frac{dW}{dt} = 5 - kW$$

Question 17

Which of the following is the correct integral for the arc length of $$y = \ln x$$ from $$x = 1 to$$ $$x = 2$$?

Accepted Answer

$$\int_{1}^{2} \sqrt{1 + \left(\frac{1}{x}\$$right)^2$$} \, dx$$

Question 18

A patient receives a constant infusion of medication at rate $$R mg/hr$$, and the medication is eliminated at rate proportional to the amount present, $$M(t). $$Which equation models the medication amount?

Accepted Answer

$$\frac{dM}{dt} = R - kM$$

Question 19

Which of the following is the correct formula for the sum of a geometric series $$\sum_{n=0}^{\infty} ar^n$$ when $$|r| \u003c 1$$?

Accepted Answer

$$\frac{a}{1 - r}$$

Calculus II Course Schedule: Topics and Study Guide

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