Question 1

Which of the following best describes a first-order linear differential equation?

Accepted Answer

An equation of the form $$\frac{dy}{dt} + P(t)y = Q(t)$$

Question 2

Consider the differential equation $$\frac{dy}{dt} = y(y-3). $$Which of the following are equilibrium solutions?

Accepted Answer

$$y = 0$$ and $$y = 3$$

Question 3

Given the initial value problem $$\frac{dy}{dt} = 2y$$, $$y(0) = 5$$, what is the solution?

Accepted Answer

$$y(t) = 5e$$^{2t}$$

Question 4

Which of the following statements about Euler's Method is TRUE?

Accepted Answer

It approximates solutions to differential equations by using tangent line slopes at discrete points.

Question 5

Suppose $$y' = y(y-3)$$ with $$y(0) = 1. $$What happens to the solution as $$t 	o \infty$$?

Accepted Answer

It approaches $$0$$

Question 6

Which of the following is the general solution to $$\frac{dy}{dx} = \frac{3y}{2x+1}$$?

Accepted Answer

$$y = Ce$$^{\frac{3}$${2} \ln|2x+1|}$$

Question 7

A student borrows $$50,000 at 0.5\%$$ interest per month and pays $$500$$ per month. Which differential equation models the loan balance $$L(t)$$?

Accepted Answer

$$\frac{dL}{dt} = 0.005L - 500$$

Question 8

Which of the following sequences is bounded above and below but does NOT converge?

Accepted Answer

$$a_n = (-1)^n$$

Question 9

Evaluate the limit: $$\lim_{n 	o \infty} e$$^{-n}$$ 2 \sin(e$$^{-n}$$).$$

Accepted Answer

$$0$$

Question 10

Which of the following is the explicit formula for the sequence $$2, 5, 8, 11, \ldots$$?

Accepted Answer

$$a_n = 2 + 3(n-1)$$

Question 11

Which of the following best describes a geometric series?

Accepted Answer

A series where each term is a constant multiple of the previous term.

Question 12

Given the recursive sequence $$a_1 = 2$, a$$_{n+1} = a_n + 3$$, what is a_4$?

Accepted Answer

$$11$$

Question 13

Which of the following is a solution to Newton's Law of Cooling $$\frac{dT}{dt} = -k(T - T_{env})$$?

Accepted Answer

$$T(t) = T_{env} + (T_0$ - T_{env})e$$^{-kt}$$

Question 14

Which of the following series is telescoping?

Accepted Answer

$$\sum_{k=2}^{\infty} \left( \frac{1}{k-1} - \frac{1}{k+1} \right)$$

Question 15

Which of the following best describes a sequence that is increasing and bounded above?

Accepted Answer

$$a_n = 1 - \frac{1}{n}$$

Question 16

Suppose $$y' = ky(M - y)$$ models a learning curve, where $$k \u003e 0$$ and $$M is $$the maximum performance. What does the solution $$y(t)$$ approach as $$t 	o \infty$$?

Accepted Answer

$$M$$

Question 17

Which of the following is the general solution to $$\frac{dy}{dx} = x e$$^{x^2}$$ - \ln(y^2$)$$?

Accepted Answer

$$\ln|y| = \frac{1}{2} e$$^{x^2}$$ - x + C$$

Question 18

Which of the following is a sequence that is decreasing and converges?

Accepted Answer

$$a_n = \frac{1}{n}$$

Question 19

For the Sierpinski triangle, what is the area remaining after $$n$$ iterations if the original area is $$A$$?

Accepted Answer

$$A \left( \frac{3}{4} \right)^n$$

Differential Equations and Sequences & Series: Study Notes

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