Unit 5: Differential Equations and Series

Answer: D — This series is an arithmetic series because each term is obtained by adding a constant value (1) to the previous term. The correct answer is A. Options B and C are incorrect because a geometric series requires a common ratio between terms, which is not present in this series. Option D is incorrect because the series does exhibit a clear pattern, specifically that of an arithmetic sequence.

Answer: A — The correct answer is A because the series is a geometric series with a common ratio of 1/2. Since the absolute value of the common ratio (|1/2|) is less than 1, the series converges. Option B is incorrect because the series converges, not diverges. Options C and D are incorrect because the series is a geometric series, not an arithmetic series.

Answer: C — The correct answer is C because the series is a geometric series with first term a = 1/2 and common ratio r = 1/2. The formula for the sum of an infinite geometric series is S = a / (1 - r), which yields S = (1/2) / (1 - 1/2) = 1. This verifies the student's claim. The other options are incorrect because the ratio test and root test can determine convergence but not the sum, the integral test is not applicable to this series, the comparison test is not necessary in this case, and the alternating series test is not applicable since the series is not alternating. Additionally, numerical methods are not needed since the sum can be found exactly using the formula for a geometric series.

Answer: A — To solve the differential equation, we can separate the variables: dP/P = 0.02dt. Integrating both sides, we get ln|P| = 0.02t + C, where C is the constant of integration. Taking the exponential of both sides, we get P = e^(0.02t + C) = e^(0.02t)e^C. The correct answer is A because the profit increases exponentially over time, as the exponential function grows rapidly.

Answer: B — To solve the differential equation dy/dt = 2y, we first recognize it as a separable differential equation. We can rewrite it as dy/y = 2dt. Integrating both sides gives us ∫(dy/y) = ∫2dt, which results in ln|y| = 2t + C. Applying the exponential function to both sides, we get |y| = e^(2t + C). Since C is an arbitrary constant, we can rewrite e^C as C, yielding |y| = Ce^(2t). Because y can be positive or negative, the general solution is y = Ce^(2t). The correct answer, B) y = Ce^(2t), applies the concept of separable differential equations correctly. Distractor A) y = 2t + C incorrectly applies the concept of linear equations. Distractor C) y = 2e^t + C mistakenly adds an unnecessary constant inside the exponential function. Distractor D) y = e^(2t) + C incorrectly omits the constant of integration.