Business Calculus Quiz: Present Value Continuous Compounding
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Present Value Continuous CompoundingQuestion 1 of 16

An investor is considering two projects. Project A generates revenue at a constant rate of $20,000 per year. Project B generates revenue at a rate of $f(t) = 15000e^{0.02t}$ dollars per year. Both projects run for 8 years. If the prevailing interest rate is 5% compounded continuously, which project has a higher present value, and by approximately how much?

Project A has a higher present value, by approximately $11,872.
Project A has a higher present value, by approximately $25,187.
Project B has a higher present value, by approximately $28,986.
Project A has a higher present value, by approximately $30,133.
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Business Calculus Quiz

Business Calculus Quiz: Present Value Continuous Compounding

Practice Present Value Continuous Compounding in Business Calculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Present Value Continuous Compounding, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Calculus.

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Question 1

An investor is considering two projects. Project A generates revenue at a constant rate of $20,000 per year. Project B generates revenue at a rate of $f(t) = 15000e^{0.02t}$ dollars per year. Both projects run for 8 years. If the prevailing interest rate is 5% compounded continuously, which project has a higher present value, and by approximately how much?

  1. Project A has a higher present value, by approximately $11,872.
  2. Project A has a higher present value, by approximately $25,187. (correct answer)
  3. Project B has a higher present value, by approximately $28,986.
  4. Project A has a higher present value, by approximately $30,133.
Explanation: First, calculate the present value of Project A: PVA=∫0820000e−0.05tdt=20000[e−0.05t−0.05]08=−400000[e−0.4−1]≈−400000(0.67032−1)=131,872PV_A = \int_{0}^{8} 20000e^{-0.05t} dt = 20000[\frac{e^{-0.05t}}{-0.05}]_0^8 = -400000[e^{-0.4} - 1] \approx -400000(0.67032 - 1) = 131,872. Next, calculate the present value of Project B, combining the exponents: PVB=∫0815000e0.02te−0.05tdt=∫0815000e−0.03tdt=15000[e−0.03t−0.03]08=−500000[e−0.24−1]≈−500000(0.78663−1)=106,685PV_B = \int_{0}^{8} 15000e^{0.02t}e^{-0.05t} dt = \int_{0}^{8} 15000e^{-0.03t} dt = 15000[\frac{e^{-0.03t}}{-0.03}]_0^8 = -500000[e^{-0.24} - 1] \approx -500000(0.78663 - 1) = 106,685. Comparing the two, Project A has a higher present value. The difference is $131,872 - 106,685 = 25,187.

Question 2

A project requires an initial investment of $221,000. It is expected to generate a continuous income stream of $30,000 per year for 10 years. To the nearest percent, what is the annual interest rate $r$, compounded continuously, that makes the present value of the income stream equal to the initial investment?

  1. 4%
  2. 5%
  3. 6% (correct answer)
  4. 7%
Explanation: The present value formula is PV=∫0TCe−rtdt=Cr(1−e−rT)PV = \int_{0}^{T} C e^{-rt} dt = \frac{C}{r}(1 - e^{-rT}). We are given PV=221000PV=221000, C=30000C=30000, and T=10T=10. The equation to solve is 221000=30000r(1−e−10r)221000 = \frac{30000}{r}(1 - e^{-10r}). Since this cannot be solved algebraically for rr, we must test the given interest rates. For r=0.06r=0.06 (6%), PV=300000.06(1−e−10(0.06))=500000(1−e−0.6)≈500000(1−0.54881)=225,595PV = \frac{30000}{0.06}(1 - e^{-10(0.06)}) = 500000(1 - e^{-0.6}) \approx 500000(1 - 0.54881) = 225,595. For r=0.07r=0.07 (7%), PV=300000.07(1−e−10(0.07))≈428571(1−e−0.7)≈428571(1−0.49659)=215,745PV = \frac{30000}{0.07}(1 - e^{-10(0.07)}) \approx 428571(1 - e^{-0.7}) \approx 428571(1 - 0.49659) = 215,745. The value $225,595 is closer to the target investment of $221,000 than $215,745 is. Thus, 6% is the best approximation.

Question 3

The total profit P(t)P(t) from a new product line, in millions of dollars, is projected to grow at a rate of P′(t)=2−0.2tP'(t) = 2 - 0.2t for the first five years (0≤t≤50 \le t \le 5). Find the present value of this profit stream over the five-year period, assuming an interest rate of 5% compounded continuously.

  1. $8.85 million
  2. $8.40 million
  3. $7.50 million
  4. $6.73 million (correct answer)
Explanation: The present value is PV=∫05(2−0.2t)e−0.05tdtPV = \int_{0}^{5} (2 - 0.2t)e^{-0.05t} dt. Integration by parts (u=2−0.2tu = 2 - 0.2t, dv=e−0.05tdtdv = e^{-0.05t}dt) gives the antiderivative (40+4t)e−0.05t(40 + 4t)e^{-0.05t}. Evaluating the definite integral: [(40+4(5))e−0.05(5)]−[(40+0)e0]=60e−0.25−40≈60(0.7788)−40=46.728−40=6.728[(40+4(5))e^{-0.05(5)}] - [(40+0)e^0] = 60e^{-0.25} - 40 \approx 60(0.7788) - 40 = 46.728 - 40 = 6.728. So the present value is approximately $6.73 million.

Question 4

A company is selling a patent that is expected to generate a continuous income stream modeled by f(t)=150,000e−0.04tf(t) = 150,000e^{-0.04t} dollars per year, where tt is in years. This income is expected to continue indefinitely. What is the present value of this perpetual income stream, assuming an interest rate of 6% compounded continuously?

  1. $1,500,000 (correct answer)
  2. $2,500,000
  3. $3,750,000
  4. $7,500,000
Explanation: The present value of a perpetual stream is found using an improper integral: PV=∫0∞f(t)e−rtdtPV = \int_{0}^{\infty} f(t)e^{-rt} dt. Substituting the given functions: PV=∫0∞150000e−0.04te−0.06tdt=∫0∞150000e−0.10tdtPV = \int_{0}^{\infty} 150000e^{-0.04t} e^{-0.06t} dt = \int_{0}^{\infty} 150000e^{-0.10t} dt. Evaluating the integral: PV=[150000−0.10e−0.10t]0∞=[−1500000e−0.10t]0∞PV = [\frac{150000}{-0.10}e^{-0.10t}]_0^{\infty} = [-1500000e^{-0.10t}]_0^{\infty}. As t→∞t \to \infty, e−0.10t→0e^{-0.10t} \to 0. So, the evaluation is $0 - (−1500000e0-1500000e^0) = 1,500,000.

Question 5

A new automated system generates revenue at a constant rate of $250,000 per year. Its maintenance costs are modeled by the function $C(t) = 20,000 + 5,000t,where, where t$ is in years. The system has a useful life of 10 years. If the company uses a discount rate of 7% compounded continuously, what is the present value of the net income stream generated by the system?

  1. $1,350,000
  2. $1,497,000 (correct answer)
  3. $1,800,000
  4. $2,050,000
Explanation: The net income stream is f(t)=Revenue−Cost=250000−(20000+5000t)=230000−5000tf(t) = \text{Revenue} - \text{Cost} = 250000 - (20000 + 5000t) = 230000 - 5000t. We need to find the present value: PV=∫010(230000−5000t)e−0.07tdtPV = \int_{0}^{10} (230000 - 5000t)e^{-0.07t} dt. This can be solved by splitting the integral or using integration by parts. A simpler way is to find the PV of revenue and subtract the PV of costs. PVrev=∫010250000e−0.07tdt≈1,800,058PV_{rev} = \int_{0}^{10} 250000e^{-0.07t}dt \approx 1,800,058. PVcost=∫010(20000+5000t)e−0.07tdt≈302,674PV_{cost} = \int_{0}^{10} (20000+5000t)e^{-0.07t}dt \approx 302,674. The net present value is 1,800,058−302,674=1,497,3841,800,058 - 302,674 = 1,497,384, which is approximately $1,497,000.

Question 6

A company wants to pre-fund a marketing campaign that will cost $50,000 per year, paid as a continuous stream. They have allocated a budget of $400,000, which is the present value of the total campaign cost. If the company can earn 5% interest, compounded continuously, on their funds, for how many years $T$ can they fund the campaign with this budget?

  1. 8.0 years
  2. 9.5 years
  3. 10.2 years (correct answer)
  4. 12.0 years
Explanation: The present value is given by PV=∫0TCe−rtdtPV = \int_{0}^{T} C e^{-rt} dt. We have PV=400000PV=400000, C=50000C=50000, and r=0.05r=0.05. We need to solve for TT. 400000=∫0T50000e−0.05tdt=50000[e−0.05t−0.05]0T=−1000000[e−0.05T−1]400000 = \int_{0}^{T} 50000e^{-0.05t} dt = 50000[\frac{e^{-0.05t}}{-0.05}]_0^T = -1000000[e^{-0.05T} - 1]. Dividing by -1,000,000 gives −0.4=e−0.05T−1-0.4 = e^{-0.05T} - 1, so 0.6=e−0.05T0.6 = e^{-0.05T}. Taking the natural logarithm of both sides: ln⁡(0.6)=−0.05T\ln(0.6) = -0.05T. Since ln⁡(0.6)≈−0.5108\ln(0.6) \approx -0.5108, we have −0.5108=−0.05T-0.5108 = -0.05T. Solving for TT gives T=−0.5108−0.05≈10.216T = \frac{-0.5108}{-0.05} \approx 10.216 years.

Question 7

An asset provides a continuous income stream at a constant rate CC. An analyst calculates its present value over TT years, PVTPV_T, and its present value over 2T2T years, PV2TPV_{2T}. It is found that PV2TPV_{2T} is exactly 50% larger than PVTPV_T. Based on this information, what is the value of the discount factor e−rTe^{-rT}?

  1. 0.250.25
  2. 0.500.50 (correct answer)
  3. 0.670.67
  4. 0.750.75
Explanation: The present value of a constant income stream CC over a period of length LL is PVL=∫0LCe−rtdt=Cr(1−e−rL)PV_L = \int_0^L C e^{-rt} dt = \frac{C}{r}(1 - e^{-rL}). We are given that PV2T=1.5⋅PVTPV_{2T} = 1.5 \cdot PV_T. Substituting the formula: Cr(1−e−r(2T))=1.5⋅Cr(1−e−rT)\frac{C}{r}(1 - e^{-r(2T)}) = 1.5 \cdot \frac{C}{r}(1 - e^{-rT}). The Cr\frac{C}{r} terms cancel. Let x=e−rTx = e^{-rT}. Then e−2rT=(e−rT)2=x2e^{-2rT} = (e^{-rT})^2 = x^2. The equation becomes 1−x2=1.5(1−x)1 - x^2 = 1.5(1 - x). Since 1−x2=(1−x)(1+x)1-x^2 = (1-x)(1+x), we have (1−x)(1+x)=1.5(1−x)(1-x)(1+x) = 1.5(1-x). Since T>0T>0 and r>0r>0, x=e−rTx=e^{-rT} cannot be 1, so we can divide by (1−x)(1-x). This leaves 1+x=1.51+x = 1.5, which gives x=0.5x = 0.5. Therefore, e−rT=0.5e^{-rT} = 0.5.

Question 8

Consider two perpetual income streams. Stream A provides a constant CC dollars per year. Stream B's initial rate is also CC dollars per year, but it decays at a rate kk, such that its flow is Ce−ktC e^{-kt}. Both are discounted at the same continuous interest rate rr, where r>k>0r > k > 0. What is the ratio of the present value of Stream B to the present value of Stream A?

  1. 1−kr1 - \frac{k}{r}
  2. kr\frac{k}{r}
  3. rr+k\frac{r}{r+k} (correct answer)
  4. 1+kr1 + \frac{k}{r}
Explanation: The present value of perpetual Stream A is PVA=∫0∞Ce−rtdt=CrPV_A = \int_0^\infty C e^{-rt} dt = \frac{C}{r}. The present value of perpetual Stream B is PVB=∫0∞(Ce−kt)e−rtdt=∫0∞Ce−(r+k)tdt=Cr+kPV_B = \int_0^\infty (C e^{-kt}) e^{-rt} dt = \int_0^\infty C e^{-(r+k)t} dt = \frac{C}{r+k}. The ratio of B to A is PVBPVA=C/(r+k)C/r=Cr+k⋅rC=rr+k\frac{PV_B}{PV_A} = \frac{C/(r+k)}{C/r} = \frac{C}{r+k} \cdot \frac{r}{C} = \frac{r}{r+k}.

Question 9

An investment will generate a continuous cash flow of $$$C(t) = 10,000 + 2,000tdollarsperyearfordollars per year for10years.Ifthediscountrateisyears. If the discount rate is5%$$ compounded continuously, what is the present value of the investment to the nearest thousand dollars?

  1. $$$156,000$$
  2. $$$142,000$$ (correct answer)
  3. $$$134,000$$
  4. $$$168,000$$
Explanation: PV = ∫₀¹⁰ (10,000 + 2,000t)e^(-0.05t)dt = ∫₀¹⁰ 10,000e^(-0.05t)dt + ∫₀¹⁰ 2,000te^(-0.05t)dt. The first integral equals 10,000[e^(-0.05t)/(-0.05)]₀¹⁰ ≈ 78,732. The second requires integration by parts: 2,000∫te^(-0.05t)dt ≈ 63,212. Total PV ≈ $141,944 ≈ $142,000. Choice A ignores discounting. Choice C uses wrong discount rate. Choice D incorrectly applies integration by parts.

Question 10

A perpetual income stream pays A$$ dollars continuously per year, with the first payment starting immediately. If the continuous discount rate is $$r$$, and after $$n$$ years the payment rate increases to kA$$ dollars per year forever, what is the present value of this income stream?

  1. $$$\frac{A}{r} + \frac{A(k-1)e^{-rn}}{r}$$ (correct answer)
  2. $$$\frac{A}{r} + \frac{Ak}{r}e^{-rn}$$
  3. $$$\frac{A(1 + ke^{-rn})}{r}$$
  4. $$$\frac{A}{r}(1 + k - e^{-rn})$$
Explanation: The present value consists of two parts: (1) A/r for the perpetual stream at rate A, and (2) the present value of the increase from A to kA starting at time n. The increase is (k-1)A per year starting at time n, which has present value (k-1)A/r discounted back n years: (k-1)A·e^(-rn)/r. Total PV = A/r + A(k-1)e^(-rn)/r. Choice B double-counts the base payment. Choice C has incorrect algebraic form. Choice D has wrong sign and structure.

Question 11

A business investment generates cash flows according to $$$C(t) = 30,000e^{-0.02t}dollarsperyearfordollars per year for12years.Ifaninvestorrequiresayears. If an investor requires a8%continuousreturn,butinflationisexpectedtobecontinuous return, but inflation is expected to be2%$$ continuously compounded, what is the real present value of this investment?

  1. $$$221,156$$
  2. $$$198,765$$
  3. $$$267,432$$
  4. $$$243,890$$ (correct answer)
Explanation: When you encounter present value problems with both required returns and inflation, you need to calculate the real present value using the real discount rate. The real discount rate accounts for both your required return and inflation's erosive effect on purchasing power. First, find the real discount rate. With an 8% required return and 2% inflation (both continuously compounded), the real rate is rreal=0.08+0.02=0.10r_{real} = 0.08 + 0.02 = 0.10 or 10%. You add these rates because you need extra return to compensate for inflation on top of your base requirement. Next, calculate the present value using the continuous cash flow formula: PV=∫01230,000e−0.02t⋅e−0.10t dtPV = \int_0^{12} 30,000e^{-0.02t} \cdot e^{-0.10t} \, dt Combining the exponentials: PV=30,000∫012e−0.12t dtPV = 30,000 \int_0^{12} e^{-0.12t} \, dt Evaluating: PV=30,000[e−0.12t−0.12]012=30,000⋅1−e−1.440.12=$243,890PV = 30,000 \left[\frac{e^{-0.12t}}{-0.12}\right]_0^{12} = 30,000 \cdot \frac{1 - e^{-1.44}}{0.12} = \$243,890 Choice A (221,156)likelyusesanincorrectrealratecalculation,possiblysubtractinginflationfromtherequiredreturn.ChoiceB(221,156) likely uses an incorrect real rate calculation, possibly subtracting inflation from the required return. Choice B (198,765) appears to use too high a discount rate, perhaps incorrectly combining the rates. Choice C ($267,432) might represent using only the 8% required return without properly accounting for inflation. Study tip: Remember that real discount rates require adding the required return and inflation rate when both are continuously compounded. This reflects the need for returns above inflation to maintain purchasing power.

Question 12

A lease agreement specifies rent payments of R(t) = 36,000(1.03)^t$$ dollars per year, paid continuously for $$10$$ years. However, due to a discount clause, the effective payment rate is 0.95R(t).Ifmoneyisworth. If money is worth 6%$$ compounded continuously, what is the present value of the lease payments?

  1. $$$285,670$$
  2. $$$267,389$$ (correct answer)
  3. $$$301,842$$
  4. $$$273,951$$
Explanation: Effective payment rate = 0.95 × 36,000(1.03)^t = 34,200(1.03)^t = 34,200e^(ln(1.03)t) = 34,200e^(0.02956t). PV = ∫₀¹⁰ 34,200e^(0.02956t)·e^(-0.06t)dt = ∫₀¹⁰ 34,200e^(-0.03044t)dt = 34,200[1-e^(-0.3044)]/0.03044 ≈ $267,389. Choice A ignores the 0.95 discount factor. Choice C uses discrete compounding conversion. Choice D uses ln(1.03) ≈ 0.03 instead of exact value 0.02956.

Question 13

A new tech startup projects its revenue will be generated at a rate of f(t)=50+10tf(t) = 50 + 10t thousand dollars per year, where tt is the number of years from now. Assuming an annual interest rate of 4% compounded continuously, what is the present value of this income stream over the first 10 years, rounded to the nearest thousand dollars?

  1. $797,000 (correct answer)
  2. $824,000
  3. $1,000,000
  4. $1,271,000
Explanation: The present value (PV) is calculated by the integral PV=∫0Tf(t)e−rtdtPV = \int_{0}^{T} f(t)e^{-rt} dt. Here, f(t)=50+10tf(t) = 50 + 10t, r=0.04r = 0.04, and T=10T = 10. The integral is PV=∫010(50+10t)e−0.04tdtPV = \int_{0}^{10} (50 + 10t)e^{-0.04t} dt. This requires integration by parts. Let u=50+10tu = 50 + 10t and dv=e−0.04tdtdv = e^{-0.04t} dt. Then du=10dtdu = 10 dt and v=−25e−0.04tv = -25e^{-0.04t}. The antiderivative is (−7500−250t)e−0.04t(-7500 - 250t)e^{-0.04t}. Evaluating from t=0t=0 to t=10t=10: [(−7500−250(10))e−0.04(10)]−[(−7500−0)e0]=−10000e−0.4+7500≈−10000(0.67032)+7500=−6703.2+7500=796.8[(-7500 - 250(10))e^{-0.04(10)}] - [(-7500 - 0)e^{0}] = -10000e^{-0.4} + 7500 \approx -10000(0.67032) + 7500 = -6703.2 + 7500 = 796.8. In thousands of dollars, this is $796,800, which rounds to $797,000.

Question 14

A company's cash flow is modeled as $$$F(t) = 25,000e^{0.04t}dollarsperyear.Ifthecompanywantstofindthepresentvalueofcashflowsfromyeardollars per year. If the company wants to find the present value of cash flows from year3toyearto year8usingacontinuousdiscountrateofusing a continuous discount rate of7%$$, which integral correctly represents this calculation?

  1. $$$\int_3^8 25,000e^{0.04t} \cdot e^{-0.07t} dt$$ (correct answer)
  2. $$$\int_0^8 25,000e^{0.04t} \cdot e^{-0.07t} dt - \int_0^3 25,000e^{0.04t} \cdot e^{-0.07t} dt$$
  3. $$$\int_3^8 25,000e^{0.04t} \cdot e^{-0.07(t-3)} dt$$
  4. $$$e^{-0.07 \cdot 3} \int_0^5 25,000e^{0.04t} \cdot e^{-0.07t} dt$$
Explanation: For present value of cash flows from year 3 to year 8, we integrate the cash flow function F(t) multiplied by the discount factor e^(-rt) from t=3 to t=8. This gives ∫₃⁸ 25,000e^(0.04t)·e^(-0.07t)dt. Choice B is mathematically equivalent but unnecessarily complex. Choice C incorrectly adjusts the discount factor time reference. Choice D incorrectly changes the integration limits and applies an extra discount factor.

Question 15

Two investment options are being compared. Option A provides 40,000$$ per year continuously for $$6$$ years. Option B provides 20,000peryearcontinuouslyforper year continuously for6 years, followed by $$$60,000 per year continuously for another 66 years. Using a 5%5\% continuous discount rate, what is the difference in present values (Option B minus Option A)?

  1. $$$82,914$$
  2. $$$76,183$$
  3. $$$89,247$$ (correct answer)
  4. $$$95,672$$
Explanation: When comparing investment streams with continuous cash flows, you need to calculate the present value of each option using continuous discounting. This involves integrating the cash flow function multiplied by the discount factor e−rte^{-rt}. For Option A, the present value is ∫0640000e−0.05tdt=40000⋅1−e−0.30.05=$209,424\int_0^6 40000e^{-0.05t} dt = 40000 \cdot \frac{1-e^{-0.3}}{0.05} = \$209,424 Option B has two phases: ∫0620000e−0.05tdt+∫61260000e−0.05tdt\int_0^6 20000e^{-0.05t} dt + \int_6^{12} 60000e^{-0.05t} dt The first integral equals 20000⋅1−e−0.30.05=$104,71220000 \cdot \frac{1-e^{-0.3}}{0.05} = \$104,712 For the second integral, factor out e−0.3e^{-0.3}: 60000e−0.3∫06e−0.05udu=60000e−0.3⋅1−e−0.30.05=$193,95960000e^{-0.3} \int_0^6 e^{-0.05u} du = 60000e^{-0.3} \cdot \frac{1-e^{-0.3}}{0.05} = \$193,959 Option B's total present value is $104,712+$193,959=$298,671\$104,712 + \$193,959 = \$298,671 The difference (Option B minus Option A) is $298,671−$209,424=$89,247\$298,671 - \$209,424 = \$89,247, confirming answer C. Answer A (82,914)likelyresultsfromcalculationerrorsintheexponentialterms.AnswerB(82,914) likely results from calculation errors in the exponential terms. Answer B (76,183) probably comes from incorrectly applying discrete rather than continuous discounting formulas. Answer D ($95,672) suggests errors in handling the two-phase integration for Option B, possibly from incorrect bounds or discount factor application. Study tip: For continuous cash flow problems, remember that the present value formula $∫abCe−rtdt=Cr(e−ra−e−rb)\int_a^b Ce^{-rt} dt = \frac{C}{r}(e^{-ra} - e^{-rb}) $ is your foundation. Always check that your exponential calculations are precise, as small errors compound significantly.

Question 16

An oil well's production rate is $$$P(t) = 50,000e^{-0.08t}barrelsperyear,andoilsellsforbarrels per year, and oil sells for80 per barrel. The well will be productive for 2020 years. If the required rate of return is 9%9\% compounded continuously, and extraction costs are $$25 per barrel, what is the net present value of the well?

  1. $$$10,394,118$$
  2. $$$15,823,824$$
  3. $$$12,647,059$$ (correct answer)
  4. $$$14,235,647$$
Explanation: This problem tests your understanding of net present value (NPV) calculations with continuous cash flows and exponential decay. When evaluating long-term investments like oil wells, you need to account for both changing production rates and the time value of money. Start by finding the net revenue per barrel: $80 - $25 = $55. The annual cash flow function becomes $C(t) = 55 \times 50,000e^{-0.08t} = 2,750,000e^{-0.08t}$. To find NPV, you must discount this continuous cash flow stream at the required 9% rate: NPV = \int_0^{20} 2,750,000e^{-0.08t} \cdot e^{-0.09t} dt = \int_0^{20} 2,750,000e^{-0.17t} dt Evaluating this integral: NPV = 2,750,000 \times \frac{1-e^{-0.17 \times 20}}{0.17} = 2,750,000 \times \frac{1-e^{-3.4}}{0.17} \approx 12,647,059 Answer C (12,647,059)correctlyappliesboththeexponentialproductiondeclineandcontinuousdiscounting.AnswerA(12,647,059) correctly applies both the exponential production decline and continuous discounting. Answer A (10,394,118) likely uses an incorrect discount rate or production function. Answer B (15,823,824)probablyfailstoaccountforextractioncostsorusessimpleratherthancontinuouscompounding.AnswerD(15,823,824) probably fails to account for extraction costs or uses simple rather than continuous compounding. Answer D (14,235,647) might calculate gross rather than net cash flows or use an inappropriate integration method. Remember: NPV problems with continuous cash flows require careful attention to three elements: the correct net cash flow function, proper discounting for the time value of money, and accurate integration over the specified time period.