Business Calculus Quiz: Continuity And Piecewise Functions
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Continuity And Piecewise FunctionsQuestion 1 of 17

The weekly cost, C(q)C(q), of managing inventory for a certain product depends on the quantity on hand, qq. When inventory is low (q≤50q \le 50), a special storage fee of kk is applied. The company wants to set this fee to ensure the cost function is continuous at the threshold q=50q=50.

The cost function is given by:

What must be the value of the special storage fee kk to ensure C(q)C(q) is continuous at q=50q=50?

k=−50k = -50
k=50k = 50
k=150k = 150
k=200k = 200
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Business Calculus Quiz

Business Calculus Quiz: Continuity And Piecewise Functions

Practice Continuity And Piecewise Functions 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 Continuity And Piecewise Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Calculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The weekly cost, C(q)C(q), of managing inventory for a certain product depends on the quantity on hand, qq. When inventory is low (q≤50q \le 50), a special storage fee of kk is applied. The company wants to set this fee to ensure the cost function is continuous at the threshold q=50q=50.

The cost function is given by:

What must be the value of the special storage fee kk to ensure C(q)C(q) is continuous at q=50q=50?

  1. k=−50k = -50
  2. k=50k = 50 (correct answer)
  3. k=150k = 150
  4. k=200k = 200
Explanation: For the function to be continuous at q=50q=50, the left-hand limit and the right-hand limit must be equal. Left-hand limit: lim⁡q→50−C(q)=k+5(50)=k+250\lim_{q\to50^-} C(q) = k + 5(50) = k + 250. Right-hand limit: lim⁡q→50+C(q)=2(50)+200=100+200=300\lim_{q\to50^+} C(q) = 2(50) + 200 = 100 + 200 = 300. Set them equal: k+250=300k + 250 = 300. Solving for kk gives k=300−250=50k = 300 - 250 = 50.

Question 2

The function g(t)={t2+4t−5if t≤2at+bt−1if t>2g(t) = \begin{cases} t^2 + 4t - 5 & \text{if } t ≤ 2 \\ \frac{at + b}{t - 1} & \text{if } t > 2 \end{cases} represents the growth rate of bacteria colonies over time tt (in hours). For the growth rate to be continuous at t=2t = 2, which condition must the constants aa and bb satisfy?

  1. a+b=7a + b = 7
  2. a+b=11a + b = 11
  3. 2a+b1=7\frac{2a + b}{1} = 7, which gives 2a+b=72a + b = 7 (correct answer)
  4. 2a+b1=11\frac{2a + b}{1} = 11, which gives 2a+b=112a + b = 11
Explanation: For continuity at t=2t = 2: lim⁡t→2−g(t)=g(2)=(2)2+4(2)−5=4+8−5=7\lim_{t \to 2^-} g(t) = g(2) = (2)^2 + 4(2) - 5 = 4 + 8 - 5 = 7. lim⁡t→2+g(t)=a(2)+b2−1=2a+b1=2a+b\lim_{t \to 2^+} g(t) = \frac{a(2) + b}{2 - 1} = \frac{2a + b}{1} = 2a + b. For continuity: 7=2a+b7 = 2a + b, so 2a+b=72a + b = 7. Choice C correctly states this condition and shows the substitution 2a+b1=7\frac{2a + b}{1} = 7. Choice A gives a+b=7a + b = 7, which is insufficient. Choice B gives a+b=11a + b = 11, which is incorrect. Choice D gives 2a+b=112a + b = 11, which doesn't match our calculation of 7.

Question 3

A logistics company calculates the shipping cost, C(w)C(w), for a package based on its weight ww in pounds. The fee structure is designed to offer a different rate for heavier packages.

The cost function is defined as:

Which of the following statements correctly analyzes the continuity of this function at w=5w=5 and its business implication?

  1. The function is continuous at w=5w=5, ensuring a fair price transition.
  2. The function has a removable discontinuity at w=5w=5, which can be fixed by changing the cost for a 5-pound package.
  3. The function has a jump discontinuity at w=5w=5, creating a sudden price increase for packages just over 5 pounds. (correct answer)
  4. The function has a jump discontinuity at w=5w=5, creating a sudden price discount for packages just over 5 pounds.
Explanation: To analyze continuity at w=5w=5, we check the value of the function and the one-sided limits. The value at w=5w=5 is C(5)=10+2.5(5)=22.5C(5) = 10 + 2.5(5) = 22.5. This is also the left-hand limit. The right-hand limit is lim⁡w→5+C(w)=15+2(5)=25\lim_{w\to5^+} C(w) = 15 + 2(5) = 25. Since the left-hand limit (22.5)doesnotequaltheright−handlimit(22.5) does not equal the right-hand limit (25), there is a jump discontinuity. This implies that a package weighing slightly more than 5 pounds costs $25, a sudden increase from the $22.50 cost for a package weighing exactly 5 pounds.

Question 4

An electric utility company designs a two-tier billing system for residential customers. The total monthly bill, B(x)B(x), is a function of the kilowatt-hours (kWh), xx, consumed.

The billing function is:

What is the financial consequence for a customer at the 500 kWh threshold?

  1. The bill has a jump discontinuity, resulting in an immediate $5 surcharge for using just over 500 kWh. (correct answer)
  2. The marginal cost of electricity jumps by $5, but the total bill is continuous.
  3. The bill is continuous, and the customer simply starts paying a higher marginal rate for additional usage.
  4. The bill has a removable discontinuity, indicating a flaw in the rate for exactly 500 kWh.
Explanation: When you encounter piecewise functions in business calculus, always check for continuity at the boundary points—this reveals important economic implications like pricing jumps or rate changes. Let's examine what happens at exactly 500 kWh. Using the first piece: B(500)=0.12(500)=$60B(500) = 0.12(500) = \$60. Now check what happens just above 500 kWh, say at 500.01 kWh. Using the second piece: B(500.01)=65+0.15(500.01−500)=65+0.15(0.01)=$65.0015B(500.01) = 65 + 0.15(500.01-500) = 65 + 0.15(0.01) = \$65.0015. This creates a jump discontinuity—the bill instantly jumps from $60 to approximately $65 when usage exceeds 500 kWh, even by a tiny amount. That's a $5 surcharge for crossing the threshold. Answer A correctly identifies this jump discontinuity and the immediate $5 financial consequence. Answer B incorrectly claims the total bill is continuous—we just showed it's not. While the marginal rate does increase from $0.12 to $0.15 per kWh (a $0.03 increase, not $5), the total bill itself jumps. Answer C makes the same error, claiming continuity where none exists. Answer D mischaracterizes this as a "removable discontinuity," but this is intentional rate design, not a flaw—the jump is a deliberate feature of the two-tier system. Study tip: For piecewise functions, always test continuity at boundary points by calculating the function value from both pieces. In business contexts, discontinuities often represent real economic consequences like penalty fees or rate structure changes.

Question 5

A simplified progressive tax system calculates the total tax owed, T(i)T(i), based on an individual's income, ii. The system is designed to be continuous to prevent a 'tax cliff,' where earning one additional dollar pushes an individual into a new bracket and causes a large, sudden increase in their tax liability.

The tax function is:

In this function, AA represents the total tax paid on income from all lower brackets. What must be the value of AA for the tax function to be continuous at i=40,000i=40,000?

  1. A=0A = 0
  2. A=4000A = 4000
  3. A=10000A = 10000
  4. A=6000A = 6000 (correct answer)
Explanation: When you encounter a piecewise function that needs to be continuous, you're looking for the point where both pieces give the same output value. Continuity means there's no sudden jump or gap in the function. To find the value of AA, you need to ensure that both pieces of the tax function give the same result at the boundary point i=40,000i = 40,000. From the left side (using the first piece), the tax at i=40,000i = 40,000 is: T(40,000)=0.15×40,000=6,000T(40,000) = 0.15 \times 40,000 = 6,000 From the right side (using the second piece), the tax approaches: T(40,000)=A+0.25(40,000−40,000)=A+0T(40,000) = A + 0.25(40,000 - 40,000) = A + 0 For continuity, these must be equal: 6,000=A6,000 = A. Therefore, A=6,000A = 6,000. Looking at the wrong answers: Choice (A) suggests A=0A = 0, which would create a discontinuous drop from $6,000 to $0 at the boundary—clearly violating continuity. Choice (B) gives $A = 4,000,whichwouldmeanthetaxsuddenlydropsfrom$6,000to$4,000whencrossingintothehigherbracket,creatinga"taxcliff"thatdefeatsthesystem′spurpose.Choice(C)proposes$A=10,000, which would mean the tax suddenly drops from $6,000 to $4,000 when crossing into the higher bracket, creating a "tax cliff" that defeats the system's purpose. Choice (C) proposes $A = 10,000, which would cause an immediate jump from $6,000 to $10,000—again breaking continuity. Remember this pattern: for piecewise functions to be continuous, evaluate both pieces at the boundary point and set them equal. This technique appears frequently in business calculus when modeling real-world scenarios like tax systems, pricing structures, or cost functions that change at certain thresholds.

Question 6

A consulting firm offers a discount on its hourly rate for longer projects. The total cost, C(h)C(h), for a project of hh hours has a two-part structure. For projects longer than 10 hours, a lower hourly rate is applied, but a fixed fee kk is added to the bill.

The firm's cost function is:

To ensure a seamless transition in billing, what value of the fixed fee kk makes the cost function continuous at h=10h=10?

  1. k=200k = 200
  2. k=2000k = 2000 (correct answer)
  3. k=3000k = 3000
  4. k=5000k = 5000
Explanation: To ensure continuity at h=10h=10, we must equate the left-hand and right-hand limits of C(h)C(h). Left-hand limit: lim⁡h→10−C(h)=500(10)=5000\lim_{h\to10^-} C(h) = 500(10) = 5000. Right-hand limit: lim⁡h→10+C(h)=k+300(10)=k+3000\lim_{h\to10^+} C(h) = k + 300(10) = k + 3000. Setting the limits equal: 5000=k+30005000 = k + 3000. Solving for kk, we get k=5000−3000=2000k = 5000 - 3000 = 2000.

Question 7

A manufacturing process has a complex cost function, C(x)C(x), that depends on the production batch size, xx. The function's definition changes at different batch sizes.

The cost function is given by:

If the cost function is continuous for all x>0x > 0, what is the value of the marginal cost aa for the intermediate batch size?

  1. a=17.5a = 17.5 (correct answer)
  2. a=20a = 20
  3. a=25a = 25
  4. a=30a = 30
Explanation: For continuity, the pieces must meet at the boundaries. At x=100x=100: 20(100)+1000=a(100)+b  ⟹  3000=100a+b20(100) + 1000 = a(100) + b \implies 3000 = 100a + b. At x=500x=500: a(500)+b=30(500)−5000  ⟹  500a+b=15000−5000=10000a(500) + b = 30(500) - 5000 \implies 500a + b = 15000 - 5000 = 10000. We now have a system of two linear equations:
  1. 100a+b=3000100a + b = 3000
  2. 500a+b=10000500a + b = 10000 Subtract equation (1) from equation (2): (500a−100a)+(b−b)=10000−3000(500a - 100a) + (b - b) = 10000 - 3000, which gives 400a=7000400a = 7000. Solving for aa: a=7000400=704=17.5a = \frac{7000}{400} = \frac{70}{4} = 17.5.

Question 8

A company pays its sales representatives a commission, E(s)E(s), based on their total monthly sales, ss. The commission rate is 4% for sales up to and including $100,000. For sales exceeding $100,000, the commission rate is 6% on the entire sales amount, plus a fixed bonus adjustment $B$.

The earnings function is given by:

To avoid a sudden drop or jump in pay, the earnings function must be continuous at the $100,000 threshold. What must the value of the bonus adjustment $B$ be?

  1. B=2000B = 2000
  2. B=−2000B = -2000 (correct answer)
  3. B=6000B = 6000
  4. B=4000B = 4000
Explanation: For continuity at s=100,000s=100,000, the limits from both sides must be equal. Left-hand limit: lim⁡s→100000−E(s)=0.04(100,000)=4000\lim_{s\to100000^-} E(s) = 0.04(100,000) = 4000. Right-hand limit: lim⁡s→100000+E(s)=B+0.06(100,000)=B+6000\lim_{s\to100000^+} E(s) = B + 0.06(100,000) = B + 6000. Setting them equal: 4000=B+60004000 = B + 6000. Solving for BB yields B=4000−6000=−2000B = 4000 - 6000 = -2000. A negative value for BB is necessary to offset the jump caused by applying the higher 6% rate to the entire sales amount.

Question 9

A chemical supplier offers a bulk discount. The total cost, C(q)C(q), for purchasing qq pounds of the chemical is based on a per-pound price that drops at 1,000 pounds. This creates a discontinuity where it's cheaper to buy 1,000 pounds than 999 pounds.

The current cost function is:

To eliminate this illogical price drop, the company decides to add a fixed processing fee FF to all orders of 1,000 pounds or more, changing the second rule to C(q)=8q+FC(q) = 8q + F. What value of FF makes the cost function continuous at q=1,000q=1,000?

  1. F=2F = 2
  2. F=−2000F = -2000
  3. F=8000F = 8000
  4. F=2000F = 2000 (correct answer)
Explanation: When you encounter piecewise functions with discontinuities, the key to making them continuous is ensuring that the function values from both pieces match at the boundary point. For continuity at a point, the left-hand limit must equal the right-hand limit. Let's find what value of FF makes C(q)C(q) continuous at q=1000q = 1000. First, calculate the cost just before the breakpoint using the first piece: C(999)=10(999)=9990C(999) = 10(999) = 9990. As qq approaches 1000 from the left, the cost approaches 10(1000)=1000010(1000) = 10000. For continuity, the cost at q=1000q = 1000 using the second piece must equal this limit. With the processing fee, the second piece becomes C(q)=8q+FC(q) = 8q + F. At q=1000q = 1000: C(1000)=8(1000)+F=8000+FC(1000) = 8(1000) + F = 8000 + F. Setting these equal: 8000+F=100008000 + F = 10000, so F=2000F = 2000. Looking at the wrong answers: Choice A (F=2F = 2) gives C(1000)=8002C(1000) = 8002, which is far too low. Choice B (F=−2000F = -2000) actually makes the discontinuity worse by giving C(1000)=6000C(1000) = 6000, creating an even bigger price drop. Choice C (F=8000F = 8000) gives C(1000)=16000C(1000) = 16000, making it more expensive than the smaller quantity pricing. Choice D (F=2000F = 2000) correctly eliminates the discontinuity by making C(1000)=10000C(1000) = 10000. Study tip: For piecewise continuity problems, always evaluate the function at the boundary using both pieces and set them equal. The processing fee compensates for the lower per-unit rate at higher quantities.

Question 10

A company's weekly profit, P(t)P(t) in thousands of dollars, is modeled as a function of time tt in weeks from the start of the year. At the beginning of week 12 (t=12t=12), a new marketing campaign was launched, which changed the profit model.

The profit model is given by:

Which statement provides the correct analysis of the profit model at t=12t=12?

  1. The profit model is continuous at t=12t=12, suggesting the campaign had a smooth, gradual effect.
  2. The model has a jump discontinuity, implying the campaign caused an immediate, discrete increase in weekly profit. (correct answer)
  3. The model has a jump discontinuity, implying the campaign caused an immediate, discrete decrease in weekly profit.
  4. The model shows that the rate of profit growth was discontinuous at t=12t=12, but the profit itself was continuous.
Explanation: First, find the left-hand limit at t=12t=12: lim⁡t→12−P(t)=1012+4=1016=10(4)=40\lim_{t\to12^-} P(t) = 10\sqrt{12+4} = 10\sqrt{16} = 10(4) = 40. Next, find the function value at t=12t=12, which is also the right-hand limit: P(12)=−0.1(12)2+5(12)+10.4=−0.1(144)+60+10.4=−14.4+70.4=56P(12) = -0.1(12)^2 + 5(12) + 10.4 = -0.1(144) + 60 + 10.4 = -14.4 + 70.4 = 56. Since the left-hand limit (40) is not equal to the right-hand limit (56), there is a jump discontinuity. The value jumps up, indicating an immediate increase in profit from $40,000 to $56,000 per week.

Question 11

A Software-as-a-Service (SaaS) company structures its monthly pricing, C(x)C(x), based on the number of users, xx. For up to 50 users, the cost is $20 per user. For more than 50 users, the cost is $15 per user plus a fixed enterprise fee of $k.Thecompanywantstosetthefee. The company wants to set the fee ksuchthatthecostfunctioniscontinuousatsuch that the cost function is continuous atx=50$, ensuring a smooth price transition for clients as they grow.

The pricing model is given by the function:

What value must the enterprise fee kk be for the cost function C(x)C(x) to be continuous at x=50x=50?

  1. k=0k = 0
  2. k=50k = 50
  3. k=250k = 250 (correct answer)
  4. k=750k = 750
Explanation: For C(x)C(x) to be continuous at x=50x=50, the left-hand limit must equal the right-hand limit. Left-hand limit: lim⁡x→50−C(x)=20(50)=1000\lim_{x\to50^-} C(x) = 20(50) = 1000. Right-hand limit: lim⁡x→50+C(x)=k+15(50)=k+750\lim_{x\to50^+} C(x) = k + 15(50) = k + 750. Set the limits equal to each other: 1000=k+7501000 = k + 750. Solving for kk gives k=1000−750=250k = 1000 - 750 = 250.

Question 12

A car rental agency charges a flat daily fee plus a variable per-mile fee. The per-mile fee is lower for customers who drive more than 150 miles in a day, but this introduces a complexity in the pricing structure.

The total cost for a one-day rental, C(m)C(m), after driving mm miles is:

To ensure the cost function is continuous, a 'high-mileage' surcharge SS is added to the second tier. What must be the value of SS?

  1. S=−22.50S = -22.50
  2. S=0.15S = 0.15
  3. S=37.50S = 37.50
  4. S=22.50S = 22.50 (correct answer)
Explanation: When you encounter a piecewise function in business calculus, continuity is crucial for real-world applications. A discontinuous pricing function would create unfair jumps in cost, so you need to ensure the function's value is the same at the boundary point regardless of which piece you use. To find the required surcharge SS, you must set the two pieces equal at the boundary point m=150m = 150. Using the first piece: C(150)=50+0.40(150)=50+60=110C(150) = 50 + 0.40(150) = 50 + 60 = 110. Using the second piece: C(150)=50+0.25(150)+S=50+37.5+S=87.5+SC(150) = 50 + 0.25(150) + S = 50 + 37.5 + S = 87.5 + S. For continuity, these must be equal: 110=87.5+S110 = 87.5 + S, which gives us S=110−87.5=22.50S = 110 - 87.5 = 22.50. Looking at the wrong answers: Choice A (S=−22.50S = -22.50) represents the negative of the correct value, likely from incorrectly subtracting in the opposite direction. Choice B (S=0.15S = 0.15) might come from confusing the difference in per-mile rates (0.40 - 0.25 = 0.15) with the surcharge needed. Choice C (S=37.50S = 37.50) is simply the mileage cost from the second piece at m=150m = 150, ignoring the flat fee entirely. Remember: for piecewise functions to be continuous, evaluate both pieces at the boundary point and set them equal. This ensures no sudden jumps in pricing, which is essential for fair business practices and realistic mathematical models.

Question 13

Consider the function f(x)={x2−9x−3if x≠3cif x=3f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } x ≠ 3 \\ c & \text{if } x = 3 \end{cases} At how many points in the interval [0,6][0, 6] is this function discontinuous when c=5c = 5?

  1. The function is continuous everywhere on [0,6][0, 6]
  2. The function is discontinuous at exactly one point (correct answer)
  3. The function is discontinuous at exactly two points
  4. The function is discontinuous at exactly three points
Explanation: First, simplify x2−9x−3=(x−3)(x+3)x−3=x+3\frac{x^2 - 9}{x - 3} = \frac{(x-3)(x+3)}{x - 3} = x + 3 for x≠3x ≠ 3. So lim⁡x→3f(x)=lim⁡x→3(x+3)=6\lim_{x \to 3} f(x) = \lim_{x \to 3} (x + 3) = 6. For continuity at x=3x = 3, we need f(3)=lim⁡x→3f(x)f(3) = \lim_{x \to 3} f(x), which means c=6c = 6. Since c=5≠6c = 5 ≠ 6, the function is discontinuous at x=3x = 3. The function is continuous everywhere else in [0,6][0, 6] since it equals x+3x + 3 for all x≠3x ≠ 3, which is a continuous function. Therefore, there is exactly one point of discontinuity.

Question 14

A company's revenue function has a discontinuity due to a pricing policy change. The function R(x)={100x−x2if x<10ax+bif x≥10R(x) = \begin{cases} 100x - x^2 & \text{if } x < 10 \\ ax + b & \text{if } x ≥ 10 \end{cases} represents revenue in thousands of dollars for xx thousand units sold. If lim⁡x→10+R(x)=1200\lim_{x \to 10^+} R(x) = 1200 and lim⁡x→10−R(x)=900\lim_{x \to 10^-} R(x) = 900, what type of discontinuity exists at x=10x = 10?

  1. A removable discontinuity because the limits exist but are unequal
  2. No discontinuity exists because both one-sided limits exist
  3. An infinite discontinuity because the function approaches different values
  4. A jump discontinuity because the one-sided limits exist but are unequal (correct answer)
Explanation: When analyzing discontinuities in piecewise functions, you need to examine the behavior of one-sided limits and determine whether the function "jumps" between different values. Let's verify the given limit information. For x<10x < 10, we use R(x)=100x−x2R(x) = 100x - x^2. As xx approaches 10 from the left: lim⁡x→10−R(x)=100(10)−102=1000−100=900\lim_{x \to 10^-} R(x) = 100(10) - 10^2 = 1000 - 100 = 900. This matches the given information. For the right-hand limit, we're told lim⁡x→10+R(x)=1200\lim_{x \to 10^+} R(x) = 1200, which means the linear piece ax+bax + b approaches 1200 as xx approaches 10 from the right. Since both one-sided limits exist but are finite and unequal (900 ≠ 1200), this creates a jump discontinuity. The function literally "jumps" from a revenue of $900,000 to $1,200,000 at the policy change point. Choice A incorrectly calls this removable—removable discontinuities occur when limits are equal but the function value differs, allowing you to "remove" the discontinuity by redefining the function at that point. Here, the unequal one-sided limits make removal impossible. Choice B misses the key point: having one-sided limits doesn't guarantee continuity. Continuity requires the limits to be equal and match the function value. Choice C confuses this with infinite discontinuities, where limits approach ±∞. Both our limits are finite values. Study tip: Jump discontinuities occur when one-sided limits exist, are finite, but unequal—visualize the graph literally "jumping" between two different y-values.

Question 15

A manufacturer's marginal cost changes based on production volume. The marginal cost function, C′(q)C'(q), for producing qq units is given. The company's fixed costs are $1,000, meaning $C(0)=1000.Thetotalcostfunction. The total cost function C(q)$ must be continuous.

The marginal cost is:

Given that the total cost function C(q)C(q) is continuous and C(0)=1000C(0) = 1000, which of the following is the correct total cost function?

  1. C(q)={30q−0.05q2+1000if 0≤q≤10015q+1000if q>100C(q) = \begin{cases} 30q - 0.05q^2 + 1000 & \text{if } 0 \le q \le 100 \\ 15q + 1000 & \text{if } q > 100 \end{cases}
  2. C(q)={30q−0.05q2+1000if 0≤q≤10015q+3500if q>100C(q) = \begin{cases} 30q - 0.05q^2 + 1000 & \text{if } 0 \le q \le 100 \\ 15q + 3500 & \text{if } q > 100 \end{cases}
  3. C(q)={30q−0.05q2+1000if 0≤q≤10015q+2000if q>100C(q) = \begin{cases} 30q - 0.05q^2 + 1000 & \text{if } 0 \le q \le 100 \\ 15q + 2000 & \text{if } q > 100 \end{cases} (correct answer)
  4. C(q)={30−0.1q+1000if 0≤q≤10015q+1000if q>100C(q) = \begin{cases} 30 - 0.1q + 1000 & \text{if } 0 \le q \le 100 \\ 15q + 1000 & \text{if } q > 100 \end{cases}
Explanation: First, find the total cost function for the first piece by integrating C′(q)C'(q): C1(q)=∫(30−0.1q)dq=30q−0.05q2+KC_1(q) = \int (30 - 0.1q) dq = 30q - 0.05q^2 + K. Using C(0)=1000C(0)=1000, we find K=1000K=1000. So, C1(q)=30q−0.05q2+1000C_1(q) = 30q - 0.05q^2 + 1000. Next, find the cost at the boundary q=100q=100: C1(100)=30(100)−0.05(100)2+1000=3000−500+1000=3500C_1(100) = 30(100) - 0.05(100)^2 + 1000 = 3000 - 500 + 1000 = 3500. Now, integrate the second piece: C2(q)=∫15dq=15q+K2C_2(q) = \int 15 dq = 15q + K_2. For continuity, C2(100)C_2(100) must equal C1(100)C_1(100). So, 15(100)+K2=350015(100) + K_2 = 3500, which simplifies to 1500+K2=35001500 + K_2 = 3500. Solving gives K2=2000K_2 = 2000. Therefore, the second piece is 15q+200015q + 2000.

Question 16

Consider the piecewise function h(x)={2x2−8x−2if x≠2mif x=2h(x) = \begin{cases} \frac{2x^2 - 8}{x - 2} & \text{if } x ≠ 2 \\ m & \text{if } x = 2 \end{cases} Which statement about the continuity of h(x)h(x) at x=2x = 2 is correct?

  1. h(x)h(x) is continuous at x=2x = 2 only when m=8m = 8 (correct answer)
  2. h(x)h(x) is continuous at x=2x = 2 for any value of mm
  3. h(x)h(x) is continuous at x=2x = 2 only when m=4m = 4
  4. h(x)h(x) cannot be made continuous at x=2x = 2 for any value of mm
Explanation: When you encounter a piecewise function with a potential discontinuity, you need to check three conditions for continuity: the function must be defined at the point, the limit must exist, and the limit must equal the function value. First, let's find the limit as xx approaches 2. The expression 2x2−8x−2\frac{2x^2 - 8}{x - 2} appears to give 00\frac{0}{0} when x=2x = 2, so we need to simplify. Factor the numerator: 2x2−8=2(x2−4)=2(x−2)(x+2)2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2). Therefore: 2x2−8x−2=2(x−2)(x+2)x−2=2(x+2)\frac{2x^2 - 8}{x - 2} = \frac{2(x - 2)(x + 2)}{x - 2} = 2(x + 2) for x≠2x ≠ 2. Now lim⁡x→2h(x)=lim⁡x→22(x+2)=2(2+2)=8\lim_{x \to 2} h(x) = \lim_{x \to 2} 2(x + 2) = 2(2 + 2) = 8. For continuity at x=2x = 2, we need h(2)=lim⁡x→2h(x)h(2) = \lim_{x \to 2} h(x), which means m=8m = 8. Looking at the choices: A is correct because continuity requires m=8m = 8. B is wrong because continuity depends on a specific value of mm, not any value. C incorrectly claims m=4m = 4, but our limit calculation shows the limit equals 8, not 4. D is wrong because we can indeed make the function continuous by choosing the right value of mm. Study tip: For removable discontinuities (holes in graphs), always factor and cancel common terms to find the limit, then set the function value equal to that limit for continuity.

Question 17

A logistics company's shipping cost function transitions between two different rate structures at 50 pounds: S(w)={2w+15if 0≤w<501.5w+cif w≥50S(w) = \begin{cases} 2w + 15 & \text{if } 0 ≤ w < 50 \\ 1.5w + c & \text{if } w ≥ 50 \end{cases} where ww is weight in pounds. If the function is continuous at w=50w = 50, but a customer ships exactly 50 pounds, which statement correctly describes the situation?

  1. The cost is $115 and represents a jump discontinuity
  2. The cost is $100 and the function is continuous but not differentiable
  3. The cost is $115 and the function is continuous but not differentiable (correct answer)
  4. The cost is $100 and the function is both continuous and differentiable
Explanation: First, find cc for continuity. lim⁡w→50−S(w)=2(50)+15=115\lim_{w \to 50^-} S(w) = 2(50) + 15 = 115. For continuity: S(50)=1.5(50)+c=75+cS(50) = 1.5(50) + c = 75 + c. Setting 115=75+c115 = 75 + c gives c=40c = 40. So S(50)=1.5(50)+40=115S(50) = 1.5(50) + 40 = 115. The cost for 50 pounds is $115. Check differentiability: Left derivative: $S′(50−)=2S'(50^-) = 2 .Rightderivative:. Right derivative: S′(50+)=1.5S'(50^+) = 1.5 .Since. Since 2≠1.52 ≠ 1.5 ,thefunctionisnotdifferentiableat, the function is not differentiable at w=50w = 50 $. The function is continuous (we made it so) but not differentiable due to the change in slope. Choice A is wrong about discontinuity. Choice B has the wrong cost. Choice D claims differentiability, which is false.