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Unit 4 of 5

Unit 4: Applications

Study guide for CLEP CLEP Calculus — Unit 4: Applications. Practice questions, key concepts, and exam tips.

17

Practice Questions

15

Flashcards

5

Key Topics

Key Concepts to Study

related rates
optimization
area between curves
volumes of revolution
exponential growth and decay modeling

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 17.

Q1MEDIUM

Economics: A company's cost is C(x) = $x^{2}$ + 2x + 5. What is marginal cost when x = 3?

A) 4
B) 6
C) 8
D) 10
E) 12
Show Answer

Answer: C — 8 is correct because marginal cost is derivative of cost, C'(x) = 2x + 2, so C'(3) = 8.

Q2HARD

A function f(x) is approximated by its linearization at x = 2, where f(2) = 3 and f'(2) = 4. Which of the following is a correct statement about the approximation?

A) The approximation is an overestimate of f(x) when x > 2 and an underestimate when x < 2.
B) The approximation is an overestimate of f(x) for all x except x = 2.
C) The approximation is always an overestimate of f(x) for all x.
D) The approximation is always an underestimate of f(x) for all x.
E) The approximation is an underestimate of f(x) when x > 2 and an overestimate when x < 2.
Show Answer

Answer: E — Since f'(2) is positive, the linearization has a positive slope, meaning it underestimates f(x) for x > 2 and overestimates for x < 2.

Q3EASY

A company has a cost function C(x) = $2x^{2}$ + 5x + 100. What is the cost when x = 10?

A) 250
B) 300
C) 350
D) 400
E) 450
Show Answer

Answer: C — 350 is correct because C(10) = 2(10)^2 + 5(10) + 100 = 200 + 50 + 100 = 350.

Q4HARD

A particle moves in the plane according to the parametric equations x(t) = $t^{2}$ and y(t) = 2t. What does the definite integral of the velocity vector represent?

A) The particle's total distance traveled over the interval of time
B) The particle's displacement over the interval of time
C) The particle's speed over the interval of time
D) The particle's acceleration over the interval of time
E) The particle's average velocity over the interval of time
Show Answer

Answer: B — The definite integral of the velocity vector represents the particle's displacement, or net change in position, over the interval of time.

Q5EASY

The half-life of a certain radioactive substance is 10 years. If we start with 100 grams of this substance, how much will be left after 20 years?

A) 60 grams
B) 30 grams
C) 40 grams
D) 50 grams
E) 25 grams
Show Answer

Answer: E — The correct option is "25 grams" because after one half-life, 50 grams remain, and after another half-life, half of 50 grams, which is 25 grams, remains, illustrating the concept of exponential decay. This process occurs due to the substance's radioactive nature, where the amount of substance decreases by half every 10 years.

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Study Tips for Unit 4: Applications

  • Focus on understanding concepts, not memorizing facts — CLEP tests application
  • Practice with timed questions to build exam-day speed
  • Review explanations for wrong answers — they reveal common misconceptions
  • Use flashcards for key terms, practice questions for deeper understanding

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