All flashcards
Flashcard 1: What does the second derivative indicate about the concavity of f(x)?
Answer: Concave up if f′′(x)>0, concave down if f′′(x)<0. Second derivative sign determines whether graph curves up or down.
Flashcard 2: Identify the economic meaning of f′(x) if f(x) is profit.
Answer: Marginal profit. Derivative shows additional profit from one more unit.
Flashcard 3: What does f′(x)>0 imply about a company's profit function f(x)?
Answer: Profit is increasing. Positive derivative means profit grows with more production.
Flashcard 4: What does f′(x)<0 imply about a company's profit function f(x)?
Answer: Profit is decreasing. Negative derivative means profit falls with more production.
Flashcard 5: Identify the units of f′(x) if f(x) is in dollars and x is in units.
Answer: Dollars per unit. Units follow quotient rule: dependent variable over independent.
Flashcard 6: What does the concavity of f(x) tell us about f′(x)?
Answer: Concave up: f′(x) is increasing; Concave down: f′(x) is decreasing. Concavity describes whether first derivative is rising or falling.
Flashcard 7: If f(x) is a cost function, what does a positive f′′(x) imply?
Answer: Increasing marginal costs. Positive second derivative means costs accelerate upward.
Flashcard 8: What does a change in sign of f′(x) indicate?
Answer: A possible local extremum. Sign change in first derivative indicates peak or valley.
Flashcard 9: What does the derivative of a position function represent?
Answer: The velocity of the object. Position derivative gives instantaneous rate of change in location.
Flashcard 10: What does f′(t) represent if f(t) is the height of a ball?
Answer: The velocity of the ball. Height derivative gives upward or downward speed.
Flashcard 11: What is the significance of the derivative in optimization problems?
Answer: Used to find maximum and minimum values. Setting derivative to zero finds optimal solutions.
Flashcard 12: What does f′(x)=2x signify for f(x)=x2 at x=3?
Answer: The slope of the tangent line is 6. Substituting x=3 into derivative formula gives slope.
Flashcard 13: Determine the meaning of f′′(a)<0 at a critical point.
Answer: Local maximum at x=a. Negative second derivative at critical point confirms maximum.
Flashcard 14: What does f′(x)>0 indicate about the function f(x)?
Answer: f(x) is increasing. Positive derivative means function values rise as x increases.
Flashcard 15: What is the economic interpretation of f′(x) if f(x) is revenue?
Answer: The marginal revenue. Derivative shows additional revenue from one more unit sold.
Flashcard 16: What does f′(x)=0 typically indicate about f(x)?
Answer: A potential local maximum or minimum. Zero derivative indicates horizontal tangent line at that point.
Flashcard 17: If f(x) is distance, what does f′(x) represent?
Answer: The velocity. Distance derivative gives rate of distance change over time.
Flashcard 18: If f(x) is a cost function, what does a negative f′′(x) imply?
Answer: Decreasing marginal costs. Negative second derivative means cost growth slows down.
Flashcard 19: Find the meaning of f′(a) if f(x) is a cost function.
Answer: The marginal cost at x=a. Derivative of cost function shows additional cost per unit.
Flashcard 20: Determine the meaning of f′′(a)>0 at a critical point.
Answer: Local minimum at x=a. Positive second derivative at critical point confirms minimum.
Flashcard 21: What does f′(x)<0 indicate about the function f(x)?
Answer: f(x) is decreasing. Negative derivative means function values fall as x increases.
Flashcard 22: If f(x) is a temperature function, what does f′(x) represent?
Answer: The rate of change of temperature. Shows how temperature changes with respect to input variable.
Flashcard 23: What is the second derivative of a position function with respect to time?
Answer: The acceleration. Second derivative of position gives rate of velocity change.
Flashcard 24: What is the derivative of a velocity function with respect to time?
Answer: The acceleration. Velocity derivative gives rate of change of velocity over time.
Flashcard 25: What is the relationship between speed and the derivative of position?
Answer: Speed is the absolute value of the velocity. Speed is magnitude of velocity, ignoring direction.
Flashcard 26: If f(x) is a demand curve, what does f′(x) indicate?
Answer: The rate of change of demand. Shows how demand responds to price changes.
Flashcard 27: What is the derivative of a velocity function with respect to time?
Answer: The acceleration. Velocity derivative gives rate of change of velocity over time.
Flashcard 28: What does f′(x)>0 imply about a company's profit function f(x)?
Answer: Profit is increasing. Positive derivative means profit grows with more production.
Flashcard 29: What does f′′(x)>0 indicate about f(x)?
Answer: f(x) is concave up. Positive second derivative means graph curves upward.
Flashcard 30: What does f′(x)=0 typically indicate about f(x)?
Answer: A potential local maximum or minimum. Zero derivative indicates horizontal tangent line at that point.