1If a function represents the population at time , what represents the instantaneous rate of growth of the population?
A.
B.
C.
D.
Correct Answer:
Explanation:The instantaneous rate of growth is given by the first derivative of the function with respect to time, or .
Incorrect! Try again.
2Given the total cost function , what is the Marginal Cost (MC)?
A.
B.
C.
D.
Correct Answer:
Explanation:Marginal Cost is the derivative of the Total Cost function. .
Incorrect! Try again.
3What is the formula for Average Cost (AC) given a total cost function ?
A.
B.
C.
D.
Correct Answer:
Explanation:Average Cost is defined as the total cost divided by the number of units produced ().
Incorrect! Try again.
4If the demand function is given by , what is the Total Revenue (TR) function?
A.
B.
C.
D.
Correct Answer:
Explanation:Total Revenue is Price multiplied by Quantity (). Therefore, .
Incorrect! Try again.
5For a profit function , profit is maximized when:
A.Marginal Revenue > Marginal Cost
B.Marginal Revenue < Marginal Cost
C.Marginal Revenue = Marginal Cost
D.Average Cost = Average Revenue
Correct Answer: Marginal Revenue = Marginal Cost
Explanation:Profit maximization occurs at the quantity where the additional revenue from selling one more unit equals the additional cost of producing it ().
Incorrect! Try again.
6Find the critical points of the function .
A.
B. and
C.
D. only
Correct Answer: and
Explanation:Find and set it to 0. . Solving gives , so .
Incorrect! Try again.
7Which of the following conditions confirms a local maximum at a critical point ?
A.
B.
C.
D.
Correct Answer:
Explanation:By the Second Derivative Test, if the second derivative is negative () at a critical point, the function is concave down, indicating a local maximum.
Incorrect! Try again.
8If , what is the partial derivative ?
A.
B.
C.
D.
Correct Answer:
Explanation:When differentiating partially with respect to , treat as a constant. The derivative of is and the derivative of (constant) is 0.
Incorrect! Try again.
9If , what is ?
A.
B.
C.
D.
Correct Answer:
Explanation:Treat as constant. The derivative of wrt is . The derivative of wrt is .
Incorrect! Try again.
10The Marginal Revenue (MR) is the slope of which curve?
A.Demand Curve
B.Total Revenue Curve
C.Average Cost Curve
D.Total Cost Curve
Correct Answer: Total Revenue Curve
Explanation:Marginal Revenue is the derivative of Total Revenue (), which represents the slope of the Total Revenue curve.
Incorrect! Try again.
11Given , what is the behavior of the Average Fixed Cost (AFC) as increases?
A.It increases
B.It remains constant
C.It decreases
D.It becomes zero immediately
Correct Answer: It decreases
Explanation:Fixed Cost is 5000. AFC = . As increases, the denominator increases, causing the fraction to decrease.
Incorrect! Try again.
12For a function , the notation represents:
A.
B.
C.
D.
Correct Answer:
Explanation: means first differentiate with respect to , then differentiate the result with respect to .
Incorrect! Try again.
13According to Young's Theorem (or Schwarz's Theorem), for a continuous function , which equality holds?
A.
B.
C.
D.
Correct Answer:
Explanation:The mixed partial derivatives are equal if the function and its partial derivatives are continuous.
Incorrect! Try again.
14Find the critical point of the function .
A.
B.
C.
D.
Correct Answer:
Explanation:Set partials to 0: . .
Incorrect! Try again.
15In the context of two variables, is used to test for extrema. What does represent?
A.
B.
C.
D.
Correct Answer:
Explanation:In the Hessian determinant notation for the second partial derivative test, , , and .
Incorrect! Try again.
16If and at a critical point , then the point is a:
A.Local Maximum
B.Local Minimum
C.Saddle Point
D.Point of Inflection
Correct Answer: Local Minimum
Explanation:If the discriminant and the leading second partial derivative , the surface is concave up in all directions, indicating a relative minimum.
Incorrect! Try again.
17If at a critical point, the point is a:
A.Local Maximum
B.Local Minimum
C.Saddle Point
D.Test Inconclusive
Correct Answer: Saddle Point
Explanation:A negative discriminant indicates that the function curves up in one direction and down in another, forming a saddle point.
Incorrect! Try again.
18Find the partial derivative for .
A.
B.
C.
D.
Correct Answer:
Explanation:Using the chain rule, derivative of is . Here . Partial of wrt is . So, .
Incorrect! Try again.
19A cost function is . At what production level is the Average Cost minimized?
A.
B.
C.
D.
Correct Answer:
Explanation:. Set : .
Incorrect! Try again.
20If the total revenue function is , at what value of is revenue maximized?
A.40
B.20
C.10
D.
Correct Answer: 20
Explanation:Find and set to 0. . .
Incorrect! Try again.
21The Relative Growth Rate of a function is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:The relative (or proportional) rate of change is the derivative divided by the function itself, which is also the derivative of .
Incorrect! Try again.
22Given , what is ?
A.
B.
C.
D.
Correct Answer:
Explanation:Chain rule: derivative of is . Here . Partial of wrt is . Result: .
Incorrect! Try again.
23In economics, the partial derivative of a utility function with respect to good is known as:
A.Marginal Cost
B.Marginal Revenue
C.Marginal Utility of x
D.Average Utility
Correct Answer: Marginal Utility of x
Explanation:Marginal utility is the additional satisfaction gained from consuming one more unit of a good, mathematically represented by the partial derivative of the utility function.
Incorrect! Try again.
24If , the Marginal Cost is:
A.Increasing
B.Decreasing
C.Constant
D.Zero
Correct Answer: Constant
Explanation:. The value 3 is a constant.
Incorrect! Try again.
25For a production function where is labor and is capital, the Marginal Productivity of Capital is:
A.
B.
C.
D.
Correct Answer:
Explanation:Marginal productivity of a factor is the partial derivative of production with respect to that specific factor.
Incorrect! Try again.
26What is the critical point for ?
A.
B.
C.
D.No critical point
Correct Answer:
Explanation:. Set , so , which implies .
Incorrect! Try again.
27If the demand function is , calculate elasticity of demand when . (Use point elasticity formula )
A.-1
B.-0.5
C.-2
D.1
Correct Answer: -1
Explanation:. If , . .
Incorrect! Try again.
28Calculate for .
A.
B.
C.
D.
Correct Answer:
Explanation:. .
Incorrect! Try again.
29Which of the following functions represents a constant returns to scale Cobb-Douglas production function?
A.
B.
C.
D.
Correct Answer:
Explanation:In Cobb-Douglas , constant returns to scale implies . Here .
Incorrect! Try again.
30If Marginal Revenue is 20 and Marginal Cost is 15, to maximize profit, the firm should:
A.Reduce output
B.Increase output
C.Maintain current output
D.Shut down
Correct Answer: Increase output
Explanation:Since , producing an additional unit adds more to revenue than it does to cost, increasing total profit. The firm should produce more.
Incorrect! Try again.
31Determine the stationary points of .
A.
B.
C.
D.
Correct Answer:
Explanation:. . Multiply first eq by 2: . Subtract second eq: . If , .
Incorrect! Try again.
32The second derivative of is:
A.
B.
C.
D.
Correct Answer:
Explanation:. .
Incorrect! Try again.
33A rectangle has a fixed perimeter of 20. What dimensions maximize the area?
A.5 by 5
B.4 by 6
C.2 by 8
D.1 by 9
Correct Answer: 5 by 5
Explanation:. Area . . So .
Incorrect! Try again.
34For , calculate the mixed partial derivative .
A.
B.
C.
D.
Correct Answer:
Explanation:. Differentiate this with respect to : .
Incorrect! Try again.
35If and , what is the profit function ?
A.
B.
C.
D.
Correct Answer:
Explanation:Profit .
Incorrect! Try again.
36Find the minimum value of .
A.7
B.3
C.
D.4
Correct Answer: 3
Explanation:. Substitute into : .
Incorrect! Try again.
37What is if ?
A.7
B.5
C.3
D.4
Correct Answer: 7
Explanation:. Evaluate at : .
Incorrect! Try again.
38If , what is the Total Cost ?
A.
B.
C.
D.
Correct Answer:
Explanation:.
Incorrect! Try again.
39If , calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:. .
Incorrect! Try again.
40The point where the Total Revenue curve and the Total Cost curve intersect is called the:
A.Break-even Point
B.Shut-down Point
C.Maximum Profit Point
D.Inflection Point
Correct Answer: Break-even Point
Explanation:At the break-even point, Total Revenue equals Total Cost, meaning Profit is zero.
Incorrect! Try again.
41If (a constant), what are and ?
A.Both are 0
B.1 and 1
C. and
D.Undefined
Correct Answer: Both are 0
Explanation:The derivative of a constant is always 0, regardless of the variable.
Incorrect! Try again.
42For , the point is a:
A.Point of Inflection
B.Local Maximum
C.Local Minimum
D.Discontinuity
Correct Answer: Point of Inflection
Explanation: (0 at ). (0 at ). The concavity changes from negative () to positive (). It is a saddle/inflection point, not a max/min.
Incorrect! Try again.
43If the marginal cost function is , find the variable cost of producing 2 units.
A.8
B.6
C.12
D.4
Correct Answer: 8
Explanation:Variable Cost is the integral of Marginal Cost. . At , .
Incorrect! Try again.
44If , what is ?
A.
B.
C.
D.
Correct Answer:
Explanation:. Then differentiate wrt : .
Incorrect! Try again.
45Which variable is held constant when calculating for ?
A.
B.
C.
D.None
Correct Answer:
Explanation:In partial differentiation with respect to , all other independent variables (like ) are treated as constants.
Incorrect! Try again.
46The slope of the tangent to the curve at represents the:
A.Instantaneous rate of change at x=3
B.Average rate of change
C.Total Area under the curve
D.The y-intercept
Correct Answer: Instantaneous rate of change at x=3
Explanation:The derivative evaluated at a point gives the slope of the tangent, which is the instantaneous rate of change.
Incorrect! Try again.
47For a two-variable function, if at a critical point, the Second Derivative Test is:
A.Inconclusive
B.Indicating a Maximum
C.Indicating a Minimum
D.Indicating a Saddle Point
Correct Answer: Inconclusive
Explanation:If the discriminant (Hessian determinant) is zero, the test fails, and other methods must be used to classify the critical point.
Incorrect! Try again.
48If , what is the growth rate (instantaneous) at ?
A.0.1
B.10
C.1
D.
Correct Answer: 0.1
Explanation:. At , .
Incorrect! Try again.
49Given , find the marginal cost at .
A.8
B.10
C.9
D.12
Correct Answer: 8
Explanation:. At , .
Incorrect! Try again.
50Which theorem allows us to swap the order of partial differentiation () under continuity conditions?