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This exam consisted of 13 numbered pages.

ERASMUS UNIVERSITY ROTTERDAM
Erasmus School of Economics
International Bachelor Economics & Business Economics, Bachelor 1
Exam Mathematics 1 (FEB11003X-10)

Monday 11 July 2011, 9:30 – 12:30 hrs

Instructions ESE • You are not allowed to use a calculator. • You are not allowed to use a programmable calculator. • You are not allowed to use notes (except a cheat sheet, see below). • You are not allowed to use books. • You are not allowed to use a dictionary. • You are allowed to take the examination papers with you (but you have to turn in the separate answer sheet).

Additional information
You are allowed to use a cheat sheet, for which the following rules apply: • Two-sided A4 • Your name and student number in the upper right corner
All material that does not satisfy these rules will be taken away from you and may be considered a fraud attempt.

Use the separate answer sheet to indicate your answers.

The exam consists of 19 problems grouped in four parts, each with different types of problems. For the basic problems, multiple choice problems and calculation problems you will score 3, 4 and 5 points, respectively, per correct answer and no points for incorrect answers. For the open problems (8 points per problem) your score will depend on the answer and the calculation. The exam grade is the result from the formula (10 + number of scored points)/10. So 63 scored points result in grade 7.3, since (10+63)/10= 7.3.

Good luck!
Part I: Basic problems
3 points per problem

Problem 1
Suppose [pic] and[pic], can we say y is a function of x? In addition, can we say x is a function of y? Explain your answer.
|Final answer: |
|We can say y is a function of x, but we can’t say x is a function of y. |
|We can rewrite the equation as [pic]so that for each x within its domain, the function only assigns one value to y. However, when we |
|express x in terms of y, it becomes[pic], so that there are 2 values of x associated with each value of y. Therefore, we can say y is a |
|function of x, but we can’t say x is a function of y. |

Problem 2
Suppose the price elasticity of demand for potato is 1.5, if the price of potato changes from 10 to 20, and the original demand for potato is 120, what is the final demand for potato?
|Final answer: |
|The price elasticity of 1.5 means if the price of potato increases by 1%, the demand of it increases by 1.5%, as the price of potato |
|increases from 10 to 20, there is 100% increase of price, so that the demand of potato increases by 150%, which is 180. Therefore, the |
|final demand of potato is 120+180=300. |

Problem 3
Let[pic]be a function that is twice-differentiable on its domain. If the second derivative of a stationary point on the function is zero, is it possible that such stationary point is also an extreme point. If yes, then give an example, otherwise explain your reason.

|Final answer: |
|Yes, an example could be:[pic] |
|When the second derivative of a stationary point is positive, this point is local minimum. |
|When the second derivative of a stationary point is negative, this point is local maximum. |
|However, when the second derivative of a stationary point is zero, this point is undetermined. |
|For the function[pic], [pic],[pic]. Therefore, its stationary point is zero and the second derivative of point zero is also zero. If we |
|draw a graph, we can see zero is a minimal point (both local and global) on the graph. |

Problem 4
If a function is continuous and twice-differentiable on its domain, and the global maximum point exists. Is it true that the second derivative of the global maximum point must be non-positive? If yes, then give reasons, otherwise give a counterexample.
|Final answer: |
|A counter example is [pic] |
|Note that for this function, the global maximum point is x=-5, while the second derivative at this point is 10. |
|Generally speaking, for twice-differentiable function, the global maximum point could either be the stationary point or the boundary |
|point. If it is stationary point, then the second derivative must be non-positive. However, for the boundary point, there is not |
|restriction on the second derivatives. |

Part II: Multiple choice problems
4 points per problem

Problem 5
Solve [pic]for x. Which of the following answers is correct?

A. [pic] or [pic] B. [pic] C. [pic] D. [pic] or [pic]
|Final answer: |
|C |
|[pic] |

Problem 6
The derivative of the function [pic] is

A. [pic] B. [pic]
C. [pic] D. [pic]

|Final answer: |
|D |
|[pic] |

Problem 7
Calculate the derivative of[pic]. The answer is:

A. [pic] B. [pic] C. [pic] D. [pic]

|Final answer: |
|C |
|[pic] |

Problem 8
Let[pic]. The interval on which this function is increasing is

A. [pic] B. The whole domain C. Does not exist D. [pic]
|Final answer: |
|C |
|[pic] |
|The function is increasing when [pic], so that |
|[pic], so that such interval doesn’t exist. |

Problem 9
Let [pic]. Calculate the partial elasticity of f with respect to y. Which of the following answers is correct?

A. [pic] B. [pic]
C. [pic] D. [pic]
|Final answer: |
|B |
|[pic] |
|The partial elasticity is: [pic] |

Problem 10
Consider the function[pic]. The degree of homogeneity is

A. 4 B. 8 C. 12 D. 16
|Final answer: |
|B |
|[pic] |
|Therefore, the degree of homogeneity is 8. |

Part III: Calculation problems
5 points per problem

Problem 11
Determine the remainder of the polynomial division[pic].
|Final answer: |
|-62 |
|[pic] |
|Therefore, the remainder of the polynomial division is -62. |

Problem 12
Consider the function [pic]with domain [10, +∞). Determine the inverse and its domain. If the inverse does not exist, report ‘no inverse’.
|Final answer: |
|The inverse function is [pic], and the domain is [pic] |
|[pic] |
|As the domain of the function is [10, ∞], x must be positive, therefore |
|[pic] |
|The domain of the inverse is the range of the origin. |
|As [pic], [pic], therefore, the range of the origin function is: |
|[pic] |

Problem 13
Consider the equation[pic]. Use implicit differentiation to compute the value of[pic]in the point [pic]

|Final answer: |
|0.4 |
|First, by differentiating both sides of equation w.r.t. x, we get |
|[pic] |
|Therefore, in the point [pic], [pic] |

Problem 14
Find the equation for the tangent to the graph of [pic] at the point (1, 26)
|Final answer: |
|[pic] |
|The general equation of the tangent line at a certain point [pic] is |
|[pic] where [pic] is the slope or the first derivative at that point. |
|[pic], therefore, the slope at (1,26) is -23 and the equation of the function is |
|[pic] |

Problem 15
Find the maximum and minimum value of the function [pic] on the interval [1, 4].
|Final answer: |
|Stationary points |
|[pic](x=0 is out of the interval [1, 4]) |
|Endpoints [pic], [pic] |
|[pic], [pic], [pic] |
|So the maximum value is 0 at[pic], the minimum value is -27 at[pic]. |

Problem 16
Let[pic]. Find all first- and second-order partial derivatives.
|Final Answer: |
|[pic] |

Part IV: Open problems
8 points per problem

Problem 17 (2+2+2+2=8 points)
Consider the function[pic]

a) Determine the first derivative and the stationary point(s) of f.
|Final answer: |
|[pic] |

b) Classify the stationary point(s) using the first derivative (local minimum, local maximum or not an extreme point).
|Final answer: |
|[pic] |
|So the sign diagram of [pic] is: |
|[pic] |
|Therefore, both [pic]and [pic] are local minimum points, and [pic] is a local maximum point. |

c) Determine the second derivative [pic]and all value(s) of x for which[pic].
|Final answer: |
|[pic] |

d) Determine on which interval(s) the function f is concave.(It may be useful to know [pic])
|Final answer: |
|[pic] |
|The function is concave when its second derivative is non-positive, so we could pick the following 3 points to determine the signs of |
|[pic] |
|[pic] |
|So [pic] , i.e. [pic] is concave on the intervals [pic]. |

Problem 18 (2+2+1+3=8 points)
Consider the function [pic]

a) Calculate all first order partial derivatives.
|Final answer: |
|[pic] |

b) Calculate all second order partial derivatives.
|Final answer: |
|[pic] |

c) Find all stationary points of f.
|Final answer: |
|[pic] |
|Substitute [pic] into [pic] |
|Substitute [pic] into [pic] |
|Therefore, the corresponding y is 0 and 128. |
|So there are 2 stationary points (0, 0), (8, 128). |

d) Use the second-derivative test to classify all the stationary points (local minimum, local maximum, saddle point or inconclusive).
|Final answer: |
|Let [pic] where |
|[pic] |
|Stationary points |
|A |
|B |
|C |
|[pic] |
|Classification |
| |
|(0, 0) |
|0 |
|0 |
|-12 |
|=0 |
|Inconclusive |
| |
|(8, 128) |
|-2048 |
|128 |
|4 |
|

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