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Date: October 9, 2014 Time: 6:00 p.m. to 7:30 p.m.

Number of pages: 9 (including cover page) Exam type: Closed book Aids: No calculators or other electronic aids Rules governing formal examinations: Each candidate must be prepared to produce, upon request, a UBC card for identification.

No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.

Candidates suspected of any of the following, or similar, dis honest practices shall be immediately dismissed from the exam ination and shall be liable to disciplinary action: • Having at the place of writing any books, papers or memo randa, calculators, computers, sound or image players/record ers/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; • Speaking or communicating with other candidates; • Purposely exposing written papers to the view of other candi dates or imaging devices. The plea of accident or forgetfulness shall not be received.

Candidates must not destroy or mutilate any examination ma terial; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.

Candidates must follow any additional examination rules or di rections communicated by the instructor or invigilator.

For examiners’ use only Question Mark Possible marks 1 10 2 3 4 5 Total 5 5 4 6 30

Note that your answers must be in “calculator-ready” form, but they do not have to be simplified.

In general, you may use any result proven in class or on assignments. However, in question 2, you must use the definition of limit; and in question 4, you must use the definition of derivative.

This page may be used for rough work. It will not be marked.

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1. (a) [2 marks] Evaluate the limit lim x → 1 1 1 − − x √ x .

(b) [2 marks] Define what it means for a function f to be continuous at a point c .

(c) [2 marks] Suppose that f ( x + h ) − f ( x ) = − 5 hx 2 − 2 hx + 5 h 2 x + 4 h 2 − h 3 for all h . Find f 0 ( x ).

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(d) [2 marks] Let g ( x ) = h ( x ) , x h (2) = − 8 and h 0 (2) = − 7. Find g 0 (2).

(e) [2 marks] (1 , 1).

Find the slope of the line tangent to the curve x 3 + y 3 = 2 xy at the point 4

2.

[5 marks] Let f ( x ) = − 2 if x has a decimal expansion containing the digit “1” 2 otherwise .

For example, lim x → 0 f ( x ) = 2.

f (0 .

3) = 2 and f (0 .

31) = − 2. Prove using the definition of limit that 5

3.

[5 marks] the y Consider the shape pictured below. Prove that there exists a line parallel to -axis that divides the area of the shape exactly in half (that is, such that half of the area is on the left side of the line).

*y x *

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4.

[4 marks] Let f ( x ) be a differentiable function, and let the definition of derivative that g 0 ( x ) = f ( x ) + xf 0 ( x g ( x ) = xf ( x ). Prove using ). (You must use the definition of derivative; no credit will be given for proving the result using the Power Rule, Product Rule, Quotient Rule or Chain Rule.) 7

1 5. Let f ( x ) = x k − 1 where k is a positive integer.

(a) [1 mark] Calculate f 0 ( x ).

(b) [3 marks] Note that f 0 ( x ) is of the form ( x k p 1 ( x ) − 1) 1+1 where p 1 ( x ) is a polynomial.

Assume the n th derivative of f ( x ) may be written in the form ( x k p n ( x ) − 1) n +1 where p n ( x ) is a polynomial. Prove, by differentiating the above, that the ( n derivative of f ( x ) may be written in the form + 1) th where p n +1 ( x ) is another polynomial.

p n +1 ( x ) ( x k − 1) n +2 8

(c) [2 marks] Explain briefly why you may use the answers to part (a) and part (b) to conclude that the 100 th derivative of f ( x ) may be written in the form ( x p 100 ( x ) k − 1) 101 where p 100 ( x ) is a polynomial.

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