MTH100 Midterm Past Papers

Midterm 2

This test has 5 questions on 7 pages, for a total of 40 points.

• Read all the questions carefully before starting to work.
• Q1 and Q2 are short-answer questions; put your answer in the boxes provided.
• All other questions are long-answer; you should give complete arguments and explanations

for all your calculations; answers without justifications will not be marked.

• Continue on the back of the previous page if you run out of space.
• Attempt to answer all questions for partial credit.
• This is a closed-book examination. None of the following are allowed: documents,

cheat sheets or electronic devices of any kind (including calculators, cell phones, etc.)

First Name: Last Name:

Student-No: Section:

Signature:

Question: Total

Points:

Score:

Student Conduct during Examinations

1. Each examination candidate must be prepared to produce, upon the request of the invigilator or examiner, his or her UBCcard for identification.

2. Examination candidates are not permitted to ask questions of the examiners or invigilators, except in cases of supposed errors or ambiguities in examination questions, illegible or missing material, or the like.

3. No examination 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. Should the examination run forty-five (45) minutes or less, no examination candidate shall be permitted to enter the examination room once the examination has begun.

4. Examination candidates must conduct themselves honestly and in accordance with established rules for a given examination, which will be articulated by the examiner or invigilator prior to the examination commencing. Should dishonest behaviour be observed by the examiner(s) or invigilator(s), pleas of accident or forgetfulness shall not be received.

5. Examination candidates suspected of any of the following, or any other similar practices, may be immediately dismissed from the examination by the examiner/invigilator, and may be subject to disciplinary action:

(i) speaking or communicating with other examination candidates, unless otherwise authorized; (ii) purposely exposing written papers to the view of other examination candidates or imaging devices;

(iii) purposely viewing the written papers of other examination candidates;

(iv) using or having visible at the place of writing any books, papers or other memory aid devices other than those authorized by the examiner(s); and,

(v) using or operating electronic devices including but not limited to telephones, calculators, computers, or similar devices other than those authorized by the examiner(s) (electronic devices other than those authorized by the examiner(s) must be completely powered down if present at the place of writing).

6. Examination candidates must not destroy or damage any examination material, must hand in all examination papers, and must not take any examination material from the examination room without permission of the examiner or invigilator.

7. Notwithstanding the above, for any mode of examination that does not fall into the traditional, paper-based method, examination candidates shall adhere to any special rules for conduct as established and articulated by the examiner.

8. Examination candidates must follow any additional examination rules or directions communicated by the examiner(s) or invigilator(s). Short-Answer Questions. Questions 1 and 2 are short-answer questions. Put your answer

in the box provided. Full marks will be given for a correct answer placed in the box. Show

your work also, for part marks. Each part is worth 2 to 3 marks, and not all parts are of equal

difficulty. Simplify your answers as much as possible in Questions 1 and 2.

1. (a) 2 marks Let y = 101−x2 . Find dy dx .

Answer: dy dx = −2 ln(10)x · 101−x2

Solution: Use the chain rule

dy dx = ln(10) · 101−x2 · (−2x)

(b) 2 marks Evaluate tan−1 � tan � 11π

Answer: − π

Solution: As done in class, one way to solve this problem is to draw a unit circle

and mark down the angle of 11π in the second quadrant. Shifting the angle back

into the range of − π 2 , π yields − π 4 because tan x is negative in the second and

fourth quadrant.

(c) 2 marks y = (sin x)ln x. Find dy dx .

Answer: y′ = ln(sin x) x + ln x cos x sin x y

Solution: Use logarithmic differentiation.

ln y = ln x · ln(sin x)

y y′ = 1 x · ln(sin x) + ln x · sin x · cos x

Multiplying through by y gives the answer in the box. 2. (a) 3 marks Simplify sin(sec−1 x). Note: sec−1 x is the inverse function of sec x.

Answer: x2−1 x

Solution: Let y = sec−1 x then sec y = x or equivalently cos y = x . Drawing a

triangle with angle y as done in class gives sin y = x2−1 x

(b) 3 marks Find all x-coordinates at which the tangent line to the curve x2y2 = 16 has slope −1.

Answer: x = ±2

Solution: Implicit differentiation gives

2xy2 + x2 · 2yy′ = 0.

Plugging in y′ = −1 and factoring gives

2xy(y − x) = 0.

This implies x = y since there are no points on the curve with x = 0 or y = 0. The curve’s equation thus becomes x4 = 16 or x = ±2.

(c) 3 marks Which of the following functions satisfies dp dt = p3?

A) p(t) = 1

Q4. t4

1. p(t) = − 1

t+3

C) p(t) = √1 − 2t

D) None of these.

Answer: C)

Solution: By differentiating each expression for p(t) separately, we conclude that

C) is the correct answer, i.e,

dp dt = − 2(1 − 2t)3/2 · (−2) = (1 − 2t)3/2 = √1 − 2t 3 = p3