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Simon Fraser University, Department of Mathematics, Burnaby Campus
Math 157, Summer 2010
Midterm 2
July 14, 2010, 11:30 a.m.–12:20 p.m., AQ 3182
Last Name (please print):
First Name (please print):
Student Number:
SFU Email (please print): @sfu.ca
Instructor: Roland Wittler
Instructions
- DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO.
- Fill in the above box.
- This exam contains 7 pages with a total of 5 questions. Once the exam begins please check to make sure your exam is complete.
- SHOW ALL YOUR WORK!
- If you run out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues.
- Only scientific, non-programmable calculators with no differentiation and integration capabilities are allowed.
- No book, paper, or device, other than the usual wri- ting instruments, this booklet and an acceptable cal- culator, shall be within reach of a student during the examination.
- During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.
- Try your Best!
Do not write in this table.
Question Marks
1 /
2 /
Total /
Question 1: Fill the gaps in the following statements. (6 marks)
(a) The increment ∆y measures the change in y, whereas
the differential dy measures the change in y.
(b) The graph of the function f (x) = bx^ has the -axis as a
asymptote.
(c) The graph of the function f (x) = logb x is decreasing on its domain if.
(d) If f (x) = eg(x), than f ′(x) =.
(e) Pythagorean Identity: sin^2 θ + cos^2 θ =.
(f) Converting Degrees to Radians: 360◦= radians.
Question 3: The Newton-Raphson Method. (4 marks) Estimate the value of 3
40 using the Newton-Raphson method for the function f (x) = x^3 −40.
(a) Give the formula for xn− 1 with respect to xn in its explicit form (without f or f ′).
(b) For the initial guess x 1 = 3, compute x 2 , x 3 and x 4. Give the values rounded to 5 decimals, but calculate with highest possible accuracy.
Question 4: Exponential Functions as Mathematical Models. (6 marks) To brew a pot of tea, boiling water is filled into a pot containing a few tea bags. The tempe- rature of the cooling water can be described by the following exponential decay function:
T (t) = A e−kt^ + 20 ,
where A and k are constants, the temperature T is measured in ◦C, and the elapsed time t is measured in minutes.
(a) Let t = 0 correspond to the time point where the boiling water (100◦C) is filled into the pot. Compute the constant A.
(b) When the tea bags are removed after 3 minutes, the temperature is 75◦C. Compute the decay constant k.
(c) Compute the time t at which the tea is cooled down to a comfortable temperature of 40◦C.
(c) g(θ) = cos (θ) eθ ,
dg dθ
(d) 2x^3 + 4y^2 = 5 ,
dy dx
