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Be sure this exam has 12 pages including this cover The University of British Columbia Sessional Examinations – December 2009 Mathematics 102 — Differential Calculus with applications to Life Sciences Closed book examination Time: 2 12 hours
Name: Signature:
Student Number: Section:
No calculators, notes or other aids. You must show your work to obtain full credit. Express answers in terms of fractions or constants such as
3 or ln(4) rather than decimals. The last page contains some helpful formulae.
Rules governing examinations
- Each candidate should be prepared to produce his or her library/AMS card upon request.
- Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or electronic devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
- Smoking is not permitted during examinations.
Problem total possible score
- 18
total 100
Section 101 (MWF 10:00): J. Macdonald Section 102 (MWF 8:00): J. Allard Sections 103 (MWF 11:00) and 104 (MWF 1:00): Y-X. Li Section 105 (TuTh 9:30): R. Israel Section 106 (MWF 8:00): A. Duncan
- For this short-answer question, only the answers (placed in the boxes) will be marked.
(3 points) (a)Find the average rate of change of f (x) = x^2 /(1 + x^2 ) on the interval 0 ≤ x ≤ 2.
(3 points) (b) Find the global (absolute) minimum of the function f (x) = x^2 e−x^ on the interval − 3 ≤ x ≤ 3.
(3 points) (c)Find the y-intercept of the tangent line at the point (1, −1) to the graph of the function y(x) defined implicitly by 2x^3 + y^3 + xy = 0.
(3 points) (d) Which of the following differential equations has y(t) = 4e^2 t^ as a solution?
(i) y′^ = − 4 y (ii) y′^ = − 2 y (iii) y′′^ = 4y (iv) y′′^ = − 4 y
(3 points) (e)Given the curve y = V x^2 /(x^2 + K^2 ), find new variables X = X(x) and Y = Y (y) so that the graph of Y as a function of X is a straight line.
(3 points) (f)Find an approximate value for cos(π/3 +
3 /100), based on known values of functions at x = π/3.
d
y
x
- A navy vessel is located at 25 km from the enemy shoreline where an enemy helicopter base is located. The vessel is moving toward the base at a constant speed of vv = 30 km/h. At this time (referred to as t = 0 for convenience), its early warning radar detected that an enemy helicopter is located on the ground in the base and is taking off vertically with a constant speed vh = 40 km/h (see figure). To determine the optimal time to fire a laser beam to shoot down the helicopter, the following information must be provided to the commander.
(6 points) (a)At what time will the distance between the vessel and the helicopter be shortest? What is this distance? Assume that the speeds and directions of both the vessel and helicopter are constant.
t = hours, distance = km
(6 points) (b) The laser is kept pointing directly at the helicopter at all times. At the time found in (a), what is the rate of change of the angle θ at which the laser is aimed (see figure)?
dθ dt
= radians/h
(7 points) 4. Find all points on the graph of y^2 = 4x + 12 where the tangent line to that graph is parallel to y = x + 1.
- Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference in temperature between the environment and the object. In a room at 20◦^ C, suppose a cup of coffee cools from 80◦^ C to 60◦^ C in 5 minutes.
(5 points) (a)How long does it take to cool from 80◦^ C to 30◦^ C?
It takes minutes.
(5 points) (b) What is its temperature 15 minutes after it was at 75◦^ C?
Temperature = ◦^ C.
- Consider the function f (x) = 10 ln(x^2 + 1) − x^2
(7 points) (a)Find all the critical points of f (x) and classify each one as a local maximum, local minimum or neither.
(4 points) (b) Does this function have a global maximum on the interval (−∞, ∞)? Does it have a global minimum there? Circle Yes or No in each case.
Global maximum? Yes No Global minimum? Yes No
(4 points) (c)f (1) is close to 6. Suppose we use Newton’s method to find the solution of f (x) = 6, starting with the initial guess x 0 = 1. Find x 1.
x 1 =
(6 points) 9. Two quantities x and y, both depending on time t, are related by the equation x^2 − y^3 = 1. If dx/dt = 4, what is dy/dt at the moment when x = 3 and y = 2?
dy dt
- Consider the differential equation dy dx
y^2 − 1 y^2 + 1
(4 points) (a)Find all stable and unstable steady states (equilibrium values), if any, for this differential equation.
Stable : Unstable:
(3 points) (b) What value will the solution of this differential equation with initial condition y(0) = 0 approach as t increases?
t→lim+∞ y(t) =
