Definition - Differential Calculus - Exam, Exams of Calculus

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Marks

[8] 1. Using the definition of derivative, find f ′(2) where f (x) =

1 + x^2

[12] 2. Use implicit differentiation to find the points on the curve

3 y^3 − 2 x^3 − 6 x^2 y + 5y = 0

where the tangent line to the curve is horizontal. Write your answers in the box in the form (x, y).

Points:

[6] 4. Suppose the number of bacteria in a colony doubles every 20 minutes. If there are 2^39 cells (about 5. 5 × 1011 ) after 12 hours, how many were there at the beginning?

Initial number =

[6] 5. The tide on a certain shore on the planet Outer Thebulon IV has a period of 36.5 hours, and the high tide level is 8 m above the low tide level. At t = 0 the water level is 2 m above the low tide level and rising. Using trigonometric functions, find a function to describe the height H(t) of the water above the low tide level.

H(t) =

[12] 6. Sketch the graph of f (x) =

x 1 + x^2

on the grid provided, showing all of the following if they are present: i) x and y intercepts ii) critical points iii) intervals where f is increasing or decreasing iv) points of inflection v) intervals where f is concave up or down.

[6] 8. Find a and b such that f (x) = ax^3 + bx^2 + 1 has an inflection point at (− 1 , 2).

a = , b =

[10] 9. Okapis have two types of food (A and B) available in their environment. These animals spend 10 hours every day looking for food. When looking for A, the okapi gets 2 kilograms of A per hour it spends on this; when looking for B, the okapi gets 1 kilogram of B per hour. The nutrition value obtained from x kilograms of A is given by x^3 − 16 x^2 + 25x + 500 and from y kilograms of B is given by y^2. How should an okapi divide its time between the two types of food to maximize the total daily nutrition value, and what is the total nutrition value when it does so?

Time spent looking for A:

Total nutrition value:

Useful Formulae

Law of cosines: c^2 = a^2 + b^2 − 2 ab cos θ

Trig identities:

sin^2 θ + cos^2 θ = 1 sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B − sin A sin B

tan θ =

sin θ cos θ

Values: θ sin θ cos θ 0 0 1 π/ 6 1 / 2

π/ 4

π/ 3

π/ 2 1 0

The End

Be sure that this examination has 11 pages including this cover

The University of British Columbia Sessional Examinations - December 2006

Mathematics 102 Differential Calculus with applications to Life Sciences

Closed book examination Time: 2 12 hours

Name Signature

Student Number Section

Special Instructions:

Non-graphing calculators allowed, no other aids. Show your work. The last page contains some helpful formulae.

Rules governing examinations

1. Each candidate should be prepared to produce his or her library/AMS card upon request.
2. 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 autho- rized 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.
3. Smoking is not permitted during examinations.

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