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The final examination for mathematics 101 at the university of british columbia, held on april 24, 2009. The examination consists of short-answer and full-solution problems covering various topics in calculus and integration. Students were not allowed to use books, notes, or calculators during the exam, which lasted 2.5 hours.
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The University of British Columbia Final Examination - April 24, 2009
Mathematics 101
All Sections
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
Section :
Instructor :
Special Instructions:
No books, notes, or calculators are allowed. Unless it is otherwise specified, answers may be left in “calculator-ready” form, where calculator means basic scientific calculator. Show all your work, little or no credit will be given for a numerical answer without the correct accompanying work. If you need more space than the space provided, use the back of the previous page. Where boxes are provided for answers, put your final answers in them.
Rules governing examinations
Total 100
Page 1 of 13 pages
(a) Evaluate
3 + x^5 √ x
[3] dx
Answer
(b) What integral is defined by the following expression? lim n→∞
∑^ n
i=
π 4 n
tan
iπ 4 n [3] Do not evaluate the integral.
[3] (e) Find the length of the curve^ x^ = 100 + 2y
3 / (^2) , 0 ≤ y ≤ 11.
Answer
[3] (f) What is the average value of | sin θ − cos θ| over the interval 0 ≤ θ ≤ π/2?
[3]^ (g) Give the first three nonzero terms of the Maclaurin series (power series in^ x) for ∫ e−x
2 − 1 x
dx
Answer
(h) For what values of r does the function y = xr^ satisfy the following differential equation (for x > 0)? x^2 y′′^ + 4xy′^ + 2y = 0
Full-Solution Problems. In the remaining questions, justify your answers and show all your work. If a box is provided, write your final answer there. If you need more space, use the back of the previous page. Unless otherwise indicated, simplification of answers is not required.
e
dx x(ln x)p^
[6] converge?
[10] 3. Let R be the finite region in the xy-plane bounded by x = 0, y = 0 and y = cos x.
(a) Calculate the centroid of R.
Answer
(b) Express the volume of the solid obtained by rotating R about the line x = −2 as an integral. Do not evaluate the integral.
[10] (^) 5. Let X be a random variable with probability density function
f (x) =
kx(1 − x^4 ) if 0 ≤ x ≤ 1 , 0 if x < 0 or x > 1.
(a) Find the value of k.
Answer
(b) Find the mean μ.
Answer
(c) Find an algebraic equation satisfied by the median m.
[6] (a) y′^ = xy^2 with initial conditions y(0) = 1.
Answer
[6] (b) y′^ − 2 y = 4 + e^3 t. (Give the general solution)
[10] (^) 8. Consider the chemical reaction
A + B → C + D.
Suppose at time t = 0 sec the concentration of chemical A is 0.1 mol/L, the concentration of chemical B is 0.2 mol/L, and the concentrations of chemicals C and D are both 0. For t ≥ 0, let x(t) be the concentration of chemical D in mol/L. It can be shown that x(t) is the solution to the initial-value problem
dx dt
= k(0. 1 − x)(0. 2 − x), x(0) = 0,
where k is a positive constant whose value can be determined by experiment.
(a) Solve the initial-value problem to find x(t) explicitly.
Answer
(b) What value does the concentration of chemical D approach as t approaches ∞?