All public logs

Combined display of all available logs of UBC Wiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).

22:03, 24 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019 (Created page with "{{MER Exam page|creation_box=true}}   Category:MER Exam Page")
21:24, 24 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A12/Hint 2 (Created page with "Consider the null-space of the matrix <math> A </math>, that is the set <math> \mathrm{Null}(A) = \{x \in \mathbb{R}^7 : Ax = 0\}.</math> What can you say about its dimension? Could the null-space be trivial? 1-dimensional? 2-dimensional? Vectors in the null-space are solutions only if <math> b </math> is the <math> 0 </math> vector. However, given a solution, can you use knowledge about the null-space to say something about the full set of solutions?")
20:58, 24 March 2025 MihaiMarian talk contribs created page User:MihaiMarian (Created page with "Category:MER Participant")
18:23, 20 March 2025 MihaiMarian talk contribs uploaded File:RevolVol.png (Uploaded own work with UploadWizard) Tag: Upload Wizard
18:23, 20 March 2025 MihaiMarian talk contribs created page File:RevolVol.png (Uploaded own work with UploadWizard)
23:49, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 18 (a)/Statement (Created page with "In a particular market, the demand and supply functions for a quantity <math> q </math> of a good are given below: <math display="block"> D(q) = \frac{1}{2^q} + 1, \qquad S(q) = \begin{cases} 0 &\text{ for } q \leq 10 \\ \frac{q}{10}-1 &\text{ for } q > 10. \end{cases} </math> (a) Aside from its name, how can you tell that <math> D(q) </math> is the demand function (and not the supply function)? Circle the best way to complete the sentence, "<math> D(q) </math> is the...")
23:30, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 18 (a) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
23:29, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 15/Solution 1 (Created page with "Separation of variables tells us that <math> \int \frac{1}{B}\mathrm{d}B = \int (rt) \mathrm{d}t </math> Integrating, we find <math> \log|B| = \frac{rt^2}{2} + C,</math> which yields <math> |B| = e^{\frac{rt^2/2} + C} = e^{rt^2/2}\cdot e^C = A e^{rt^2/2}.</math> We can remove the absolute values at the expense of allowing <math>A</math> to be also negative, so the general solution is <math display="block"> B = A e^{rt^2/2}.</math> To find a specific solution, we must use...")
23:22, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 15/Hint 1 (Created page with "Try the "separation of variables" method, as in §2.4 of CLP2.")
23:21, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 15/Statement (Created page with "Let <math> B(t) </math> be the balance in a bank account that has been active for <math>t</math> years. Due to a special promotional interest rate calculation, <math> B </math> satisfies the differential equation <math display="block"> \frac{\mathrm{d}B}{\mathrm{d}t} = (rt)B. </math> If <math> B(0) = \$ 100</math>, find an explicit equation for <math> B(t),\;\; t \geq 0</math>. Your answer may only depend on <math>r</math> and <math>t</math>.")
23:17, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 15 (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
23:17, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C/April 2024 (Created page with "{{MER Exam page|creation_box=true}}   Category:MER Exam Page")
23:15, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18/Solution 1 (Created page with "We wish to optimize <math display="block"> F(T) = \frac{1}{4} \int_T^{T+4} \left( 1 + e^{-(t-4)^2} + \cos(\pi t) \right) \mathrm{d}t.</math> Ultimately, we wish to solve <math> F'(T) = 0</math> and then verify that the solution is indeed a local maximum. A first attempt might be to try integrating the above expression in order to obtain another expression for <math>F(T)</math>, but note the presence of <math> e^{-(t-4)^2} </math> in the integrand. The function <math>x \m...")
22:50, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18/Hint 2 (Created page with "We are tasked with finding the time <math> T </math> for which the average population from <math> T </math> to <math> T+4</math> is as large as possible. This function is given by <math display="block"> F(T) = \frac{1}{T+4-T} \int_T^{T+4} P(t) \mathrm{d}t .</math> Can you proceed from here?")
22:47, 17 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18/Hint 1 (Created page with "This is an optimization problem. Start by writing down an expression for the objective function.")
00:13, 13 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18/Statement (Created page with "A key statistic for describing how a population changes is the average population over a fixed time interval, such as a season, a year, or a generation. For any given function <math>f</math>, the average value from <math>t = a</math> to <math>t = b</math> is <math display="block"> \frac{1}{b - a} \int^b_a f(t) \mathrm{d}t. </math> Suppose the population (in thousands) of a species at time <math>t</math> (in months) is given by the function <math display="block"> P(t) =...")
00:08, 13 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18 (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
00:04, 13 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 01/Solution 1 (Created page with "{{Science:Math_Exam_Resources/Courses/MATH101_A/April_2024/Question_01/Solution_1}}")
00:04, 13 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 01/Hint 1 (Created page with "{{Science:Math_Exam_Resources/Courses/MATH101_A/April_2024/Question_01/Hint_1}}")
23:59, 12 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 01/Statement (Created page with "{{Science:Math_Exam_Resources/Courses/MATH101_A/April_2024/Question_01}}")
23:55, 12 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 01 (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
23:54, 12 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B/April 2024 (Created page with "{{MER Exam page|creation_box=true}}   Category:MER Exam Page")
23:04, 4 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Solution 2 (Created page with "Here is a solution using the limit comparison test, Theorem 1.12.22 in [https://personal.math.ubc.ca/~CLP/CLP2/clp_2_ic_text.pdf CLP2] . We expect the integrand to behave like <math> \frac{1}{x} </math>, whose integral from 1 to <math> \infty </math> diverges, so we apply the limit comparison test with <math> f(x) = \frac{x^2+1}{x^3+2} </math> and <math> g(x) = \frac{1}{x}</math> (notation as in the theorem statement in CLP). We compute th...")
22:49, 4 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Hint 2 (Created page with "For a more elementary solution that doesn't use the limit comparison test, consider the following rewriting of the integrand: <math display="block"> \frac{x^2+1}{x^3+2} = \frac{x^2}{x^3+2} + \frac{1}{x^3+2} = \frac{1}{x+\frac{2}{x^2}} + \frac{1}{x^3+2}. </math> Can you use this to show that the integrand is bounded by <math>\frac{K}{x}</math> for some <math>K</math>? Alternatively, the second solution applies the limit comparison test.")
01:01, 1 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (c)/Hint 1 (Created page with "What is the relationship between the power series representation <math> f(x) = \sum_{n=0}^\infty A_nx^n </math> and the Taylor series representation <math> f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n </math>?")
01:00, 1 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (c)/Solution 1 (Created page with "What is the relationship between the power series representation <math> f(x) = \sum_{n=0}^\infty A_nx^n </math> and the Taylor series representation <math> f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n </math>?")
00:53, 1 March 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (b)/Hint 1 (Created page with "In part (a) we found an expression of <math> f(x) </math> as a sum of two geometric series. Can you find the radii of convergence of the two series? What do they say about the radius of convergence of the sum?")
00:46, 26 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (b)/Solution 1 (Created page with "We saw in part (a) that <math display="block"> f(x) = \frac{2}{2+x} + \frac{2}{1-x} = \sum_{n=0}^\infty \left(\frac{-x}{2} \right)^n + \sum_{n=0}^\infty 2x^n,</math> where the first series converges as long as <math> -1 < -x/2 < 1 </math>, and the second series converges as long as <math> -1 < x < 1</math>. Since have the following sequence of equivalences <mat...")
00:37, 26 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (a)/Solution 1 (Created page with "Following the hint, let's look for <math> A, B </math> such that <math display="block"> f(x) = \frac{A}{2+x} + \frac{B}{1-x}. </math> Following steps similar to those of [https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH101_A/April_2024/Question_15 Question 15], we find <math> A = B = 2</math>. We must now notice that each summand on the right-hand side of <math display="block"> f(x) = \frac{2}{2+x} + \frac{2}{1-x} </math> is...")
23:58, 25 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 12/Solution 1 (Created page with "Since the series <math> \sum_{n=1}^\infty \cos(\theta_n) </math> converges, it must be that the terms of the sum converge to 0: <math display="block"> \lim_{n \to \infty} \cos(\theta_n) = 0. </math> This is a consequence of Theorem 3.3.1 in [https://personal.math.ubc.ca/~CLP/CLP2/clp_2_ic_text.pdf CLP] . Note in particular that it does not matter what precise value the series converges to. We see then that, for large <math> n </math>, th...")
23:27, 25 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 12/Hint 1 (Created page with "If an infinite series <math> \sum_{n=1}^\infty a_n </math> converges, what does this tell you about <math> \lim_{n \rightarrow \infty} a_n </math>?")
06:56, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 15/Solution 1 (Created page with "We want to find <math> A, B</math> so that <math display="block"> \frac{1}{(x+1)(x-3)} = \frac{A}{x+1} + \frac{B}{x-3}.</math> Since the right-hand side is equal to <math> \frac{A(x-3) + B(x+1)}{(x+1)(x-3)}</math>, we must have <math> 1 = A(x-3) + B(x+1) = (A+B)x + B - 3A</math>, as an equation that holds for all <math> x </math>. Therefore, we must have <math display="block"> \begin{cases} A+B &= 0 \\ B-3A &= 1 \end{cases}</math> From the first equation, <math> B = -A<...")
06:43, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 15/Hint 1 (Created page with "Recall the method of partial fraction decomposition: try to find <math> A, B</math> so that <math display="block"> \frac{1}{(x+1)(x-3)} = \frac{A}{x+1} + \frac{B}{x-3}.</math>")
06:40, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Solution 1 (Created page with "Following the hint, we have <math display="block"> \frac{x^2+1}{x^3 + 2} = \frac{1}{x + \frac{2}{x^2}} + \frac{1}{x^3 + 2}. </math> Given that the first term in the sum above looks close to <math>\frac{1}{x}</math>, and we know that <math> \int_1^\infty \frac{1}{x}\; \mathrm{d}x </math> diverges, we should try to prove that the integral above diverges as well. To do this, let us try to bound the integrand ''below'' by a function that diverges. Since <math> x \in [1, \in...")
06:19, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Hint 1 (Created page with "Note first that, since the denominator does not vanish on the interval <math> [1, \infty) </math>, the integral is improper because one of the integration bounds is infinity (as opposed to there being a singular point in the domain of integration). Thinking about using a comparison test, consider the following rewriting of the integrand: <math display="block"> \frac{x^2 +1}{x^3+2} = \frac{x^2}{x^3 + 1} + \frac{1}{x^3+2} = \frac{1}{x + \frac{2}{x^2}} + \frac{1}{x^3 + 2}....")
06:11, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 07/Solution 1 (Created page with "Since the question is asking about the area of a region in the plane, we must be careful not to compute a singed area. The easiest way to do this is to use an absolute value. <math display = "block"> A = \int_0^\pi |\sin(x) - \cos(x)|\; \mathrm{d}x. </math> By drawing a picture, we see that <math> \sin(0) - \cos(0) = -1</math> and <math> \sin(\pi) - \cos(\pi) = 1 </math>, which shows us that there is some <math> x \in (0, \pi) </math> for which <math> \sin(x) - \cos(x) =...")
05:54, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 07/Hint 1 (Created page with "Draw a picture and notice that the question is asking about the area of a region, something that should be positive.")
05:40, 24 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 18 (a)/Hint 1 (Created page with "The denominator of <math> f(x) </math> factors as <math> -(x+2)(x-1) </math>. Can you write <math> f(x) </math> as a sum <math> \frac{A}{x+2} + \frac{B}{x-1}</math>, for some choices of <math> A, B </math>? Do the summands look familiar?")
00:46, 15 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 05/Solution 1 (Created page with "Following the hint, we wish to perform the substitution <math> u = \sin(x) </math>, but we must first rewrite the integrand by using the fundamental identity <math> \sin^2(x) + \cos^2(x) = 1</math>: <math display="block"> \int_0^{\pi/2} \sin^4(x) \cos^3(x) \mathrm{d}x = \int_0^{\pi/2}\sin^4(x)\left(1-\sin^2(x)\right) \cos(x) \mathrm{d}x = \int_0^{\pi/2} \left(\sin^4(x) - \sin^6(x) \right) \cos(x)\mathrm{d}x. </math> We now perform the <math>u</math>-substitution above....")
00:37, 15 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 05/Hint 1 (Created page with "Note that <math> \sin' = \cos </math> and that trigonometric functions are related through lots of identities. Can you rewrite the integrand so that the substitution <math> u = \sin(x) </math> becomes useable?")
01:24, 11 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 03/Solution 1 (Created page with "If we imagine that the integrand is the side of a right-angled triangle, then the other sides of the triangle have length <math> x </math> and 3, and 3 is the length of the hypotenuse. We use this picture to guess the following substitution: <math display="block>\begin{align} \frac{x}{3} &= \sin(\theta)\\ \mathrm{d}x &= 3\cos(\theta) \mathrm{d}\theta \end{align}</math> We thus have <math display="block"> \int \sqrt{9-x^2} \mathrm{d}x = \int \sqrt{9 - 9\sin^2(\theta)...")
00:48, 11 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 03/Hint 1 (Created page with "Because the integrand is of a difference of squares, try using a trigonometric substitution.")
00:44, 11 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 01/Solution 1 (Created page with "One version of the fundamental theorem of calculus is <math display="block"> \frac{\mathrm{d}}{\mathrm{d}x} \int_a^x f(t) \mathrm{d}t = f(x).</math> In our case, <math>F</math> is a composite <math>H \circ g</math> of the two functions <math display="block">\begin{align} H(x) &= \int_0^{x} \frac{4-t}{1-\cos^2(t)} \mathrm{d}t,\\ g(x) &= x^2. \end{align}</math> By the chain rule, <math display="block">F'(x) = H'(g(x)) \cdot g'(x) = 2x\cdot \frac{4-x^2}{1-\cos^2(x^2)}.</...")
00:33, 11 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 01/Hint 1 (Created page with "What does the fundamental theorem of calculus say? Note that the upper bound of the integral defining <math>F</math> is <math>x^2</math>, not <math>x</math>.")
21:03, 3 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 C (Created page with "<!--- Comments created in February 2014 In the first section below, we are defining variables that hold meta information for the course. This is useful, eg, in an overview listing of all courses. The parameters are 1. A course description (eg. "Differential Calculus with Applications to Physical Sciences and Engineering") 2. The "English" name of the course (eg. "Math 100/180", particularly useful for cross listed courses) 3. course_url: for where the course page i...")
20:52, 3 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 B (Created page with "<!--- Comments created in February 2014 In the first section below, we are defining variables that hold meta information for the course. This is useful, eg, in an overview listing of all courses. The parameters are 1. A course description (eg. "Differential Calculus with Applications to Physical Sciences and Engineering") 2. The "English" name of the course (eg. "Math 100/180", particularly useful for cross listed courses) 3. course_url: for where the course page i...")
22:47, 2 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A/April 2024 (Creation)
22:37, 2 February 2025 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH101 A (Creation)
21:12, 15 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B5 (c)/Hint 2 (Created page with "The matlab command <math>[P,D]=\mathrm{eig}(A)</math> yields the matrix <math>P</math>, whose columns are the eigenvectors of <math>A</math>, and the diagonal matrix <math>D</math>, whose entries are the eigenvalues of <math>A</math>.")
21:14, 14 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Solution 2 (Created page with "We will follow the second hint. Let <math>v \in \mathbb{R}^2</math> be a vector that makes an angle of <math>30^\circ</math> with the <math>x</math>-axis. Then the first reflection fixed <marth>v</math>, so the composite <math>\mathit{Ref}_{50^\circ}\circ\mathit{Ref}_{30^\circ}</math> takes <math>v</math> into <math>\mathit{Ref}_{50^\circ}(v)</math>, which is a vector that makes an angle of 40º with <math>v</math>, in the counterclock-wise direction. Therefore, <math>\m...")