# User contributions

- 19:28, 28 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 03 (b)
- 19:27, 28 November 2017 (diff | hist) . . (+2) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 03 (b)/Hint 1
- 19:27, 28 November 2017 (diff | hist) . . (+420) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 03 (b)/Hint 1 (Created page with "The maximum and the minimum can be attained only at critical points or the boundary. Compute the critical points of the function and the value the function takes at those poin...")
- 19:25, 28 November 2017 (diff | hist) . .
**(+1,572)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 03 (b)/Solution 1 (Created page with "The minimum or maximum of the function can only occur at a critical points of the function (points where both <math>\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y...") - 00:25, 28 November 2017 (diff | hist) . . (+40) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (d)/Hint 1 (current)
- 00:24, 28 November 2017 (diff | hist) . . (+2) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (h)
- 00:23, 28 November 2017 (diff | hist) . . (+1) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (h)/Hint 2 (Formatting) (current)
- 00:21, 28 November 2017 (diff | hist) . . (-121) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (d)/Solution 1 (current)
- 00:15, 28 November 2017 (diff | hist) . . (+28) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (d)/Solution 1
- 00:14, 28 November 2017 (diff | hist) . . (-4) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (d)/Solution 1
- 17:49, 24 November 2017 (diff | hist) . . (-197) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 01 (n)/Solution 1 (Simplified L'Hopital step + fixed some formatting issues)
- 23:33, 23 November 2017 (diff | hist) . . (+281) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Hint 1
- 17:01, 22 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)
- 17:01, 22 November 2017 (diff | hist) . . (+196) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Hint 2 (Created page with "Find the value of <math>\lim_{n \rightarrow \infty} a_n</math> to help you evaluate the limit <math>\lim_{n \rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right|</math> appeari...") (current)
- 16:59, 22 November 2017 (diff | hist) . . (+117) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Hint 1
- 16:56, 22 November 2017 (diff | hist) . . (+80) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Hint 1 (Created page with "Apply the ratio test to determine the radius of convergence of the power series.")
- 16:55, 22 November 2017 (diff | hist) . .
**(+914)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Solution 1 (Created page with "We will apply the ratio test to determine the radius of convergence of the power series <math>\sum_{n=0}^\infty a_n x^n</math>. The ratio test says that the power series conv...") - 16:47, 22 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (a)
- 16:46, 22 November 2017 (diff | hist) . . (+123) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (a)/Hint 2 (Created page with "Use the fact that if <math>\lim_{n \rightarrow \infty} b_n \neq 0</math>, then the series <math>\sum_n b_n</math> diverges.")
- 16:46, 22 November 2017 (diff | hist) . . (+119) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (a)/Hint 1 (Created page with "Use the fact that if a series <math>\sum_n b_n</math> converges, then <math>\lim_{n \rightarrow \infty} b_n = 0</math>.")
- 16:35, 22 November 2017 (diff | hist) . . (+481) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (a)/Solution 1 (Created page with "Since <math>\sum_{n=0}^\infty (1-a_n)</math> converges, then <math>\lim_{n \rightarrow \infty} (1-a_n) = 0</math>, which implies that <math>\lim_{n \rightarrow \infty} a_n = 1...")
- 15:17, 22 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (a)
- 15:16, 22 November 2017 (diff | hist) . . (+156) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (a)/Hint 2 (Created page with "Once a series for <math>\frac{df}{dx}</math> is obtained, integrate it and use the condition <math>f(0) = 1</math> to obtain a series for <math>f(x)</math>.")
- 15:15, 22 November 2017 (diff | hist) . . (+172) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (a)/Hint 1 (Created page with "Expand <math>\frac{df}{dx}</math> as a Maclaurin series by applying the geometric series <math>\frac{1}{1+y} = 1 - y + y^2 - y^3 + \cdots = \sum_{n=0}^\infty (-y)^n.</math>")
- 15:13, 22 November 2017 (diff | hist) . .
**(+1,251)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (a)/Solution 1 (Created page with "Since <math>\frac{df}{dx} = x\left(\frac{1}{1 + 3x^3}\right)</math>, a Maclaurin series for <math>\frac{df}{dx}</math> can be computed from the geometric series ::<math>\fra...") - 12:02, 22 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (a)
- 12:02, 22 November 2017 (diff | hist) . . (+235) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (a)/Hint 1 (Created page with "Recall the definition of expected value. If <math>f(x)</math> is the probability density function of a random variable <math>X</math>, its expected value is defined as the int...")
- 11:59, 22 November 2017 (diff | hist) . . (+232) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (a)/Solution 1
- 11:53, 22 November 2017 (diff | hist) . .
**(+908)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (a)/Solution 1 (Created page with "To compute <math>E(X)</math>, we will compute the integral <math>E(X) = \int_{-\infty}^\infty xf(x) \, dx</math> where <math>f(x)</math> is the probability density function of...") - 23:06, 21 November 2017 (diff | hist) . . (+11) . . m Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (b)/Solution 1
- 22:55, 21 November 2017 (diff | hist) . . (-409) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (b)/Solution 1
- 19:42, 21 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (b)
- 19:42, 21 November 2017 (diff | hist) . . (+205) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (b)/Hint 1 (Created page with "Recall that if <math>f(y)</math> is the probability density function of a random variable <math>X</math>, its cumulative distribution function is defined as <math>F(x) = \int_...")
- 19:41, 21 November 2017 (diff | hist) . .
**(+1,694)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 04 (b)/Solution 1 (Created page with "The cumulative density function <math>F(x)</math> of a random variable <math>X</math> is defined as <math>F(x) = \int_{-\infty}^x f(y) \, dy</math> where <math>f(y)</math> is...") - 19:20, 21 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (d)
- 19:20, 21 November 2017 (diff | hist) . . (+166) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (d)/Hint 1 (Created page with "The function <math> \frac{1}{x (\ln x)^3}</math> appears in the series and has a closed form integral. This means that applying the integral test can be helpful here.")
- 19:18, 21 November 2017 (diff | hist) . .
**(+970)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (d)/Solution 1 (Created page with "We will apply the integral test. The test states that if the integral <math>\int_n^\infty f(x) \, dx </math> converges, the series <math>\sum_{k=n}^\infty f(k)</math> converg...") - 19:07, 21 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)
- 19:07, 21 November 2017 (diff | hist) . . (+1) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)/Hint 2 (current)
- 19:07, 21 November 2017 (diff | hist) . . (+117) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)/Hint 2 (Created page with "Use the fact that <math>sin \sqrt{m} \geq -1 </math> for all m to help you find a series to compare the given one to.")
- 19:06, 21 November 2017 (diff | hist) . . (+40) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)/Hint 1 (Created page with "Apply the comparison and p-series tests.") (current)
- 19:05, 21 November 2017 (diff | hist) . .
**(+835)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)/Solution 1 (Created page with "We will apply the comparison test to show that the series diverges. The comparison test states that if <math>a_n</math> and <math>b_n</math> are sequences with <math>0 \leq a_...") - 19:00, 21 November 2017 (diff | hist) . . (-9) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (b)/Solution 1
- 18:54, 21 November 2017 (diff | hist) . . (0) . . Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (b)
- 18:53, 21 November 2017 (diff | hist) . . (+40) . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (b)/Hint 1 (Created page with "Apply the comparison and p-series tests.") (current)
- 18:51, 21 November 2017 (diff | hist) . .
**(+917)** . . N Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (b)/Solution 1 (Created page with "We will apply the comparison test to see if the series converges or diverges. The comparison test states that if <math>a_n</math> and <math>b_n</math> are two sequences with <...")