drmthornber.bsky.social
Retired A level maths teacher, current examiner. Enjoy maths puzzles and cryptic crosswords.
33 posts
75 followers
67 following
Getting Started
Active Commenter
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For polynomials, tangent lines can be found by looking for repeated roots. No need for limits.
We find m so that p(x)-p(a)=m(x-a) has a repeated root.
This gives m=p’(a)
For quadratics it’s good old b^2-4ac=0
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It’s definitely a US thing. Just check any of their doorstep-sized textbooks.
I always thought it was easier to remember than concave/convex, but confusion arises later when convex is used to mean “the curve is under the chord”
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There are a number of similar triangles. The ratios of corresponding lengths will be equal.
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See my previous reply.
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Lists are arrays. You can access individual items with indexing. For example L1[2]
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I usually buy direct from the Double Two website. Lots of colours available
https://www.doubletwo.co.uk/mauve-long-sleeve-non-iron-cotton-rich-shirt/
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The first 3 terms are (z-3)^2+1. Try the substitution u=z-3, followed by completing the square.
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You should try the Guardian quick cryptic. Each one uses 4 different clue types and they give examples of each.
https://www.theguardian.com/crosswords/series/quick-cryptic
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There’s a typo in 2a) …
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#FridayFive
1. On the Dark Side - Eddy and the Cruisers
2. Power In The Darkness - Tom Robinson Band
3. Because the Night - Patti Smith
4. Corcovado (Quiet Nights of Quiet Stars) - Getz, Gilberto
5. Dancing In The Dark - Cannonball Adderley
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We have seen issues with this in the past. Some courses at some universities want to make offers on 3 A levels taken at the same time. Medicine has been a problem here.
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I do this in Excel
1) create a scatter graph with a single point
2) use series properties to make the point invisible
3) use axis properties to get scales and labels correct
4) use major and minor grid line properties to get the grid lines looking nice
You can export to Word
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#FridayFive
1. The Only Living Boy in New York - Simon and Garfunkel
2. My Friend Stan - Slade
3. Whatever Happened To You - Highly Likely
4. There’s a Light Beyond These Woods (Mary Margaret) - Nanci Griffith
5. Woody and Dutch on the Slow Train to Peking - Rickie Lee Jones
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The contract had a penalty clause of £12000 per day for every day it ran over the agreed finish date. They finished on time!
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When my school was rebuilt we took our FM classes to visit the site manager. He showed us a huge stack of Gantt charts for every part of the project. They tracked the number of workers used by every subcontractor every day and updated the charts every night.
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#FridayFive
River - Joni Mitchell
Gaudete - Steeleye Span
Mary’s Boy Child - Harry Belafonte
I Believe In Father Christmas- Greg Lake
Merry Christmas Darling- The Carpenters
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I integrate by parts
https://www.youtube.com/watch?v=-reFBJ4R9iA
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#FridayFive
1. Winter Song - Lindisfarne
2. The Snow It Melts The Soonest - Anne Briggs
3. Mother Russia - Renaissance
4. Snowbound - Genesis
5. Under Ice - Kate Bush
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The box and whiskers plot was first introduced in 1970, after all the hypothesis testing stuff.
en.wikipedia.org/wiki/Box_plot
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Not needed on lines 1 or 2 because it is contained in/implied by the integral on the right.
Needed on line 3 because the LHS is indefinite, but the RHS as written is definite.
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Simple numerical approximation:
1.4^2=1.96 so sqrt(2)>1.4
pi/4>3/4=0.75
sum>2.15>2
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This is what I did. It works very nicely - leads to a linear and a quadratic.
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#FridayFive
1 Les Feuilles Mortes - Yves Montand
2 As Falls Wichita, So Falls Wichita Falls - Pat Metheny & Lyle Mays
3 California Dreamin’ - The Mamas & The Papas
4 Who Knows Where The Time Goes? - Fairport Convention
5 (Don’t Fear) The Reaper - Blue Oyster Cult
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2/2
Solving by elimination, as suggested by others in this thread, gives formulas for x and y in terms of theta. Now check that these formulas are well defined for every theta (e.g. no dividing by sine theta etc.)
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1/2
You have found the correct answer, but your reasoning is not valid.
What you have shown is that IF a solution exists, then it must lie on the circle. This does not guarantee that a solution exists for every theta.
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#FridayFive
1) Don’t You Rock Me Daddy O - The Vipers Skiffle Group
2) Brand New Key - Melanie
3) Grey Cortina - Tom Robinson Band
4) The City of New Orleans - Arlo Guthrie
5) White Horses - Jacky
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Here are some very old papers from OCR
www.cambridgeassessment.org.uk/our-research...
The 1967 S paper is particularly brutal!
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You have to do it in the right order
Step 1 Divide by the coefficient of x^2
Step 2 Move the constant term to the RHS
Step 3 Add an appropriate number to CTS
Step 4 Take the square root and solve
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We should start with CTS. It’s guaranteed to work and easy for anyone confident with fractions. Formula comes next and factorising last (if at all). Even better - factorise by CTS plus diff of 2 squares!
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Try x^(-2/5) from -1 to 1. Both infinitely tall parts have positive area.