MentDB WeakMQL Recipes
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MT

Math teacher & student

Equations, tables of values, fractions and probabilities: exact answers and their decimal check.

10 recipes on this page · open in the interactive view

Recipes
  1. Solve a quadratic equation, exact and decimal
  2. Study a function: derivative and table of values
  3. A fractions word problem
  4. Probability of at least two sixes
  5. A cubic from its roots: expand, factor, study
  6. Solve a right triangle and its six trig ratios
  7. A class test: mode, frequency table, two variances
  8. Arithmetic, geometric and harmonic means
  9. Rounding, truncating and the sign of negative numbers
  10. Simplify an expression, then solve with roots
executed on MentDB

Solve a quadratic equation, exact and decimal

Solve 3x^2 - 5x - 4 = 0: discriminant, exact roots, decimal roots rounded to 4 places, and a numerical check.

Pipeline
01 · deltaDiscriminant b^2 - 4acmath calc→02 · exactExact rootscas solve→03 · roundRounded rootsmath round→04 · checkNumerical rootsnum poly_roots
Script
admin
#1. Discriminant of 3x^2 - 5x - 4 = 0;
-> "[a]" 3; -> "[b]" -5; -> "[c]" -4;
-> "[delta]" (math calc "[b]^2-4*[a]*[c]");
#2. Exact roots;
-> "[exact]" (cas solve "3*x^2-5*x-4 = 0" "x");
#3. Decimal roots;
-> "[x1]" (math calc "(-[b]-sqrt([delta]))/(2*[a])");
-> "[x2]" (math calc "(-[b]+sqrt([delta]))/(2*[a])");
concat "delta = " [delta] " > 0, two roots: " [exact] " ~ " (math round [x1] 4) " and " (math round [x2] 4) "; check: " (num poly_roots "[-4, -5, 3]")
Result
mentdb
delta = 73 > 0, two roots: [{"x":"1\/6*(5-sqrt(73))"},{"x":"1\/6*(5+sqrt(73))"}] ~ -0.5907 and 2.2573; check: ["-0.5906672908862551","2.257333957552922"]
Go further
executed on MentDB

Study a function: derivative and table of values

For f(x) = x^3 - 3x^2 - 9x + 5, compute f'(x), find where it vanishes, and build the table of values on [-3, 5] to draw the variations.

Pipeline
01 · deriveDerivative f'(x)math diff→02 · zerosSolve f'(x) = 0cas solve→03 · tableTable of valuesnum table
Script
admin
#1. f and its derivative;
-> "[f]" "x^3-3*x^2-9*x+5";
-> "[df]" (math diff [f] "x");
#2. Where f' = 0;
-> "[crit]" (cas solve (concat [df] " = 0") "x");
#3. Table of values from -3 to 5;
concat "f'(x) = " [df] "; f' = 0 for " [crit] "; table: " (num table [f] "x" -3 5 1)
Result
mentdb
f'(x) = 3*x^2 - 6*x - 9; f' = 0 for [{"x":"-1"},{"x":"3"}]; table: [
  [-3,-22],
  [-2,3],
  [-1,10],
  [0,5],
  [1,-6],
  [2,-17],
  [3,-22],
  [4,-15],
  [5,10]
]
Go further
executed on MentDB

A fractions word problem

Lily eats 1/4 of a pizza, then Tom eats 2/5 of what is left. Which fraction did Tom eat, and what part of the pizza remains (as a fraction and a percentage)?

Pipeline
01 · restWhat Lily leavesarith frac_sub→02 · tom2/5 of the restarith frac_mul→03 · percentRemaining part in %math calc
Script
admin
#1. Lily eats 1/4 of the pizza;
-> "[rest]" (arith frac_sub 1 "1/4");
#2. Tom eats 2/5 of the rest;
-> "[tom]" (arith frac_mul "2/5" [rest]);
#3. What is left;
-> "[left]" (arith frac_sub [rest] [tom]);
concat "Tom ate " [tom] " of the pizza, " [left] " is left = " (math calc (concat [left] "*100")) " %"
Result
mentdb
Tom ate 3/10 of the pizza, 9/20 is left = 45 %
Go further
executed on MentDB

Probability of at least two sixes

A die is thrown 5 times. Compute P(at least two sixes) as an exact fraction with combinations, then check it with the binomial law.

Pipeline
01 · ncrCombinations C(5, k)arith ncr→02 · exactExact fractionsarith frac_sub→03 · checkBinomial law B(5, 1/6)stat dist_cdf
Script
admin
#1. 5 throws of a die, X = number of sixes. P of 0 and of 1 six, as exact fractions;
-> "[p0]" (arith frac_mul (arith ncr 5 0) "3125/7776");
-> "[p1]" (arith frac_mul (arith ncr 5 1) "625/7776");
#2. At least two sixes;
-> "[p]" (arith frac_sub 1 (arith frac_add [p0] [p1]));
#3. Decimal value and check with the binomial law;
concat "P(X >= 2) = " [p] " = " (math round (math calc [p]) 4) "; check with the binomial law: " (math calc (concat "1-" (stat dist_cdf "binomial" "[5, 0.16666666666666666]" 1)))
Result
mentdb
P(X >= 2) = 763/3888 = 0.1962; check with the binomial law: 0.1962448559670782
Go further
executed on MentDB

A cubic from its roots: expand, factor, study

Build p(x) = (x-1)(x-2)(x+3) from its roots, check the expansion and the factorisation with the CAS, find the critical points (decimal and exact), the area between the roots 1 and 2, and where the curve meets the line y = 6 - x.

Pipeline
01 · buildMultiply the factorsnum poly_mul→02 · checkExpand and factorcas factor→03 · studyDerivative and its rootsnum poly_derive→04 · areaPrimitive evaluated at 1 and 2num poly_integrate→05 · meetp minus the linenum poly_add
Script
admin
#1. Is the CAS available on this server;
-> "[cas]" (json load select "rx_cas" (cas status) /installed);
#2. Cubic with roots 1, 2 and -3, coefficients from x^0 upwards;
-> "[p]" (num poly_mul (num poly_mul "[-1, 1]" "[-2, 1]") "[3, 1]");
-> "[expanded]" (cas expand "(x-1)*(x-2)*(x+3)");
-> "[factored]" (cas factor "x^3-7*x+6");
#3. Critical points: roots of p prime, decimal then exact;
-> "[crit]" (num poly_roots (num poly_derive [p]));
-> "[critx]" (cas solve "3*x^2-7 = 0" "x");
#4. Area under the x axis between the roots 1 and 2;
-> "[P]" (num poly_integrate [p]);
-> "[area]" (math calc (concat "abs(" (num poly_eval [P] 2) "-(" (num poly_eval [P] 1) "))"));
#5. Intersections with the line y = 6 - x: roots of p - line;
-> "[inter]" (num poly_roots (num poly_add [p] "[-6, 1]"));
concat "{\"cas\": " [cas] ", \"p\": " [p] ", \"expanded\": \"" [expanded] "\", \"factored\": \"" [factored] "\", \"critical_x\": " [crit] ", \"critical_exact\": " [critx] ", \"area_1_2\": " [area] ", \"meets_line_at\": " [inter] "}"
Result
mentdb
{"cas": true, "p": [6,-7,0,1], "expanded": "6-7*x+x^3", "factored": "(-2+x)*(-1+x)*(3+x)", "critical_x": ["-1.5275252316519465","1.5275252316519465"], "critical_exact": [{"x":"-sqrt(7\/3)"},{"x":"sqrt(7\/3)"}], "area_1_2": 0.75, "meets_line_at": ["-2.449489742783178","0","2.449489742783178"]}
Go further
  • Draw p and the line with plot function and a JSON array of two expressions.
  • cas solve "x^3-7*x+6 = 6-x" "x" gives the exact intersections.
executed on MentDB

Solve a right triangle and its six trig ratios

A ladder reaches 12 m up a wall with its foot 5 m from the wall. Find its length, the angle with the ground in degrees, and check the six ratios sin, cos, tan, csc, sec, cot of that angle (13, 5, 12 triangle).

Pipeline
01 · lengthHypotenusemath hypot→02 · angleAngle from the two legsmath atan2→03 · degreesRadians to degreesmath rad_to_deg→04 · ratiostan, sec, csc, cotmath sec
Script
admin
#1. Legs of the triangle in m;
-> "[a]" 5;
-> "[b]" 12;
-> "[L]" (math hypot [a] [b]);
#2. Angle with the ground, atan2 of opposite and adjacent;
-> "[t]" (math atan2 [b] [a]);
-> "[deg]" (math round (math rad_to_deg [t]) 2);
#3. The six ratios, expected 12/13 5/13 12/5 13/12 13/5 5/12;
concat "length = " [L] " m, angle = " [deg] " deg, same with atan 12/5: " (math round (math rad_to_deg (math atan 2.4)) 2) "; sin = " (math round (math sin [t]) 4) ", cos = " (math round (math cos [t]) 4) ", tan = " (math round (math tan [t]) 4) ", csc = " (math round (math csc [t]) 4) ", sec = " (math round (math sec [t]) 4) ", cot = " (math round (math cot [t]) 4)
Result
mentdb
length = 13 m, angle = 67.38 deg, same with atan 12/5: 67.38; sin = 0.9231, cos = 0.3846, tan = 2.4, csc = 1.0833, sec = 2.6, cot = 0.4167
Go further
  • Ask the class to explain why atan2 is safer than atan when the adjacent side is negative or zero.
  • Use arith to_fraction on the ratios to get 12/13, 5/13...
executed on MentDB

A class test: mode, frequency table, two variances

Twenty pupils got marks out of 10. Build the frequency table with cumulative frequencies, find the mode, the mean and the median, and compare the population variance (divide by n) with the sample variance (divide by n - 1).

Pipeline
01 · sortOrdered seriesmath order_double→02 · tableFrequency tablestat frequency→03 · modeMost frequent markstat mode→04 · spreadPopulation vs sample variancestat pvariance
Script
admin
-> "[m]" "[7, 5, 8, 6, 7, 9, 4, 7, 6, 8, 5, 7, 10, 6, 7, 8, 3, 6, 7, 9]";
#1. Sorted marks and frequency table;
-> "[sorted]" (math order_double [m]);
-> "[freq]" (stat frequency [m]);
#2. Centre and spread;
-> "[pv]" (stat pvariance [m]);
-> "[sv]" (stat variance [m]);
concat "sorted: " [sorted] "; mode = " (stat mode [m]) ", mean = " (math avg [m]) ", median = " (stat median [m]) ", population variance = " [pv] ", sample variance = " [sv] ", ratio = " (math calc "[sv]/[pv]") " = 20/19; table " [freq]
Result
mentdb
sorted: [3.0,4.0,5.0,5.0,6.0,6.0,6.0,6.0,7.0,7.0,7.0,7.0,7.0,7.0,8.0,8.0,8.0,9.0,9.0,10.0]; mode = [7], mean = 6.75, median = 7, population variance = 2.7875, sample variance = 2.9342105263157894, ratio = 1.0526315789473684 = 20/19; table [
  {"value": 3, "count": 1, "frequency": 0.05, "cumulative_count": 1, "cumulative_frequency": 0.05},
  {"value": 4, "count": 1, "frequency": 0.05, "cumulative_count": 2, "cumulative_frequency": 0.1},
  {"value": 5, "count": 2, "frequency": 0.1, "cumulative_count": 4, "cumulative_frequency": 0.2},
  {"value": 6, "count": 4, "frequency": 0.2, "cumulative_count": 8, "cumulative_frequency": 0.4},
  {"value": 7, "count": 6, "frequency": 0.3, "cumulative_count": 14, "cumulative_frequency": 0.7},
  {"value": 8, "count": 3, "frequency": 0.15, "cumulative_count": 17, "cumulative_frequency": 0.85},
  {"value": 9, "count": 2, "frequency": 0.1, "cumulative_count": 19, "cumulative_frequency": 0.95},
  {"value": 10, "count": 1, "frequency": 0.05, "cumulative_count": 20, "cumulative_frequency": 1}
]
Go further
  • Read the median on the cumulative_frequency column: the first mark where it reaches 0.5.
  • Draw the marks with plot histogram.
executed on MentDB

Arithmetic, geometric and harmonic means

Three classic problems: a round trip at 60 then 40 km/h (harmonic mean), an investment growing +10 %, +50 %, -20 % (geometric mean of the factors), and the inequality AM >= GM >= HM on the same numbers.

Pipeline
01 · amArithmetic meanmath avg→02 · hmAverage speedstat harm_mean→03 · gmAverage growth factorstat geo_mean→04 · rootSame with a cube rootmath nroot
Script
admin
#1. Round trip: the average speed is the harmonic mean;
-> "[v]" "[60, 40]";
-> "[hm]" (stat harm_mean [v]);
#2. Yearly growth factors;
-> "[f]" "[1.10, 1.50, 0.80]";
-> "[gm]" (stat geo_mean [f]);
-> "[gm2]" (math nroot (math calc "1.10*1.50*0.80") 3);
#3. AM, GM, HM of the factors;
concat "speed: AM = " (math avg [v]) " km/h is wrong, HM = " (math round [hm] 2) " km/h; growth: GM = " (math round [gm] 5) " = cube root " (math round [gm2] 5) " so " (math round (math calc "([gm]-1)*100") 2) " % a year, not " (math round (math calc (concat "(" (math avg [f]) "-1)*100")) 2) "; AM " (math avg [f]) " >= GM " (math round [gm] 4) " >= HM " (math round (stat harm_mean [f]) 4)
Result
mentdb
speed: AM = 50.0 km/h is wrong, HM = 48 km/h; growth: GM = 1.09696 = cube root 1.09696 so 9.7 % a year, not 13.33; AM 1.1333333333333335 >= GM 1.097 >= HM 1.0617
Go further
  • Prove AM >= GM for two numbers with (sqrt a - sqrt b)^2 >= 0.
  • Use stat weighted_mean for a trip with different distances.
executed on MentDB

Rounding, truncating and the sign of negative numbers

A lesson on rounding: compare floor, ceil, trunc, rint (round half to even) and round on 2.5, 3.5 and -2.7, take the fractional part, and the sign functions.

Pipeline
01 · truncToward zeromath trunc→02 · rintHalf to evenmath rint→03 · fracFractional partmath frac_part→04 · signSignmath signum
Script
admin
-> "[x]" -2.7;
#1. Four ways to get an integer from -2.7;
-> "[r1]" (concat "floor " (math floor [x]) ", ceil " (math ceil [x]) ", trunc " (math trunc [x]) ", round " (math round [x] 0));
#2. Ties: rint rounds half to even, banker's rounding;
-> "[r2]" (concat "rint 2.5 = " (math rint 2.5) ", rint 3.5 = " (math rint 3.5) ", round 2.5 = " (math round 2.5 0));
#3. Fractional part keeps the sign;
-> "[r3]" (concat "frac_part -2.7 = " (math frac_part [x]) ", frac_part 3.14 = " (math frac_part 3.14));
concat [r1] "; " [r2] "; " [r3] "; sign -2.7 = " (math sign [x]) ", signum 0 = " (math signum 0)
Result
mentdb
floor -3, ceil -2, trunc -2, round -3; rint 2.5 = 2, rint 3.5 = 4, round 2.5 = 3; frac_part -2.7 = -0.7, frac_part 3.14 = 0.14; sign -2.7 = -1, signum 0 = 0
Go further
  • Explain why banker's rounding avoids a bias when summing many rounded values.
  • Compare with math round_sig for significant figures.
executed on MentDB

Simplify an expression, then solve with roots

Simplify x^2*x + 3*x^3 - x*x^2 + 0*y, evaluate it at x = 2, then solve 3x^3 = 81 with a cube root and x^5 = 7776 with a fifth root, checking with powers.

Pipeline
01 · simplifyGroup like termsmath simplify→02 · evaluateValue at x = 2math eval→03 · cubeCube rootmath cbrt→04 · checkBack with a powermath power
Script
admin
#1. Simplify;
-> "[e]" (math simplify "x^2*x+3*x^3-x*x^2+0*y");
-> "[v]" (math eval [e] "{\"x\": 2}");
#2. 3x^3 = 81 so x = cube root of 27;
-> "[x1]" (math cbrt (math calc "81/3"));
#3. x^5 = 7776;
-> "[x2]" (math nroot 7776 5);
concat "simplified: " [e] ", value at 2 = " [v] "; 3x^3 = 81 gives x = " [x1] ", check 3*x^3 = " (math calc (concat "3*" (math power [x1] 3))) "; x^5 = 7776 gives x = " [x2] ", check " (math power [x2] 5) "; the real cube root of -8 is " (math nroot -8 3)
Result
mentdb
simplified: 3*x^3, value at 2 = 24; 3x^3 = 81 gives x = 3, check 3*x^3 = 81; x^5 = 7776 gives x = 6, check 7776; the real cube root of -8 is -2
Go further
  • cas factor can factor the polynomial before solving.
  • Show that an even root of a negative number has no real value.

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