MatheAss 10.0 − Calculus
Sequences and series (New in version 9.0 from May 2021)
The program determines the first n terms of a sequence (ai) and of the corresponding series (sum of the sequence terms), when the first terms of the sequence and an explicit function ai=ƒ(i) or a recursion formula ai=ƒ(a0, a1, ... , ai-1) are given.
Sequence ¯¯¯¯¯¯¯¯ ( a[ i ] ) = (1; 3; 5; 7; 9; 11; 13; 15; 17; 19) Series ¯¯¯¯¯ ( Σ a[ i ] ) = (1; 4; 9; 16; 25; 36; 49; 64; 81; 100)
Divide polynomials
The program calculates the product and quotient of two polynomials.
1st polynomial: 3·x4 - 2 x + 1
2nd polynomial: 2·x + 5
Product: 6·x5 + 15·x4 - 4·x2 - 8·x + 5
Quotient: 3/2·x3 - 15/4·x2 + 75/8·x - 391/16
Remainder: 1971/16
Factor polynomials (New in version 9.0)
The program calculates the rational zeros and the linear factorization of a polynomial.
p(x) = x5 - 9·x4 - 82/9·x3 + 82·x2 + x - 9
= (1/9)·(9·x5 - 81·x4 - 82·x3 + 738·x2 + 9·x - 81)
= (1/9)·(3·x - 1)·(3·x + 1)·(x - 9)·(x - 3)·(x + 3)
Rational zeros: 1/3, -1/3, 9, 3, -3
Transform polynomials (New in version 9.0)
A polynomial function ƒ(x) can be shifted or stretched in the x‑direction and y‑direction.
ƒ(x) = - 1/4·x4 + 2·x3 - 16·x + 21 Shift by dx = -2 , dy = 0 ƒ(x + 2) = - 1/4·x4 + 6·x2 + 1
GCD and LCM of polynomials (New in version 9.0 from February 2021)
The program calculates the greatest common divisor (GCD) and the least common multiple (LCM) of two polynomials p1(x) and p2(x).
p1(x) = 4·x6 - 2·x5 - 6·x4- 18·x3 - 2·x2 + 24·x + 8 p2(x) = 10·x4- 14·x3 - 22·x2 + 14·x + 12 GCD(p1,p2) = x2 - x - 2 LCM(p1,p2) = 40·x8 - 36·x7 - 76·x6 - 144·x5 + 88·x4+ 356·x3 - 4·x2 - 176·x - 48
Function plotter 1
Up to ten functions can be drawn simultaneously in one coordinate system. Combinations or derivatives of already defined functions are also allowed.
If ƒ1(x) = sin(x) and ƒ2(x) = 3*sqrt(x), then ƒ3(x) = 2*y1^2-y2 substitudes ƒ3(x) = 2*sin(x)^2-3*sqrt(x) ƒ4(x) = f2(y1) substitudes ƒ4(x) = 3*sqrt(sin(x)) ƒ5(x) = y2' substitudes ƒ5(x) = 3/(2*sqrt(x))
Example: ƒ₁(x)=sin(x), ƒ₂(x)=x and ƒ₃(x)=y1+y2
Function plotter 2
A piecewise defined function consisting of up to nine subfunctions is drawn. For each subfunction, the domain, the type of interval, and the color are entered. It can also be specified whether the boundary points are drawn or not.
Example:
Parametric curves
This program can draw curves that are not given by an explicit function term, but by two functions describing the horizontal and vertical displacement.
Example: Lissajous figures
x(k) = sin(3*k)
y(k) = cos(5*k)
k from -Pi to Pi
Lissajous figures are obtained when two alternating voltages of different frequencies are applied to an oscilloscope.
Families of curves
The program draws graphs of arbitrary functions that contain a parameter k. The values for k can be listed explicitly or defined by start value, end value, and step size.
ƒ(x,k) = sin(x + k)
k from -2 to 2 with step size Pi/4
Polynomial functions (New in version 9.0)
The program performs curve sketching for a polynomial function. This means it determines the derivatives and the antiderivative, checks the function for rational zeros, extrema, inflection points, and symmetry.
Function:
¯¯¯¯¯¯¯¯
ƒ(x) = 3·x4 - 82/3·x2 + 3
= 1/3·(9·x4 - 82·x2 + 9)
= 1/3·(3·x - 1)·(3·x + 1)·(x - 3)·(x + 3)
Derivations:
¯¯¯¯¯¯¯¯¯¯
ƒ'(x) = 12·x3 - 164/3·x
ƒ"(x) = 36·x2 - 164/3
ƒ'"(x) = 72·x
Antiderivative:
¯¯¯¯¯¯¯¯¯¯¯
F(x) = 3/5·x5 - 82/9·x3 + 3·x + c
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.
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Rational functions (New in version 9.0)
The program performs curve sketching for a (rational) function. It determines the derivatives, discontinuities, and continuous extensions. The function is examined for zeros, extrema, inflection points, and its behavior as |x| → ∞.
Function :
¯¯¯¯¯¯¯¯
3·x3 + x2 - 4 (x - 1)·(3·x2 + 4·x + 4)
ƒ(x) = —————— = ———————————
4·x2 - 16 4·(x - 2)·(x + 2)
Singularities:
¯¯¯¯¯¯¯¯¯¯¯
x = 2 Pole with change of sign
x =-2 Pole with change of sign
Derivatives:
¯¯¯¯¯¯¯¯¯¯
3·(x4 - 12·x2) 3·(x2·(x2 - 12))
ƒ'(x) = ———————— = ————————
4·(x4 - 8·x2 + 16) 4·(x - 2)2·(x + 2)2
6·(x3 + 12·x) 6·(x·(x2 + 12))
ƒ"(x) = ——————————— = ———————
x6 - 12·x4 + 48·x2 - 64 (x - 2)3·(x + 2)3
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.
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Curve sketching
The program performs curve sketching for an arbitrary function. This means it determines the derivatives, examines the function for zeros, extrema, and inflection points, draws the graphs of ƒ, ƒ' and ƒ'', and outputs a value table.
Function: ‾‾‾‾‾‾‾‾‾‾‾‾ ƒ(x) = x^4-2*x^3+1 Examination in the range from -10 to 10 Derivations: ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ ƒ'(x) = 4*x^3-6*x^2 ƒ"(x) = 12*x^2-12*x Zeros: ‾‾‾‾‾‾‾‾ N1(1|0) m = - 2 N2(1,83929|0) m = + 4,5912 Extremes: ‾‾‾‾‾‾‾‾‾‾‾‾‾ T1(1,5|-0,6875) m = 0 Pts of inflection: ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ W1(0|1) m = + 0 W2(1|0) m = - 2
Newton iteration
Newton iteration is an approximation method for computing a zero of ƒ(x).
If you enter a starting value x₀ that is sufficiently close to the desired zero,
the next approximation is obtained by computing the intersection of the tangent
to the graph of ƒ at the point
ƒ(x) = x-cos(x)
x ƒ(x) ƒ'(x)
———————— —————— ——————
x0 = 1
x1 = 0,75036387 0,45969769 1,841471
x2 = 0,73911289 0,018923074 1,681905
x3 = 0,73908513 0,00004646 1,6736325
x4 = 0,73908513 0,00000000 1,673612
Integral calculus
(since February 2021 with arc lengths)
The program calculates the oriented and absolute area between two function curves
over a chosen interval. It also determines:
- the moments of rotation around the x‑ and y‑axis,
- the resulting volumes of revolution,
- the arc lengths in the interval [a;b], and
- the centroid of the area (when A₁ = A₂).
ƒ₁(x) = cosh(x) ƒ₂(x) = x^2 + 1 Integration interval [a;b] from -2 to 2 Oriented area : A₁ = -2.07961 Absolute area : A₂ = 2.07961 Arc lengths : L₁[a;b] = 7.254 L₂[a;b] = 9.294
Series expansion
A function given as a series is drawn. Series expansions for different parameter ranges can be compared and shifted vertically for better distinction.
The first 16 terms of the Taylor series for the sine function. ƒ(x,k) = x^(2*k-1)/fac(2*k-1) * (-1)^(k+1), k = 4, 8 and 16
Surface functions
A surface function ƒ(x,y) is drawn — that is, the three‑dimensional graph of a function with two variables.
Example:
ƒ(x, y) = sin(u) / u
u(x, y) = sqrt(x * x + y * y)
-9 ≤ x ≤ 9
-9 ≤ y ≤ 9;
-0.5 ≤ z ≤ 1.5
