aosor (Function)

Command last updated 6/3/2024.

aosor(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

Returns the area of the surface obtained by rotating the graph of f between x = a and x = b around the x-axis.

Example

Find the integral that gives the area of the surface of revolution obtained by rotating the graph of \(y = \cos^{-1}(x)\) between \(y = 0\) and \(y = \dfrac{\pi}{2}\) around the \(y\)-axis. (2023 Specialist Exam 2 Section A Question 11)

Contributed by Nhan

aosort (Function)

Command last updated 6/3/2024.

aosort(x, y, t, a, b)

• x is an expression in terms of t
• y is an expression in terms of t
• t is a variable name
• a is the t-value of the starting point
• b is the t-value of the ending point

Returns the area of the surface obtained by rotating the curve between t = a and t = b around the x-axis.

Example

Find the area of the surface of revolution obtained by rotating the curve defined by \(x = t + 1\) and \(y = t^2\) between \(t=0\) and \(t=2\) around the \(x\)-axis.

Contributed by Nhan

arcLenv (Function)

Command last updated 23/11/2022.

arcLenv(r, t, a, b)

• r is a vector in terms of t
• t is a variable name
• a is the t-value of the starting point
• b is the t-value of the ending point

Returns the arc length of r between t = a and t = b.

Example

Find the arc length of the curve described by \(x = 2 \cos(t) + 2\) and \(y = (e-2) \sin(t)\), between \(t = \dfrac{\pi}{2}\) and \(t = \pi\). (2023 Specialist Exam 2 Section B Question 1f)

Contributed by Nathan Chen

aroc (Function)

Command last updated 4/7/2022.

aroc(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

Returns the average rate of change of f between x=a and x=b.

Example

Find the average rate of change of \(h(t) = -60 \cos \left( \dfrac{\pi t}{15} \right) + 75\), between \(t=0\) and \(t=\dfrac{30}{4}\). (2023 Methods Exam 2 Section B Question 2c)

Contributed by Nhan

avalue (Function)

Command last updated 4/7/2022.

avalue(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

Returns the average value of f between x=a and x=b.

Example

Find the average value of \(h(t) = -60 \cos \left( \dfrac{\pi t}{15} \right) + 75\), between \(t=0\) and \(t=\dfrac{30}{4}\). (2023 Methods Exam 2 Section B Question 2b)

Contributed by Nhan

axisints (Program)

Command last updated 9/7/2022.

IMPORTANT - requires printpts stored in the same folder as your current folder or in the “library” folder (see the Variable Manager).

axisints(f, l, u)

• f is an expression in terms of x
• l is the lower boundary of x (l should not be in terms of unknown variables, and should be -∞ if f contains other unknown variables besides x)
• u is the upper boundary of x (u should not be in terms of unknown variables, and should be ∞ if f contains other unknown variables besides x)

Returns a list of coordinates of x-axis and y-axis intercepts on f in the domain xϵ[l, u].

Example

Find the axis intercepts of \(y = x(x - 2)(x + 1)\). (2023 Methods Exam 2 Section B Question 1a)

Contributed by Nhan

biCDf (Function)

Command last updated 4/7/2022.

biCDf(l, u, n, p)

• l is the query lower boundary
• u is the query upper boundary
• n is the number of trials
• p is the probability of success each trial

l and u should evaluate to numbers, not expressions in terms of unknown variables. Prefer to use discrete if it would be more appropriate.

Equivalent to the builtin binomialCDf(l, u, n, p), but returns exact values in Standard mode whereas binomialCDf can return only decimal answers.

Returns the probability of between and including l and u successes from n trials with p probability of success each trial.

Example

Let \(X \sim \mathrm{Bi}(7, p)\). Find \(\mathrm{Pr}(2 \leq X \leq 3)\) as a polynomial in terms of \(p\). (2017 Methods Exam 2 Section B Question 3f)

Contributed by Nhan

biPDf (Function)

Command last updated 4/7/2022.

biPDf(x, n, p)

• x is the query integer
• n is the number of trials
• p is the probability of success each trial

Equivalent to the builtin binomialPDf(x, n, p), but returns exact values in Standard mode whereas binomialPDf can return only decimal answers.

Returns the probability of x successes from n trials with p probability of success each trial.

Example

A box contains \(n\) green balls and \(m\) red balls. A ball is selected at random, then replaced in the box. Find the probability that in 8 such selections a green ball is selected at least once. (2023 Methods Exam 2 Section A Question 8)

Contributed by Nhan

bmeth (Program)

Command last updated 7/2/2023.

bmeth(f, xl, xh, n)

• f is an expression in terms of x (execution of Program may fail if there are other unknown variables in the expression)
• [xl, xh] is an interval containing a root of f
• n is the number of iterations

Applies the bisection method for n iterations. Prints the details of each iteration and returns the coordinates following the n-th iteration.

Example

Find the \(x\)-value for a root of \(\sin(x)\) estimated by the bisection method, with an initial range of \([3, 5]\), after 2 iterations.

Contributed by Nhan

cis (Function)

Command last updated 23/11/2022.

cis(θ)

• θ is an angle

Returns cos(θ) + i sin(θ).

Example

Find \(\mathrm{cis} \left( -\dfrac{2\pi}{7} \right) + \mathrm{cis} \left( -\dfrac{12\pi}{7} \right)\) in the form \(A \cos(B \pi)\). (2023 Specialist Exam 2 Section B Question 2fi)

Contributed by Nhan

cosec (Function)

Command last updated 23/11/2022.

cosec(θ)

• θ is an angle

Returns the cosecant of θ.

Example

Find the largest set of values of \(x\) for which \(\cos(x) > \dfrac{1}{4} \csc(x)\). (2017 Specialist Exam 2 Section A Question 2)

Contributed by Nhan

cot (Function)

Command last updated 23/11/2022.

cot(θ)

• θ is an angle

Returns the cotangent of θ.

Example

Find the cotangent of \(-\dfrac{2\pi}{3}\).

Contributed by Nhan

difft (Function)

Command last updated 23/11/2022.

difft(x, y, t, n)

• x is an expression in terms of t
• y is an expression in terms of t
• t is a variable name
• n is the derivative order

n must be 0, 1 or 2.

Returns \(\dfrac{ d^{n} y }{ d x^{n} }\).

Example

Find \(\dfrac{dy}{dx}\) when \(t=2\) given that \(x = \dfrac{6t}{t+1}\) and \(y = \dfrac{-8}{t^2 + 4}\). (2023 Specialist Exam 2 Section A Question 9)

Contributed by Nhan

discrete (Function)

Command last updated 4/7/2022.

Note - this program does not work in all cases, such as biCDf (use the builtin binomialCDf instead). Answers should be tested by the user.

discrete(”f ”, x, l, u, h)

• f is an expression in terms of x, enclosed in ” ”
• x is a variable name
• l is the x-value of the starting point
• u is the x-value of the ending point
• h is the step size

Returns a matrix with x-values in the left column and f-values at the corresponding x-values in the right column, for discrete values of x with step size h in the domain [l,u].

Example

\(X \sim \mathrm{Bi}(n, 0.1)\). Find the least \(n\) such that \(\mathrm{Pr}(X \geq 2) \geq 0.5\). (2021 Methods Exam 2 Section A Question 17)

Contributed by Nhan

hCDf (Function)

Command last updated 4/7/2022.

hCDf(l, u, n, M, N)

• l is the query lower boundary
• u is the query upper boundary
• n is the number of draws from the population
• M is the number of successes in the population
• N is the population size

Parameters should evaluate to numbers, not expressions in terms of unknown variables.

Equivalent to the builtin hypergeoCDf(l, u, n, M, N), but returns exact values in Standard mode whereas hypergeoCDf can return only decimal answers.

Returns the probability of between and including l and u successes from n draws, without replacement between draws, from a population size N that contains M successes.

The hypergeometric distribution is not on the VCE Mathematical Methods Study Design, but can be useful for certain examination questions. See the Example. Students are expected to, and should be able to, answer these questions using alternative Mathematical Methods. This function should be used only for checking, and only with an understanding of the circumstances in which the hypergeometric distribution applies.

Example

A bag contains 5 blue marbles and 4 red marbles. 4 balls are taken from the bag, without replacement between draws. Find the probability that the proportion of blue marbles in the 4 draws is greater than half. (2017 NHT Methods Exam 2 Section A Question 11)

Contributed by Nhan

hPDf (Function)

Command last updated 4/7/2022.

hPDf(x, n, M, N)

• x is the query integer
• n is the number of draws from the population
• M is the number of successes in the population
• N is the population size

Equivalent to the builtin hypergeoPDf(x, n, M, N), but returns exact values in Standard mode whereas hypergeoPDf can return only decimal answers.

Returns the probability of x successes from n draws, without replacement between draws, from a population size N that contains M successes.

The hypergeometric distribution is not on the VCE Mathematical Methods Study Design, but can be useful for certain examination questions. See the Example. Students are expected to, and should be able to, answer these questions using alternative Mathematical Methods. This function should be used only for checking, and only with an understanding of the circumstances in which the hypergeometric distribution applies.

Example

A box contains n marbles, of which k marbles are coloured red and the remainder n-k marbles are coloured green. 2 marbles are drawn from the box, without replacement between draws. Find the probability that the 2 marbles drawn are the same colour. (2019 Methods Exam 2 Section A Question 17)

Contributed by Nhan

infpts (Program)

Command last updated 23/11/2022.

IMPORTANT - requires printpts stored in the same folder as your current folder or in the “library” folder (see the Variable Manager).

infpts(f, l, u)

• f is an expression in terms of x
• l is the lower boundary of x (l should not be in terms of unknown variables, and should be -∞ if f contains other unknown variables besides x)
• u is the upper boundary of x (u should not be in terms of unknown variables, and should be ∞ if f contains other unknown variables besides x)

Returns a list of coordinates of points on f in the domain xϵ[l, u] where the second derivative equals 0.

Each point must be verified to determine whether it is really a point of inflection.

Example

Find the coordinates of any points of inflection of the graph of \(\dfrac{x}{1+x^{3}}\). (2017 Specialist Exam 2 Section B Question 1aiii)

Contributed by Nhan

intpts (Program)

Command last updated 9/7/2022.

IMPORTANT - requires printpts stored in the same folder as your current folder or in the “library” folder (see the Variable Manager).

intpts(f, g, l, u)

• f is an expression in terms of x
• g is an expression in terms of x
• l is the lower boundary of x (l should not be in terms of unknown variables, and should be -∞ if either f or g contains other unknown variables besides x)
• u is the upper boundary of x (u should not be in terms of unknown variables, and should be ∞ if either f or g contains other unknown variables besides x)

Returns a list of coordinates of points of intersection of f and g.

Example

Find the coordinates of the points of intersection of the graphs of \((x-1)^3 (x+2)^3\) and \((x-1)^2 (x+2)^3\). (2019 NHT Methods Exam 2 Section B Question 1a)

Contributed by Nhan

keypts (Program)

Command last updated 9/7/2022.

IMPORTANT - requires printpts stored in the same folder as your current folder or in the “library” folder (see the Variable Manager).

keypts(f, l, u)

• f is an expression in terms of x (execution of Program may partially fail if there are other unknown variables in the expression)
• l is the lower boundary of x (l should not be in terms of unknown variables)
• u is the upper boundary of x (u should not be in terms of unknown variables)

Prints and categorises coordinates of key points on y=f in the domain xϵ[l, u].

• y-axis intercepts where x=0
• x-axis intercepts where y=0
• stationary points where f \('\)=0
  • local minima where f \(''\)>0
  • local maxima where f \(''\)<0
  • stationary points of inflection where f \(''\)=0 and f \('''\)≠0
  • stationary points of unknown nature where f \(''\)=0 and f \('''\)=0 (users should verify the nature of these points themselves)
• points of inflection where f \(''\)=0 and f \('''\)≠0
• possible points of inflection where f \(''\)=0 and f \('''\)=0 (users should verify the nature of these points themselves)
• endpoints where x=l and x=u

The 2nd and 3rd derivative tests are not on the VCE Mathematical Methods Study Design.

Example

Find key points on \(y = \dfrac{1}{5} x^5 + \dfrac{3}{4} x^4 - 2x^2 + \dfrac{12}{5}\) for \(x \in (-\infty, 2]\).

Contributed by Nhan

lineof (Function)

Command last updated 4/7/2022.

lineof(a, b, c, d, x)

• (a, b) are the coordinates of a point
• (c, d) are the coordinates of a point
• x is a variable name

Returns an expression for the line connecting points (a, b) and (c, d) in terms of x.

Example

Let \(f(x) = x^3 - 5x\). Find the equation of the line through \((-1, f(-1))\) and \((1, f(1))\). (2017 Methods Exam 2 Section B Question 1bi)

Contributed by Nhan

mu (Function)

Command last updated 4/7/2022.

mu(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

For fastest calculation time, the domain [a, b] should be as small as possible.

If f is a piecewise function it should be entered raw not as a user-defined function.

The integral of f over the domain [a, b] must equal 1, as this is assumed by the mu function.

Returns the mean of the continuous probability distribution defined by the probability density function f over the domain [a, b].

Example

Find the expected value of \(v\) for the probability density function

[f(v) = \left{ \begin{array}{ll} \dfrac{1}{6 \pi} \sin \left( \sqrt{ \dfrac{v-30}{3} } \right) & 30 \leq v \leq 3 \pi^2 + 30
\ 0 & \mathrm{elsewhere} \end{array} \right.]

(2023 Methods Exam 2 Section B Question 4i)

Contributed by Nhan

nPDf (Function)

Command last updated 4/7/2022.

nPDf(x, σ, μ)

• x is a number or variable name
• σ is the standard deviation
• μ is the mean

Equivalent to the builtin normPDf(x, σ, μ), but more convenient to obtain the specific probability density function and to graph the function.

Returns the value of the normal distribution function with a standard deviation of σ and a mean of μ at the value x.

Example

Let \(M \sim \mathrm{N}(68, 8^2)\). Find \(\mathrm{Pr}(60<M<90)\). (2018 Methods Exam 2 Section B Question 4a)

Contributed by Nhan

nmeth (Program)

Command last updated 7/2/2023.

nmeth(f, xa, n)

• f is an expression in terms of x (execution of Program may fail if there are other unknown variables in the expression)
• xa is the initial guess for the x-value of a root of f
• n is the number of iterations

Applies Newton’s method to approximate an x-intercept of f for n iterations. Prints the details of each iteration and returns the coordinates following the n-th iteration. Column i is the iteration number, x is \(x_i\), y is \(f(x_i)\), dy/dx is \(f'(x_i)\) and δx is \(x_{i+1} - x_i = - \dfrac{f(x_i)}{f'(x_i)}\).

Example

Find \(x_1\), \(x_2\) and \(x_3\) using Newton’s method with \(x_0 = 0\) to approximate an x-intercept of \(h(x) = 2^x - x^2\). (2023 Methods Exam 2 Section B Question 3f)

Contributed by Nhan

normalt (Function)

Command last updated 23/11/2022.

normalt(x, y, t, to)

• x is an expression in terms of t
• y is an expression in terms of t
• t is a variable name
• to is a value of t

Returns the expression in terms of x for the normal line to the curve at the point t = to.

Example

Find the normal to the curve defined by \(x = 2\sin(2t)\) and \(y = 3\cos(t)\) when \(t = \dfrac{\pi}{3}\).

Contributed by Nhan

pcheck (Program)

Command last updated 6/3/2024.

Note - this program may throw an error if in Decimal mode. Use Standard mode instead.

Define f(x) = {a, b, c, d, e}

pcheck(f(x), l, r)

• f(x) is defined as a list of any length containing expressions {a, b, …} in terms of x
• l is the left hand side of the equation
• r is the right hand side of the equation

Returns a matrix with each row corresponding to an option of f(x), with the columns containing for that option, from left to right, f(x), l, r, judge(l=r).

Example

\(f(x+f(x)) = f(2x)\). Which option could possibly be \(f(x)\)? (2018 Methods Exam 2 Section A Question 10)

Contributed by Nhan and Kevin

printpts (Program)

Command last updated 9/7/2022.

printpts(f, {s})

• f is an expression in terms of x
• s is a list of x-values, enclosed in { }

Returns a list of coordinates of points on f corresponding to the x-values in s.

Example

Find the \(y\)-values of the points where \(x = - \infty, -1, 0, 1, \infty\) on \(y = (x-1)^2 (x+2)^3\).

Contributed by Nhan

scalares (Function)

Command last updated 23/11/2022.

scalares(a, b)

• a, b are vectors

Returns the scalar resolute of a in the direction of b.

Example

Find \(\alpha\) for which the scalar resolute of \(3\boldsymbol{\hat{\imath}} + 2\boldsymbol{\hat{\jmath}} + \alpha\boldsymbol{\hat{k}}\) in the direction of \(4\boldsymbol{\hat{\imath}} - \boldsymbol{\hat{\jmath}} + \alpha^2 \boldsymbol{\hat{k}}\) equals \(\dfrac{74}{\sqrt{273}}\). (2016 Specialist Exam 2 Section A Question 11)

Contributed by Nhan

sd (Function)

Command last updated 4/7/2022.

sd(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

For fastest calculation time, the domain [a, b] should be as small as possible.

If f is a piecewise function it should be entered raw not as a user-defined function.

The integral of f over the domain [a, b] must equal 1, as this is assumed by the sd function.

Returns the standard deviation of the continuous probability distribution defined by the probability density function f over the domain [a, b].

Example

Find the standard devation of \(x\) for the probability density function

[\left{ \begin{array}{ll} \dfrac{x}{500} & 0 \leq x < 20
\dfrac{50-x}{750} & 20 \leq x \leq 50
\ 0 & \text{elsewhere} \end{array} \right.]

(2021 Methods Exam 2 Section B Question 4g)

Contributed by Nhan

sec (Function)

Command last updated 23/11/2022.

sec(θ)

• θ is an angle

Returns the secant of θ.

Example

Find the expression for \(\int_0^{\tfrac{\pi}{6}} \tan^2(x)\sec^2(x) dx\) following a suitable substitution. (2018 Specialist Exam 2 Section A Question 8)

Contributed by Nhan

simlin (Function)

Command last updated 4/7/2022.

simlin(a, b, c, d, e, f, g)

• a, b, c are numbers or expressions in terms of g, representing a line ax+by=c
• d, e, f are numbers or expressions in terms of g, representing a line dx+ey=f
• g is a variable name

Returns a list.

• 1st item is values of g for which the 2 lines intersect 0 times.
• 2nd item is values of g for which the 2 lines intersect 1 time.
• 3rd item is values of g for which the 2 lines intersect ∞ times.

Example

Find the largest set of values of \(k\) for which the simultaneous linear equations \(kx+5y=k+5\) and \(4x+(k+1)y=0\) have ∞ solutions for \((x, y)\). (2023 Methods Exam 2 Section A Question 4)

Contributed by Nhan

statpts (Program)

Command last updated 9/7/2022.

IMPORTANT - requires printpts stored in the same folder as your current folder or in the “library” folder (see the Variable Manager).

statpts(f, l, u)

• f is an expression in terms of x
• l is the lower boundary of x (l should not be in terms of unknown variables, and should be -∞ if f contains other unknown variables besides x)
• u is the upper boundary of x (u should not be in terms of unknown variables, and should be ∞ if f contains other unknown variables besides x)

Returns a list of coordinates of stationary points on f in the domain xϵ[l, u].

Example

Find the coordinates of the stationary points of \(f(x) = x(x-2)(x+1)\). (2023 Methods Exam 2 Section B Question 1b)

Contributed by Nhan

suvat (Function)

Command last updated 23/11/2022.

suvat(b, c, d, e, f)

• b, c, d are equations of the form g=h, where each g is a different one of s, u, v, a, t and h is the corresponding value
• e, f are the remaining variables from s, u, v, a, t

s is displacement, u is initial velocity, v is final velocity, a is acceleration, t is elapsed time

Returns the solutions for e and f given constant acceleration and the provided conditions.

Example

Find the time taken for a phone to hit the ground, experiencing acceleration due to only gravity after being dropped from a balloon \(80 \mathrm{m}\) above the ground ascending at \(2.5 \mathrm{m/s}\). (2023 Specialist Exam 2 Section A Question 13)

Contributed by Nathan Chen, Nhan

tSolve (Function)

Command last updated 23/11/2022.

tSolve(Vin, QVin, Vout, Vo, Qo)

• Vin is the rate of solution volume flowing into the tank
• QVin is the concentration of solute in incoming solution
• Vout is the rate of solution volume flowing out of the tank
• Vo is the initial solution volume in the tank
• Qo is the initial quantity of solute in the tank

Returns an equation relating Q the quantity of solute in the tank and t time.

Example

A tank initially has \(20 \mathrm{kg}\) salt dissolved in \(100 \mathrm{L}\) water, with pure water flowing into the tank at \(10 \mathrm{L/minute}\) and solution flowing out of the tank at \(5 \mathrm{L/minute}\). Find the equation which relates the mass of salt and time. (2016 Specialist Exam 2 Section B Question 3a)

Contributed by Nhan

tanLinet (Function)

Command last updated 23/11/2022.

tanLinet(x, y, t, to)

• x is an expression in terms of t
• y is an expression in terms of t
• t is a variable name
• to is a value of t

Returns the expression in terms of x for the tangent line to the curve at the point t = to.

Example

Find the tangent to the curve defined by \(x = 2\sin(2t)\) and \(y = 3\cos(t)\) when \(t = \pi\). (2020 Specialist Exam 2 Section B Question 1bi)

Contributed by Nhan

tank (Function)

Command last updated 23/11/2022.

tank(Vin, QVin, Vout, Vo, Qo)

• Vin is the rate of solution volume flowing into the tank
• QVin is the concentration of solute in incoming solution
• Vout is the rate of solution volume flowing out of the tank
• Vo is the initial solution volume in the tank
• Qo is the initial quantity of solute in the tank

Returns a differential equation relating Q the quantity of solute in the tank, t time, and Q’ the derivative of Q with respect to t.

Example

A tank initially has some quantity of chemical dissolved in 8000 L water, with fresh water flowing into the tank at 15 L/minute and solution flowing out of the tank at 20 L/minute. Find the equation which relates the mass of chemical and time. (2023 Specialist Exam 2 Section A Question 8)

Contributed by Nhan

trans (Function)

Command last updated 6/3/2024.

trans(e, t, x, y)

• e is an equation in terms of x and y
• t is a matrix transformation of x and y
• x and y are variable names

Returns the equation after the transformation is applied.

Example

\(f(x) = e^x + e^{-x}\) and \(g(x) = \dfrac{1}{2} f(2-x)\). Find a possible sequence of transformations that map \(f\) to \(g\). (2023 Methods Exam 2 Section B Question 5a)

Contributed by Nhan

trsum (Function)

Command last updated 21/7/2022.

trsum(f, v, a, b, w)

• f is an expression in terms of v
• v is a variable name
• a is the v-value of the starting point
• b is the v-value of the ending point
• w is the trapezium width

Returns the approximation to \(\int_a^b f dv\) using trapeziums of width w and vertical parallel side lengths equal to the value of f at the trapezium’s left and right endpoints.

Example

Find the approximation of

[\int_0^{4} \left( \dfrac{1}{27} (x-3)^2 (x+3)^2 + 1 \right) dx]

using the trapezium rule with trapeziums of width \(1\). (Sample Methods Exam 2 Section A Question 3)

Contributed by Nathan Chen

var (Function)

Command last updated 4/7/2022.

var(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

For fastest calculation time, the domain [a, b] should be as small as possible.

If f is a piecewise function it should be entered raw not as a user-defined function.

The integral of f over the domain [a, b] must equal 1, as this is assumed by the var function.

Returns the variance of the continuous probability distribution defined by the probability density function f over the domain [a, b].

Example

Find the expected value of \(x\) for the probability density function

[\left{ \begin{array}{ll} 0.2e^{-0.2x} & x \geq 0
\ 0 & x < 0 \end{array} \right.]

(2021 NHT Methods Exam 2 Section A Question 14)

Contributed by Nhan

vectorej (Function)

Command last updated 23/11/2022.

vectorej(a, b)

• a, b are vectors

Returns the vector rejection of a perpendicular to b.

Example

Find the vector rejection of \(2\boldsymbol{\hat{\imath}} - \boldsymbol{\hat{\jmath}} + 3\boldsymbol{\hat{k}}\) perpendicular to \(\boldsymbol{\hat{\imath}} + \boldsymbol{\hat{\jmath}} - \boldsymbol{\hat{k}}\). (2019 NHT Specialist Exam 2 Section A Question 11)

Contributed by Nhan

vectores (Function)

Command last updated 23/11/2022.

vectores(a, b)

• a, b are vectors

Returns the vector resolute of a in the direction of b.

Example

Find the vector resolute of \(3\boldsymbol{\hat{\imath}} - 4\boldsymbol{\hat{\jmath}} + 12\boldsymbol{\hat{k}}\) in the direction of \(2\boldsymbol{\hat{\imath}} + 2\boldsymbol{\hat{\jmath}} - \boldsymbol{\hat{k}}\). (2017 Specialist Exam 2 Section A Question 13)

Contributed by Nhan

vosor (Function)

Command last updated 23/11/2022.

vosor(f, x, a, b)

• f is an expression in terms of x
• x is a variable name
• a is the x-value of the starting point
• b is the x-value of the ending point

Returns the volume of the solid obtained by rotating the graph of f between x = a and x = b around the x-axis.

Example

Find the volume of the solid of revolution obtained by rotating the graph of \(y^2 = x - 1\) between \(x=2\) and \(x=5\) around the \(x\)-axis. (2023 Specialist Exam 2 Section B Question 3a)

Contributed by Nhan