Table of Contents
CAS-Calculus Commands
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This section will cover the Calculus commands of the HP Prime. Unless noted, the commands are practically useful only in CAS Mode and in CAS programs.
Examples given in this section is with the CAS Simplify setting set to Maximum. The “Include complex results in variables” option is turned off. Angle Mode is set to Radians - the recommended mode for Calculus commands.
diff
Returns the derivative of the function. diff can be used to return partial derivatives. If only one argument is used, the expression is assumed to be a function of x, with all other variables treated as constants.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
For functions of x:
diff(function of x)
For other functions, including multivariate:
diff(function, variable)
Home Mode - Textbook and Algebraic Entry
Enclose the function and variable in single quotes. Use Shift + (). For powers, use the x^y key.
For functions of X:
CAS.diff('function of X')
General:
CAS.diff('function', 'variable')
Home Mode - RPN Entry
Function of X:
1: 'function of X'
CAS.diff(1), press Enter
General:
2: 'function'
1: 'variable'
CAS.diff(2), press Enter
Examples:
CAS Mode:
Functions of x - all other variables are treated as constants:
diff(a) returns 0
diff(x) returns 1
diff(a*x²+b*x+c) returns 2*a*x+b
diff using two arguments:
diff(a*x²+b*x+c, b) returns x
Partial derivatives:
diff(SIN(x*y²),x) returns y²*COS(x*y²)
diff(SIN(x*y²),y) returns 2*x*y*COS(x*y²)
Notes: You can use the single quote character (') to quickly calculate the derivative of f(x).
Example: (a*x²+b*x+c)' returns 2*a*x+b
Access: Toolbox, CAS, 2. Calculus, 1. diff
See Also: int, limit, series
int
Calculates the integral of a function. Depending on the number of arguments used, an indefinite (symbolic) or definite (numeric) integral will be calculated.
There are three ways to work with int. See the Syntax section for details.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Indefinite Integral of f(x):
int(function)
Indefinite Integral of a specified variable:
int(function, variable)
Definite Integral:
int(function, variable, lower limit, upper limit)
Home Mode - Textbook and Algebraic Entry
Enclose the function and variable in single quotes. Use Shift + (). For powers, use the x^y key.
Indefinite Integral:
CAS.int('function', 'variable')
You can leave out the 'variable' if you are working with a function of X.
Definite Integral:
CAS.int('function', 'variable', lower limit, upper limit)
Home Mode - RPN Entry
Attempting to execute CAS.int will return a Bad Argument Error.
Examples:
CAS Mode:
Indefinite Integral of f(x):
int(2*x*EXP(x)) returns 2*x*EXP(x)-2*EXP(x)
Indefinite Integral of a specified variable:
int(SIN(t),t) returns -COS(t)
Definite Integral:
int(2*x^3,x,.25,.85) returns .25905
Notes: Make sure “int” is in lower case. INT in upper case is the integer function.
Access: Toolbox, CAS, 2. Calculus, 2. int
See Also: ∫, ibpdv, ibpu, preval
limit
Calculates the limit of a function as it approaches a particular value. See the Syntax section for details.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Two sided limit:
lim(function, variable, value)
One sided limit from the right:
lim(function, variable, value, +1)
One sided limit from the left:
lim(function, variable, value, -1)
Examples:
CAS Mode:
Two sided limit:
lim(1/x, x, 1) returns 1
One sided limit from the right:
lim(1/x, x, 0, +1) returns ∞
One sided limit from the left:
lim(1/x, x, 0, -1) returns -∞
Access: Toolbox, CAS, 2. Calculus, 3. limit
See Also: int, diff
series
Calculates the Taylor series expansion of an function. The general syntax is:
series(expression, var=limit point, order, direction)
expression: algebraic expression. Required argument.
var=limit: variable to be expanded on. If limit is not specified, then the limit is assumed to be zero. Required argument.
order: the desired order for the series expansion. This argument is optional. If order is not specified, firth order is specified.
direction: the direction of the series. -1 for negative, 0 for bidirectional, 1 for positive.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Basic fifth order-series at limit = 0:
series(expression, var)
Series with specified limit and order:
series(expression, var=limit, order)
Series with specified limit, order, and direction:
series(expression, var=limit, order, direction)
Home Mode - Textbook and Algebraic Entry
Enclose the function and variable in single quotes. Use Shift + (). For powers, use the x^y key.
General Syntax:
CAS.series('expression', 'variable', 'limit', 'order', 'direction')
Optional arguments: 'limit', 'order', 'direction'
Examples:
CAS Mode:
Basic fifth order-series at limit = 0:
series(COS(x),x) returns 1 - 1/2*x^2 + 1/24*x^4 + x^6*order_size(x)
Series with specified limit and order:
series(COS(x),x=-1,3) returns
COS(1) + SIN(1) * (x+1) - COS(1)/2 * (x+1)^2 - SIN(1)/6 * (x+1)^3 + (x+1)^4*order_size(x+1)
Series with specified limit, order, and direction:
series(LN(x+1),x,4,1) returns LN(5) + (x-4)*order_size(x-4)
series(LN(x+1),x,4,1) returns LN(5) + 1/5*(x-4) + (x-4)^2*order_size(x-4)
series(LN(x+1),x,4,-1) returns LN(5) + order_size(x-4)
Notes: The last term, which contains order_size, is an arbitrary term similar to the Big O notation.
The taylor command for most purposes works in the exact same way as series.
Access: Toolbox, CAS, 2. Calculus, 4. series
See Also: diff, taylor
sum
Calculates the sum of an expression.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
sum(expression, variable, lower limit, upper limit, step)
step can be omitted. When step is omitted, the increment is assumed to be 1.
Examples:
CAS Mode:
sum(2*x,x,1,5) returns 30
sum(2*x,x,1,5,2) returns 18
sum(n^3,n,1,x) returns (x^4 + 2*x^3 + x^2)/4
Notes: An alternate way to use this function is to call the summation template, see Σ.
Access: Toolbox, CAS, 2. Calculus, 5. sum
See Also: Σ
curl
The curl of the algebraic vector. The result is a the cross product the gradient x vector field.
curl([f(x,y,z), g(x,y,z), h(x,y,z)] = [-∂g/∂z + ∂h/∂y, ∂f/∂z - ∂h/∂x, -∂f/∂y + ∂g/∂x ]
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
curl([function_1, function_2, function_3] , [variable_1, variable_2, variable_3])
Examples:
CAS Mode:
curl([5*x*y, z², 2*x*SIN(y)],[x,y,z]) returns [2*x*COS(y)-2*z, -2*SIN(y), -5*x]
Access: Toolbox, CAS, 2. Calculus, 6. Differential, 1. curl
See Also: divergence, grad, hessian, diff
divergence
The divergence of the algebraic vector. The result is a sum of the partial derivatives of each function respect to their variables.
divergence([f(x,y,z), g(x,y,z), h(x,y,z)]) = ∂f/∂x + ∂g/∂y + ∂h/∂z.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
divergence([function_1, function_2, function_3] , [variable_1, variable_2, variable_3])
Examples:
CAS Mode:
divergence([5*x*y, z², 2*x*SIN(y)],[x,y,z]) returns 5*y
Access: Toolbox, CAS, 2. Calculus, 6. Differential, 2. divergence
See Also: curl, grad, hessian, diff
grad
The gradient of trivariate function. The result is a vector of the partial derivatives of the function respect to their variables.
grad(f(x,y,z)) = [ ∂f/∂x, ∂f/∂y, ∂f/∂z ]
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
grad([function , [variable_1, variable_2, variable_3])
Examples:
CAS Mode:
grad(x²*(2*x*y+3*z*LN(x)),[x,y,z]) returns
[6*x²*y+6*x*z*LN(x)+3*x*z, 2*x^3, 3*x²*LN(x)]
Access: Toolbox, CAS, 2. Calculus, 6. Differential, 3. grad
See Also: curl, divergence, hessian, diff
hessian
Creates the Hessian matrix of second partial dervatives, dependent on the function and vector of variables.
Example, for a bivariate function f(x,y), the Hessian matrix is:
H =
| ∂²f/∂x² | ∂²f/∂x∂y |
| ∂²f/∂y∂x | ∂²f/∂y² |
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
hessian(function, vector of variables)
Examples:
CAS Mode:
hessian(8*x²-3*x*y^3, [x, y]) returns
| 16 | -9*y² |
| -9*y² | -18*x*y |
Access: Toolbox, CAS, 2. Calculus, 6. Differential, 4. hessian
See Also: curl, divergence, grad, diff
ibpdv
Integration by Parts. There are five arguments to ibpdv but only two are required. The general syntax:
ibpdv( u*v', v, variable, lower limit, upper limit)
Concept: ∫ u*v' dx = u*v - ∫u'*v dx
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Indefinite Integration by Parts of f(x):
ibpdv( u*v', v)
Definite Integration by Parts:
ibpdv( u*v', v, variable, lower limit, upper limit)
Output : [ u*v, -u*v ] If limits are supplied, then the definite integral ∫ u*v dx is calculated (first part).
Examples:
CAS Mode:
Indefinite Integration by Parts of f(x):
ibpdv(x²*COS(x), SIN(x)) returns [x²*SIN(x), -2*x*SIN(x)]
Note: v = SIN(X)
ibpdv(x*EXP(x), EXP(x)) returns [x*EXP(x), -EXP(x)]
Note: EXP(x)
Definite Integration by Parts:
ibpdv(x²*COS(x), SIN(x), x, 0, π) returns [0, -2*x*SIN(x)]
Note: v= COS(x)
Access: Toolbox, CAS, 2. Calculus, 7. Integral, 1. ibpdv
See Also: int, ibpu, ∫
ipbu
Integration by Parts. There are five arguments to ipbu but only two are required. The general syntax:
ibpu( u*v', u, variable, lower limit, upper limit)
Concept: ∫ u*v' dx = u*v - ∫u'*v dx
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Indefinite Integration by Parts of f(x):
ibpu( u*v', v)
Definite Integration by Parts:
ibpu( u*v', v, variable, lower limit, upper limit)
Output : [ u*v, -u*v ] If limits are supplied, then the definite integral ∫ u*v dx is calculated (first part).
Examples:
CAS Mode:
Indefinite Integration by Parts of f(x):
ibpu(x²*COS(x),x²) returns [x²*SIN(x), -2*x*SIN(x)]
Note: u = x²
ibpdv(x*EXP(x), EXP(x)) returns [x²*EXP(x)/2, -x²*EXP(x)/2]
Note: u = EXP(x)
Definite Integration by Parts:
ibpdv(x²*COS(x), COS(x), x, 0, π) returns [-π^3/3, x^3*SIN(x)/3]
Note: u = COS(x)
Access: Toolbox, CAS, 2. Calculus, 7. Integral, 2. ibpu
See Also: int, ∫, ibpdv
preval
Calculates F(b) - F(a). This is useful for calculating definite integrals where the anti-derivative is already known.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
F is a function of x:
preval(F, lower limit, upper limit)
F is a function of some other variable:
preval(F, lower limit, upper limit, variable)
Examples:
CAS Mode:
preval(x^3, 4, 10) = 10^3 - 4^3 returns 936
preval(9*d, 1, 9, d) returns 72
Notes: preval does not work in Home Mode
Access: Toolbox, CAS, 2. Calculus, 7. Integral, 3. preval
See Also: int, ∫, ibpdv, ibpu
sum_riemann
Calculates the sum:
Σ( f(n,k) from k=1 to n ) in the neighborhood of n = ∞, when the sum associated with a continuous function defined on the closed interval [0, 1].
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
sum_riemann( f(n,k), [n, k] )
Examples:
CAS Mode:
sum_riemann(k^3/n^5, [n,k]) returns 1/(4*n)
sum_riemann(1/k + 1/n, [n, k]) returns ∞
sum_riemann(1/(n*k-1), [n, k]) returns “Probably not a Riemann sum”
Notes: If a Riemann sum cannot be reached, the HP Prime returns the error message “it is not probably not a Riemann sum”.
Attempting to execute sum_riemann function in any of the Home modes will return an error message.
Access: Toolbox, CAS, 2. Calculus, 8. Limits, 1. sum_riemann
See Also: diff, limit
taylor
Calculates the Taylor series expansion of an function. The general syntax is:
taylor(expression, var=limit point, order, direction)
expression: algebraic expression. Required argument.
var=limit: variable to be expanded on. If limit is not specified, then the limit is assumed to be zero. Required argument.
order: the desired order for the series expansion. This argument is optional. If order is not specified, firth order is specified.
direction: the direction of the series. -1 for negative, 0 for bidirectional, 1 for positive.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Basic fifth order-series at limit = 0:
taylor(expression, var)
Series with specified limit and order:
taylor(expression, var=limit, order)
Series with specified limit, order, and direction:
taylor(expression, var=limit, order, direction)
Examples:
CAS Mode:
Basic fifth order-series at limit = 0:
taylor(COS(x),x) returns 1 - 1/2*x^2 + 1/24*x^4 + x^6*order_size(x)
Series with specified limit and order:
taylor(COS(x),x=-1,3) returns
COS(1) + SIN(1) * (x+1) - COS(1)/2 * (x+1)^2 - SIN(1)/6 * (x+1)^3 + (x+1)^4*order_size(x+1)
Series with specified limit, order, and direction:
taylor(LN(x+1),x,4,0) returns
x - 1/2*x² + 1/3*x^3 - 1/4*x^4 + x^5*order_size(x)
taylor(LN(x+1),x,4,1) returns
LN(2) + 1/2*(x-1) - 1/8*(x-1)² + 1/24*(x-1)^3 - 1/64*(x-1)^4 + (x-1)^5*order_size(x-1)
series(LN(x+1),x,4,-1) returns LN(x)
Notes: The last term, which contains order_size, is an arbitrary term similar to the Big O notation.
The series command for most purposes works almost in the exact same way as taylor. Except taylor will attempt to return a polynomial when possible.
Access: Toolbox, CAS, 2. Calculus, 8. Limits, 2. taylor
See Also: diff, series
divpc
Division by increasing power order. The result is a Taylor expansion of a(x)/b(x) at the order of n in the vicinity of x=0. a(x) and b(x) are polynomials of x.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
divpc( a(x), b(x), n )
Examples:
CAS Mode:
divpc( (x² + 2*x + 1)/x, 5) returns
±∞*x^5 ±∞*x^4 ±∞*x^3 ± ∞*x^2 ±∞*x ± ∞
divpc( x²+3*x^3, 1+x, 5) returns
2*x^5 - 2*x^4 + 2*x^3 + x^2
Access: Toolbox, CAS, 2. Calculus, 8. Limits, 3. divpc
See Also: diff, limit, taylor
laplace
Performs the Laplace Transform.
F(s) = ∫ exp(-s*t) *f(t) dt from t=0 to ∞
The HP Prime returns the Laplace Transform using the same variable unless you specify which variables are to be used.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Basic:
laplace(expression)
Variables Specified:
laplace(expression, var, lapvar)
Examples:
CAS Mode:
laplace(a) returns a/x
laplace(t², t, s) returns 2/s^3
laplace(EXP(t)*SIN(t),t,s) returns 1/(s²-2*s+2)
Access: Toolbox, CAS, 2. Calculus, 9. Transform, 1. laplace See Also: invlaplace
invlaplace
Performs the Inverse Laplace Transform.
F(s) = ∫ exp(-s*t) *f(t) dt from t=0 to ∞
If the following equation, invlaplace finds f(t). The HP Prime returns the Inverse Laplace Transform using the same variable unless you specify which variables are to be used.
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Basic:
invlaplce(expression)
Variables Specified:
invlaplace(expression, variable, invlap var)
Examples:
CAS Mode:
invlaplace(a) returns a*Dirac(x)
invlaplace(1/s^4, s, t) returns t^3/6
invlaplace(1/(s-1),s,t) returns EXP(t)
Access: Toolbox, CAS, 2. Calculus, 9. Transform, 2. invlaplace
See Also: invlaplace
fft
Discrete Fast Fourier Transform
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
Basic Fast Fourier Transform:
fft(vector)
Field of z/pz with a primitive nth root of 1:
fft(vector, A, P)
Home Mode
Normal Programming Editor
Basic Fast Fourier Transform - Numerical Only:
CAS.fft(vector)
Field of z/pz with a primitive nth root of 1 - Numerical Only:
CAS.fft(vector, A, P)
Examples:
CAS Mode:
fft([7,1,2]) returns [10, 5.5+.866025403784i, 5.5-.866025403784i]
fft([7,1,2],2,3) returns [1, -1, 0]
Access: Toolbox, CAS, 2. Calculus, 9. Transform, 3. fft
See Also: ifft
ifft
Inverse Fast Fourier Transform
Syntax
Input:
CAS Mode
Program Editor - CAS Programs
ifft(vector)
Home Mode
Normal Programming Editor
Numerical Only:
CAS.ifft(vector)
Examples:
CAS Mode:
ifft([1,2+i,2-i]) returns [1.66666666667, -.910683602523, .244016935856]
Access: Toolbox, CAS, 2. Calculus, 9. Transform, 4. ifft
See Also: fft
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