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Mathematics Formulae

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Page 1: Mathematics Formulae

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Page 3: Mathematics Formulae

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Page 7: Mathematics Formulae

Differentiation Formulas

d

dxk = 0 (1)

d

dx[f(x) ± g(x)] = f ′(x) ± g′(x) (2)

d

dx[k · f(x)] = k · f ′(x) (3)

d

dx[f(x)g(x)] = f(x)g′(x) + g(x)f ′(x) (4)

d

dx

(

f(x)

g(x)

)

=g(x)f ′(x) − f(x)g′(x)

[g(x)]2

(5)

d

dxf(g(x)) = f ′(g(x)) · g′(x) (6)

d

dxxn = nxn−1 (7)

d

dxsin x = cosx (8)

d

dxcosx = − sinx (9)

d

dxtan x = sec2 x (10)

d

dxcotx = − csc2 x (11)

d

dxsecx = secx tan x (12)

d

dxcscx = − cscx cotx (13)

d

dxex = ex (14)

d

dxln |x| =

1

x(15)

d

dxsin−1 x =

1√1 − x2

(16)

d

dxcos−1 x =

−1√1 − x2

(17)

d

dxtan−1 x =

1

x2 + 1(18)

d

dxcot−1 x =

−1

x2 + 1(19)

d

dxsec−1 x =

1

|x|√

x2 − 1(20)

d

dxcsc−1 x =

−1

|x|√

x2 − 1(21)

Trigonometry

x

y

30◦

45◦60◦120◦

135◦

150◦

240◦225◦

210◦ 330◦

315◦

300◦

(1, 0)

(0, 1)

(−1, 0)

(0,−1)

(√

3

2,

1

2)

(√

2

2,

√2

2)

( 1

2,

√3

2)(− 1

2,

√3

2)

(−√

2

2,

√2

2)

(−√

3

2,

1

2)

(−√

3

2,−

1

2)

(−√

2

2,−

√2

2)

(− 1

2,

−√

3

2)

(√

3

2,−

1

2)

(√

2

2,−

√2

2)

( 1

2,−

√3

2)

sin θ = y cos θ = x

Basic Identities

tanx =sin x

cosx(1)

cotx =1

tan x(2)

secx =1

cosx(3)

cscx =1

sin x(4)

Pythagorean Identities

sin2 x + cos2 x = 1 (5)

1 + cot2 x = csc2 x (6)

tan2 x + 1 = sec2 x (7)

Sum and Difference Formulas

sin(x ± y) = sin x cos y ± cosx sin y (8)

cos(x ± y) = cosx cos y ∓ sin x sin y (9)

Power Reducing Formulas

sin2 x =1 − cos 2x

2(10)

cos2 x =1 + cos 2x

2(11)

Page 8: Mathematics Formulae

Differentiation Formulas

d

dxk = 0 (1)

d

dx[f(x)± g(x)] = f ′(x)± g′(x) (2)

d

dx[k · f(x)] = k · f ′(x) (3)

d

dx[f(x)g(x)] = f(x)g′(x) + g(x)f ′(x) (4)

d

dx

(f(x)g(x)

)=

g(x)f ′(x)− f(x)g′(x)[g(x)]2

(5)

d

dxf(g(x)) = f ′(g(x)) · g′(x) (6)

d

dxxn = nxn−1 (7)

d

dxsinx = cos x (8)

d

dxcos x = − sinx (9)

d

dxtanx = sec2 x (10)

d

dxcot x = − csc2 x (11)

d

dxsec x = sec x tanx (12)

d

dxcsc x = − csc x cot x (13)

d

dxex = ex (14)

d

dxax = ax ln a (15)

d

dxln |x| = 1

x(16)

d

dxsin−1 x =

1√1− x2

(17)

d

dxcos−1 x =

−1√1− x2

(18)

d

dxtan−1 x =

1x2 + 1

(19)

d

dxcot−1 x =

−1x2 + 1

(20)

d

dxsec−1 x =

1|x|√

x2 − 1(21)

d

dxcsc−1 x =

−1|x|√

x2 − 1(22)

Integration Formulas

∫dx = x + C (1)

∫xn dx =

xn+1

n + 1+ C (2)

∫dx

x= ln |x|+ C (3)

∫ex dx = ex + C (4)

∫ax dx =

1ln a

ax + C (5)

∫lnx dx = x lnx− x + C (6)

∫sinx dx = − cos x + C (7)

∫cos x dx = sinx + C (8)

∫tanx dx = − ln | cos x|+ C (9)

∫cot x dx = ln | sinx|+ C (10)

∫sec x dx = ln | sec x + tanx|+ C (11)

∫csc x dx = − ln | csc x + cot x|+ C (12)

∫sec2 x dx = tan x + C (13)

∫csc2 x dx = − cot x + C (14)

∫sec x tanx dx = sec x + C (15)

∫csc x cot x dx = − csc x + C (16)

∫dx√

a2 − x2= sin−1 x

a+ C (17)

∫dx

a2 + x2=

1a

tan−1 x

a+ C (18)

∫dx

x√

x2 − a2=

1a

sec−1 |x|a

+ C (19)

Page 9: Mathematics Formulae

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Page 11: Mathematics Formulae

Summary of Convergence Tests

Test Series Converges Diverges Comments

Telescoping Series∞∑

n=1

(an − an+1) limn→∞

an+1 = L 6= ∞ limn→∞

an+1 DNE

Often requires partialfraction

decomposition.Converges to

a1 − limn→∞

an+1

Geometric Series∞∑

n=0

arn |r| < 1 |r| ≥ 1 Converges toa

1− r

nth-Term Test∞∑

n=1

anCannot be used toshow convergence.

limn→∞

an 6= 0 Cannot be used toshow convergence.

Integral Test(f is continuous, pos-itive, and decreasing)

∞∑n=1

an, an = f(n)∫ ∞

1

f(x) dx converges∫ ∞

1

f(x) dx divergesRemainder:0 < Rn <

∫ ∞N

f(x) dx

p-Series∞∑

n=1

1np

p > 1 p ≤ 1 Often used along witha comparison.

Logarithmic p-Series∞∑

n=2

1n(lnn)p

p > 1 p ≤ 1

Direct Comparison(an, bn > 0)

∞∑n=1

an

0 < an ≤ bn

and∞∑

n=1

bn converges

0 < bn ≤ an

and∞∑

n=1

bn diverges

Limit Comparison(an, bn > 0)

∞∑n=1

an

limn→∞

an/bn = L > 0

and∞∑

n=1

bn converges

limn→∞

an/bn = L > 0

and∞∑

n=1

bn diverges

Alternating Series∞∑

n=1

(−1)n−1an

0 < an+1 ≤ an

and limn→∞

an = 0Cannot be used toshow divergence.

Remainder:|RN | ≤ aN+1

Ratio Test∞∑

n=1

an limn→∞

∣∣∣∣an+1

an

∣∣∣∣ < 1 limn→∞

∣∣∣∣an+1

an

∣∣∣∣ > 1Test is inconclusive if

limn→∞

∣∣∣∣an+1

an

∣∣∣∣ = 1

nth-Root Test∞∑

n=1

an limn→∞

n√|an| < 1 lim

n→∞n√|an| > 1

Test is inconclusive iflim

n→∞n√|an| = 1

Page 12: Mathematics Formulae

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Page 22: Mathematics Formulae

Ordinary Differential Equations

1. Separable Equation

• Form:dy

dx= f(x) g(y)

• Solution obtained from∫

dy

g(y)=

∫f(x) dx + C

2. Exact Equation

• Form: M(x, y) dx + N(x, y) dy = 0 where∂M

∂y=

∂N

∂x

• Solution is defined implicitly by F (x, y) = C where

F (x, y) =∫

M(x, y) dx + g(y)

and

F (x, y) =∫

N(x, y) dy + h(x)

3. 1st Order, Linear Equation

• Form:dy

dx+ p(x)y = q(x)

• Let µ(x) = e∫

p(x) dx

• The solution follows from µ(x)y(x) =∫

µ(x)q(x) dx.

4. Bernoulli’s Equation

• Form:dy

dx+ p(x)y = q(x)yn

• Let v = y1−n to obtain the linear equation

dv

dx+ (1− n)p(x)v = (1− n)q(x)

5. Exact after Integrating Factor

• Form: M(x, y) dx + N(x, y) dy = 0

(a) If (∂M/∂y − ∂N/∂x) = 0, the equation is exact.

(b) If (∂M/∂y − ∂N/∂x)÷ (−M) = g(y) is a function of only y, then µ(y) = e∫

g(y) dy isthe integrating factor. Multiplication will make the equation exact.

(c) If (∂M/∂y − ∂N/∂x)÷N = g(x) is a function of only x, then µ(x) = e∫

g(x) dx is theintegrating factor. Multiplication will make the equation exact.

6. Homogeneous Equation

• Form:dy

dx= F

(y

x

)• Substitute v = y/x and dy/dx = v + x dv/dx.

• The new equation is separable.

Page 23: Mathematics Formulae

7. Reducible to 1st-order (Type 1)

• Form: F (x, y′, y′′) = 0

• Substitute y′ = u and y′′ = u′.

• The new equation involves only x, u, and u′. Solve for u(x) and then for y(x).

8. Reducible to 1st-order (Type 2)

• Form: F (y, y′, y′′) = 0

• Substitute y′ = u and y′′ = udu

dy.

• The new equation involves only y, u, and du/dy. Solve for u(y) and then for y(x).

9. Euler’s Method

• Givendy

dx= f(x, y), y(x0) = y0

• y(xn) ≈ yn where

yn+1 = yn + h f(xn, yn)xn+1 = xn + h (h is the constant step size.)

10. Improved Euler’s Method

• Givendy

dx= f(x, y), y(x0) = y0

• y(xn) ≈ yn where

yn+1 = yn + h2

(f(xn, yn) + f(xn+1, y

∗n+1)

)y∗n+1 = yn + h f(xn, yn)xn+1 = xn + h (h is the constant step size.)

11. Orthogonal Trajectories

• Given a one-parameter family of curves: g(x, y) = c

• Find dy/dx. (Must not contain the constant c.)

• Find a DE for the orthogonal trajectories by taking a negative reciprocal.

• Solve the new DE.

12. Homogenous, 2nd Order, Linear, Constant-Coefficient Equation

• Form: ad2y

dx2+ b

dy

dx+ cy = 0

• The characteristic equation is at2 + bt + c = 0.

• Let r = −b/2a and ω =√|b2 − 4ac|/2a.

(a) If b2 − 4ac > 0, the solution is y(x) = c1e(r+ω)x + c2e

(r−ω)x.(b) If b2 − 4ac = 0, the solution is y(x) = c1e

rx + c2xerx.(c) If b2 − 4ac < 0, the solution is y(x) = c1e

rx cos ωx + c2erx sinωx.

13. 2nd Order Cauchy-Euler Equation

• Form: x2 d2y

dx2+ bx

dy

dx+ cy = 0

• The substitution x = et transforms the original equation to

d2y

dt2+ (b− 1)

dy

dt+ cy = 0.

• Solve the new constant coefficient equation and resubstitute.

Page 24: Mathematics Formulae

14. Free Mechanical Vibrations

• Model: mx′′ + bx′ + kx = 0

(a) No damping if b = 0 (Simple harmonic motion)(b) Underdamped if b2 − 4mk < 0 (Damped oscillations)(c) Overdamped if b2 − 4mk > 0 (No oscillations)(d) Critically damped if b2 − 4mk = 0 (No oscillations)

15. Simple Harmonic Motion

• c1 cos ωt + c2 sinωt = A sin(ωt + φ), A > 0

wherec1 = A sinφ, c2 = A cos φ

A =√

c21 + c2

2

tanφ = c1/c2

• Amplitude = A, Angular frequency = ω, Frequency = f = ω/(2π), Period = T = 1/f

Page 25: Mathematics Formulae

Undetermined Coefficients for y′′ + c y′ + d y = g(x) (c and d are constants)

g(x) yp(x)

(1) pn(x) = anxn + . . . + a1x + a0 xsPn(x) = xs(Anxn + . . . + A1x + A0)

(2) aeαx xsAeαx

(3) a cos βx + b sinβx xs(A cos βx + B sin βx)

(4) pn(x)eαx xsPn(x)eαx

(5) pn(x) cos βx + qm(x) sinβx, xs{PN (x) cos βx + QN (x) sinβx},where qm(x) = bmxm + . . . + b1x + b0 where QN (x) = BnxN + . . . + B1x + B0 and N = max(n, m)

(6) aeαx cos βx + beαx sinβx xs(Aeαx cos βx + Beαx sinβx)

(7) pn(x) eαx cos βx + qm(x) eαx sinβx xseαx{PN (x) cos βx + QN (x) sinβx},where N = max(n, m)

The nonnegative integer s is chosen to be the least integer such that no term in yp(x) is asolution of the corresponding homogeneous equation y′′ + c y′ + d y = 0.

Variation of Parameters

If y1 are y2 are two linearly independent solutions of y′′+p(x)y′+q(x)y = 0, then a particularsolution of y′′ + p(x)y′ + q(x)y = g(x) is y = v1y1 + v2y2, where

v1(x) =∫ −g(x)y2(x)

W [y1, y2](x)dx, v2(x) =

∫g(x)y1(x)

W [y1, y2](x)dx,

and W [y1, y2](x) = y1(x)y′2(x) − y′1(x)y2(x).

2nd Solution from a 1st

If y1 is a nonzero solution of y′′ + p(x)y′ + q(x)y = 0, then y2 = v · y1, where

v(x) =∫ 1

[y1(x)]2· e−

∫p(x)dx dx,

is also solution. Furthermore, y1 and y2 are linearly independent.

Page 26: Mathematics Formulae

Basis Vectors

i = 〈1, 0, 0〉

j = 〈0, 1, 0〉

k = 〈0, 0, 1〉

u = 〈u1, u2, u3〉 = u1i + u2j + u3k

Magnitude

|u| =√

u21 + u2

2 + u23

Dot Product

u · w = u1w1 + u2w2 + u3w3

u · w = |u||w| cos θ

Projection

projw u =

(

u · w

w · w

)

w

Cross Product

u× w =

i j k

u1 u2 u3

w1 w2 w3

|u× w| = |u||w| sin θ

Position, Velocity, Acceleration

r(t) = x(t)i + y(t)j + z(t)k

v(t) = r′(t) = x′(t)i + y′(t)j + z′(t)k

a(t) = r′′(t) = x′′(t)i + y′′(t)j + z′′(t)k

Arc Length

L =

∫ b

a

(

dx

dt

)2

+

(

dy

dt

)2

+

(

dz

dt

)2

dt

L =

∫ b

a|v(t)| dt

s(t) =

∫ t

t0|v(τ)| dτ,

ds

dt= |v(t)|

Unit Tangent Vector

T =dr

ds=

v

|v|

Curvature

κ =

dT

ds

=1

|v|

dT

dt

=|v × a|

|v|3

y = f(x) ⇒ κ =|f ′′(x)|

[1 + (f ′(x))2]3/2

Principal Unit Normal Vector

N =1

κ

dT

ds=

dT/dt

|dT/dt|

Osculating Circle

radius: ρ =1

κ(t0)

center: C = r(t0) +1

κ(t0)N(t0)

Unit Binormal Vector

B = T × N

Torsion

τ = −dB

ds·N =

x′(t) y′(t) z′(t)x′′(t) y′′(t) z′′(t)x′′′(t) y′′′(t) z′′′(t)

|v × a|2

Acceleration

a = aT T + aN N

aT =d

dt|v| =

v · a

|v|

aN = κ|v|2 =√

|a|2 − a2T =

|v × a|

|v|

Projectile Motion

r(t) = ((v0 cos θ)t + x0) i

+

(

−1

2gt2 + (v0 sin θ)t + y0

)

j

Gradient Vector

∇f =∂f

∂xi +

∂f

∂yj +

∂f

∂zk

Directional Derivative

Duf =1

|u|(∇f · u)

Page 27: Mathematics Formulae

MuPAD Light Version 2.5.3 — Examples

MuPAD is a computer algebra system originally developed in the early 1990’s at the Uni-versity of Paderborn. It is now developed in cooperation with SciFace Software.

1. (Comments) To type a comment, start a line with two slashes.

• // It’s a great day to do math!

2. (Numeric Computation) Find the exact and approximate values of cos(π/5).

• num:=cos(PI/5)51/2

4+ 1/4

• float(num)

0.8090169944

• DIGITS:=25

25

• float(num)

0.8090169943749474241022934

• delete(DIGITS)

3. (Functions) Define f as a function and evaluate at the given point.

f(x) = 2 sin2 x cos x, x = π

• f:=x->2*sin(x)^2*cos(x)

x → 2 ∗ sin(x)ˆ2 ∗ cos(x)

• f(PI)

0

4. (Expressions) Define f as an expression and evaluate at the given point.

f(x) = 2 sin2 x cos x, x = π

• f:=2*sin(x)^2*cos(x)

2 cos(x) sin(x)2

• subs(f,x=PI)

2 cos(PI) sin(PI)2

• simplify(%)

0

5. (Graphing) Sketch the graphs of y = sin x and y = cos x for 0 ≤ x ≤ 2π.

• plotfunc2d(sin(x),cos(x),x=0..2*PI)

output omitted

1

Page 28: Mathematics Formulae

6. (Inequalities) Find the solution set for the inequality x2 − 3x ≥ 4.

• solve(x^2-3*x>=4,x)

]− infinity, −1] union [4, infinity[

7. (Linear Systems) Solve the following system of equations.

2x + 3y − z = 9−4x + y + 3z = 0

5x − 7y − 2z = −5

• solve({2*x+3*y-z=9,-4*x+y+3*z=0,5*x-7*y-2*z=-5},{x,y,z})

{[x = 121/36, y = 73/36, z = 137/36]}

8. (Limits) Compute the left- and right-hand limits at x = 2 for the function

f(x) =

x2, x < 23, x = 24− x, x > 2

.

• f:=piecewise([x<2,x^2],[x=2,3],[x>2,4-x])

piecewise(x2 if x < 2, 3 if x = 2, −x + 4 if 2 < x)

• limit(f,x=2,Left)

4

• limit(f,x=2,Right)

2

• limit(f,x=2)

undefined

9. (Continuity) Find all points at which the following rational function is discontinuous.

R(x) =x− 2

x2 − 5x− 6

• discont((x-2)/(x^2-5*x-6),x)

{2, 3}

2

Page 29: Mathematics Formulae

10. (Limit at Infinity) Evaluate: limx→∞

sin x

x.

• limit(sin(x)/x,x=infinity)

0

11. (Derivative by Definition) Use the limit definition of the derivative to find g′(x) ifg(x) =

√2x + 1.

• g:=x->sqrt(2*x+1)

x → sqrt(2x + 1)

• limit((g(x+h)-g(x))/h,h=0)1

(2x + 1)1/2

12. (Derivative of an Expression) Find the slope of the line tangent to the graph of y = x+2

xat the point (1, 3).

• y:=x+2/x

x +2

x

• dydx:=diff(y,x)

1− 2

x2

• subs(dydx,x=1)

−1

13. (2nd Derivative of a Function) Find the 2nd derivative of f(x) = x sin x.

• f:=x->x*sin(x)

x → x ∗ sin(x)

• f’’(x)

2 cos(x)− x sin(x)

14. (Graphing) Sketch the graph of y = sin(x)/x and label the x-axis in units of π.

• plotfunc2d(Ticks=[Steps=[PI,3],Automatic],sin(x)/x,x=0..20)

output omitted

3

Page 30: Mathematics Formulae

15. (Linearization) Find the linearization, L(x), of the function f(x) =√

x2 + 9 at x = −4.Sketch the graph of both f and L near x = −4.

• f:=sqrt(x^2+9); df:=diff(f,x)

(x2 + 9)1/2

x

(x2 + 9)1/2

• x0:=-4; fx0:=subs(f,x=x0); dfx0:=subs(df,x=x0)

−4

5

−4/5

• L:=fx0+dfx0*(x-x0)

9/5− 4x

5

• plotfunc2d(f,L,x=-8..0)

output omitted

16. (Newton’s Method) Starting with x0 = 1, take five steps of Newton’s method to ap-proximate a solution of x− cos x = 0.

• f:=x->x-cos(x)

x → x− cos(x)

• x:=1.0

1.0

• for i from 1 to 5 do x:=x-f(x)/f’(x); print(x) end_for:

0.7503638678

0.7391128909

0.7390851334

0.7390851332

0.7390851332

17. (Regression) Fit the following data to a model of the form y = A + B ln x + C/x2.

(1, 1), (4, 2), (11, 3), (31, 4), (83, 5), (227, 6)

• stats::reg([1,4,11,31,83,227],[1,2,3,4,5,6],A+B*ln(x)+C/x^2,[x],[A,B,C])

[[0.5971200686, 0.995516558, 0.4028178245], 0.0004391734841]

4

Page 31: Mathematics Formulae

18. (Riemann Sums) Partition the interval [1, 5] into 20 subintervals of equal width and useright endpoints to find the corresponding Riemann sum for the function f(x) = 1/x.

• export(student)

• f:=1/x1

x

• riemann(f,x=1..5,20,Right)

sum

1i1

5+ 1

, i1 = 1..20

5

• float(%)

1.532624844

• plotRiemann(f,x=1..5,20,Right)

plot::Group()

• plot(%)

output omitted

19. (Indefinite Integrals) Evaluate the indefinite integral

∫2z dz

3√

z2 + 1.

• int(2*z/(z^2+1)^(1/3),z)3z2

2+ 3/2

(z2 + 1)1/3

20. (Partial Fraction Decomposition) Find the partial fraction decomposition of

R(x) =1

x2 − x.

• partfrac(1/(x^2-x))1

x− 1− 1

x

• normal(%)1

x2 − x

21. (Finite Sums) Find the 100th harmonic number.

• sum(1/n,n=1..100)

output omitted

• float(%)

5.187377518

5

Page 32: Mathematics Formulae

22. (Vector Operations) Let ~u = 3ı̂ + 2̂− k̂ and ~v = −3ı̂ + 7k̂. Find ~u · ~v, ~u× ~v, and theangle between ~u and ~v.

• export(linalg)

• u:=matrix(3,1,[3,2,-1]); v:=matrix(3,1,[-3,0,7]) 32

−1

−3

07

• scalarProduct(u,v)

−16

• crossProduct(u,v) 14−18

6

• angle(u,v)

PI − arccos

(4 141/2 581/2

203

)

23. (Partial Derivative) Find fx(1, π, 2) if f(x, y, z) = y sin(xyz).

• diff(y*sin(x*y*z),x)

y2z cos(xyz)

• subs(%,x=1,y=PI,z=2)

2 PI2 cos(2 PI)

• simplify(%)

2 PI2

• float(%)

19.7392088

24. (Higher-Order Partial Derivative) Compute fxxyz if f(x, y, z) = sin(3x + yz).

• f:=sin(3*x+y*z)

sin(3x + yz)

• diff(f,x,x,y,z)

9yz sin(3x + yz)− 9 cos(3x + yz)

6

Page 33: Mathematics Formulae

25. (Implicit Differentiation) Given the equation xy + z3x − 2yz = 0, find ∂z/∂x at thepoint (1, 1, 1).

• f:=x*y+z^3*x-2*y*z

xy − 2yz + xz3

• dzdx:=-diff(f,x)/diff(f,z)

− y + z3

3xz2 − 2y

• subs(dzdx,x=1,y=1,z=1)

−2

26. (Critical Points) Let V (y, z) = 108yz − 2y2z − 2yz2. Find all points for which Vy(y, z)and Vz(y, z) are simultaneously zero.

• V:=108*y*z-2*y^2*z-2*y*z^2

108yz − 2yz2 − 2y2z

• solve({diff(V,y)=0,diff(V,z)=0},{y,z})

{[y = 0, z = 0], [y = 18, z = 18], [y = 0, z = 54], [y = 54, z = 0]}

27. (Gradient Vector) Compute ∇f at the given point.

f(x, y, z) = x2 + y2 − 2z2 + z ln x, (1, 1, 1)

• export(linalg)

• f:=x^2+y^2-2*z^2+z*ln(x)

z ln(x) + x2 + y2 − 2z2

• gradf:=grad(f,[x,y,z]) 2x +z

x2y

−4z + ln(x)

• subs(gradf,x=1,y=1,z=1) 3

2ln(1)− 4

• simplify(%) 3

2−4

7

Page 34: Mathematics Formulae

28. (Curvature) Find the curvature of ~r(t) = (2t− sin t)̂ı+(2−2 cos t)̂ at the point wheret = 3π/2.

• export(linalg)

• assume(t,Type::Real)

Type::Real

• r:=matrix[2,1,[2*t-sin(t),2-2*cos(t)])[2t− sin(t)−2 cos(t) + 2

]• v:=diff(r,t) [

− cos(t) + 22 sin(t)

]• mag_v:=norm(v,2)

(cos(t)2 − 4 cos(t) + 4 sin(t)2 + 4)1/2

• T:=v/mag_v

output omitted

• K:=norm(diff(T,t),2)/mag_v

output omitted

• curvature:=subs(K,t=3*PI/2)

output omitted

• simplify(curvature)

21/2

16

29. (Surface of Revolution) Sketch the graph of the surface obtained by rotating the graphof y =

√x, 0 ≤ x ≤ 1, about the x-axis.

• f:=sqrt(x)

x1/2

• plot(plot::xrotate(f,x=0..1))

output omitted

8

Page 35: Mathematics Formulae

30. (Directional Derivatives) Find the directional derivative of

f(x, y) = 2xy − 3y2

at (5, 5) in the direction of ~u = 4ı̂ + 3̂.

• export(linalg)

• f:=2*x*y-3*y^2

2xy − 3y2

• u:=matrix(2,1,[4,3]) [43

]• gradf:=subs(grad(f,[x,y]),x=5,y=5)[

10−20

]• scalarProduct(gradf,u)/norm(u,2)

−4

31. (Ordinary Differential Equations) Solve the following differential equation:

xy′ + 2y = xex/2

• eq:=ode(x*y’(x)+2*y(x)=x*exp(x/2),y(x))

ode(2 y(x) + x diff(y(x), x)− x exp

(x

2

), y(x)

)• solve(eq) (

c1

x2+ 2 exp(x)1/2 − 8 exp(x)1/2

x+

16 exp(x)1/2

x2

)

32. (Initial Value Problems) Solve the following initial value problem:

y′′ + 2y′ + y = e2x; y(0) = 2, y′(0) = 1

• eq:=ode({y’’(x)+2*y’(x)+y(x)=exp(2*x),y(0)=2,y’(0)=1},y(x))

ode({y(0) = 2, D(y)(0) = 1, y(x) + · · · − exp(2x)}, y(x))

• solve(eq) (17 exp(−x)

9+

exp(x)2

9+

8x exp(−x)

3

)

9

Page 36: Mathematics Formulae

33. (Laplace Transforms) Find the Laplace transform of f(t) = t2 − 1 + cos t.

• export(transform)

• laplace(t^2-1+cos(t),t,s)2

s3− 1

s+

s

s2 + 1

• invlaplace(%,s,t)

cos(t) + t2 − 1

34. (One Variable Statistics) Find the mean, median, mode, and standard deviation of thedata set given below.

{92.5, 43, 78, 82, 57.5, 63, 78, 91, 84.5, 68}

• export(stats)

• data:=[92.5,43,78,82,57.5,63,78,91,84.5,68]

[92.5, 43, 78, 82, 57.5, 63, 78, 91, 84.5, 68]

• mean(data)

73.75

• median(data)

78

• modal(data)

[78], 2

• stdev(data)

15.73080841

• stdev(data,Population)

14.92355521

35. (Surface Plots) Sketch the surface defined by z = sin xy.

• plotfunc3d(sin(x*y),x=-PI..PI,y=-PI..PI,Grid=[200,200])

output omitted

36. (Parametric Plots) Sketch the space curve defined by the parametric equations.

x = sin t, y = cos t, z = t

• p1:=plot::Curve3d([sin(t),cos(t),t],t=0..4*PI)

plot::Curve3d([sin(t), cos(t), t], t = 0..4 PI)

• plot(p1)

output omitted

10

Page 37: Mathematics Formulae

37. (Matrix Operations) Given the two matrices A and B as shown below, find A + 2B,AT , AT A, and B−1.

A =

1 0 43 2 −20 −1 2

B =

1 1 13 0 3

−1 −3 −5

• A:=matrix(3,3,[[1,0,4],[3,2,-2],[0,-1,2]]) 1 0 4

3 2 −20 −1 2

• B:=matrix(3,3,[[1,1,1],[3,0,3],[-1,-3,-5]]) 1 1 1

3 0 3−1 −3 −5

• A+2*B 3 2 6

9 2 4−2 −7 −8

• C:=linalg::transpose(A) 1 3 0

0 2 −14 −2 2

• C*A 10 6 −2

6 5 −6−2 −6 24

• B^(-1) 3/4 1/6 1/4

1 −1/3 0−3/4 1/6 −1/4

11

Page 38: Mathematics Formulae

38. (Gauss-Jordan Elimination) Use Gauss-Jordan elimination to solve the following sys-tem of equations.

2x + 3y − z = 9−4x + y + 3z = 0

5x − 7y − 2z = −5

• M:=matrix(3,4,[[2,3,-1,9],[-4,1,3,0],[5,-7,-2,-5]]) 2 3 −1 9−4 1 3 0

5 −7 −2 −5

• linalg::gaussJordan(M) 1 0 0 121/36

0 1 0 73/360 0 1 137/36

39. (Nonlinear Equations) Sketch the graphs of the following equations and numerically

approximate a point of intersection of the graphs.

4x2 + y2 − 4 = 0

x + y − sin(x− y) = 0

• f:=4*x^2+y^2-4; g:=x+y-sin(x-y)

4x2 + y2 − 4

x + y − sin(x− y)

• p1:=plot::implicit(f=0,x=-2..2,y=-2..2)

plot::Group()

• p2:=plot::implicit(g=0,x=-2..2,y=-2..2)

plot::Group()

• plot(p1,p2)

output omitted

• numeric::solve({f=0,g=0},{x=1,y=0})

{[y = −0.1055304923, x = 0.9986069441]}

40. (ODE Direction Fields) Sketch the direction field associated with the differential equa-tion

dy

dx= −y

x

• p1:=plot::vectorfield([1,-y/x],x=-10..10,y=-10..10)

plot::Group()

• p2:=plot::Scene(p1,Scaling=Constrained,Title="dy/dx=-y/x")

plot::Group()

• plot(p2)

output omitted

12

Page 39: Mathematics Formulae

y

z

x

�(r, θ, z)

θ r

z

Cylindrical Coordinates

x = r cos θ r2 = x2 + y2

y = r sin θ tan θ = y/xz = z z = z

dV = r dr dθ dz

y

z

x

�(ρ, θ, φ)

ρ ≥ 0

0 ≤ φ ≤ π φ

θ r

z

ρ

Spherical Coordinates

x = ρ sin φ cos θ ρ2 = x2 + y2 + z2

y = ρ sin φ sin θ tan θ = y/xz = ρ cos φ φ = cos−1(z/ρ)

θ is the same angle used in cylindrical

coordinates for r ≥ 0.

dV = ρ2 sin φ dρ dθ dφ

Page 40: Mathematics Formulae

Miscellaneous

Enter or Shift+Enter evaluates a cell

; End of command/Show output

$ End of command/Suppress output

% Most recent output

%th(5) 5th previous output

: Assignment e.g. x : 2 or f : x2 + 1

:= Function definition e.g. f(x) := sin(x)

’ Prevents evaluation

/* ... */ Comments

kill(...) Delete variables or expressions

Constants

%pi π = 3.14159 . . .

%e e = 2.71828 . . .

%gamma Euler’s constant, γ = 0.5772 . . .

%i Imaginary unit, i =√−1

%phi Golden ratio, φ = 1.618 . . .

inf Infinity, ∞

minf Negative infinity, −∞

Numerics

float(x) Decimal form, 16 digits

fpprec Floating-point precision

bfloat(x) Decimal form, fpprec digits

Equations

solve( x^2-3*x+2=0, x )

find_root( cos(x)=x, x, 0, 1 )

linsolve( [x+y=0, 2*x+3*y=2], [x,y] )

ode2( ’diff(y,x) = -x*y, y, x )

rhs( x^2+x=cos(x) ) Right-hand side

Simplifying

factor( x^2-3*x+2 )

expand( (x+3)*(2*x+1)^3 )

partfrac( 1/(x^2-x), x )

ratsimp( 1/x + 1/(x-1) )

radcan( sqrt(135) )

trigsimp( sin(5)^2 + cos(5)^2 )

trigreduce( cos(x)^3 )

Evaluation/Substitution

f(x):=x*sin(x); f(2)

subst( 2, x, f )

at( f, x=2 )

Calculus

limit( sin(x)/x, x, 0 )

limit( abs(x)/x, x, 0, plus )

diff( f, x )

diff( f, x, 2 )

depends( y, x ); diff( x*y=sin(x), x );solve( %, ’diff( y, x ) )

integrate( g, x )

integrate( g, x, 0, 5 )

sum( 1/k, k, 1, 25 )

taylor( exp(x), x, 0, 6 )

Vectors & the vect1 package

Must load vect1 package

u: [1,2,3]; v: [-1,0,3]

norm( u ) Magnitude

normalize( u ) Unit vector

dot( u, v )

cross( u, v )

angle( u, v )

proj( u, v ) projv u

Plotting

plot2d( sin(x), [x,-5,5] )

plot2d( [sin(x),cos(x)], [x,-3,3] )

plot3d( x^2-3*sin(x)*y, [x,-5,5], [y,-5,5] )

Matrices & lists

x: [3,6,9,12,15]

x[2] Element 2 of list x

A: matrix( [2,1], [-1,3] )

A[i][j] ij-element of A

A+B Matrix addition

A.B Matrix multiplication

A^^(-1) Matrix inverse

determinant(A)

Page 41: Mathematics Formulae

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Page 47: Mathematics Formulae

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x

y

(1, 0)

0◦

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2, 1

2)

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2

2,√

2

2)

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x = cos θ y = sin θ

π radians = 180◦

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