mathematics formulae
TRANSCRIPT
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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)
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)
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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
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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.
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.
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
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.
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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x = cos θ y = sin θ
π radians = 180◦
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