MATH3B Formulas

Question 1

f(x) = cotx

Answer 1

f’(x) = -csc²x

Question 2

f(x) = secx

Answer 2

f’(x) = secxtanx

Question 3

f(x) = cscx

Answer 3

f’(x) = -cscxcotx

Question 4

∫tanxdx

Answer 4

ln|secx|+C

Question 5

∫cotxdx

Answer 5

ln|sinx|+C

Question 6

∫secxdx

Answer 6

ln|secx+tanx|+C

Question 7

∫cscxdx

Answer 7

-ln|cscx+cotx|+C

Question 8

tan²x

Answer 8

sec²x-1

Question 9

sec²x

Answer 9

tan²x+1

Question 10

csc²x

Answer 10

1+cot²x

Question 11

∫√(a²-x²)dx

Answer 11

asinθ

Question 12

∫√(a²+x²)dx

Answer 12

atanθ

Question 13

∫√(x²-a²)dx

Answer 13

asecθ

Question 14

Surface Area

Answer 14

About x-axis
if it’s f(x)
∫2πy√(1+(f’(x))²dx

if it’s f(y)
∫2πy√(1+(f’(x))²dy

About y-axis
if it’s f(x)
∫2πx√(1+(f’(x))²dx

if it’s f(y)
∫2πx√(1+(f’(y))²dy

Question 15

Midpoint Rule and Error

Answer 15

Δx = (b-a)/n
Rule:
M_n = Δx * avg X value for each rectangle from Right

Error Bound:
|E_M| ≤ ^k(b-a)3/_24n²
where k = f’‘(x)

Question 16

Trapezoid Rule and Error

Answer 16

Δx = (b-a)/n
Rule:
T_n = ^R_n+L_n⁄ ₂

Error Bound:
|E_T| ≤ ^k(b-a)3⁄ _12n²
where k = f’‘(x)

Question 17

Centroid Calculations

Answer 17

m = ρ∫[f(x)-g(x)]dx
M_x = ρ∫1/2[f(x)²-g(x)²]dx
M_y = ρ∫x[f(x)-g(x)]dx

Question 18

Linear Equations

Answer 18

dy/dx + P(x)y = Q(x)

I(x) = e^∫P(x)

y = 1/I(x) [∫I(x)Q(x)dx + C]