Exam 1

Question 1

Important derivatives

tan
sec

tan⁻¹

ln(x)
logₐ(x)

sin(3x)
sin(x²)
e^(-x²)
Rule:

Answer 1

tan(x) -> sec²(x)
sec(x) -> sec(x)tan(x)

tan⁻¹(x) -> 1/[1+x²]

ln(x) -> 1/x
logₐ(x) -> 1/ [x ln(a)]

sin(3x) -> 3cos(3x)
sin(x²) -> 2x cos(x²)
e^(-x²) -> -2xe^(-x²)
Rule: multiply constant inside

Question 2

Derivative rules

Answer 2

Product:
uv = u’v + uv’

Quotient:
u/v = [u’v - uv’] / v²

Chain:
f(g(x) -> f(g’(x) * g(x)

Question 3

Important integrals

1/x

1/[1+x²]

sin(3x)
Rule:

Answer 3

1/x -> ln|x|

1/[1+x²] -> tan⁻¹(x)

sin(3x) -> 1/3 * -cos(3x)
Rule: multiply by reciprocal

Question 4

Graphs:

1/x
1/x²
ln(x)
eˣ
1/√x

Answer 4

Refer to goodnotes

Question 5

Unit circle angles

Answer 5

0, π/6, π/4, π/3, π/2

π/2, 2π/3, 3π/4, 5π/6, π

π, 7π/6, 5π/4, 4π/3, 3π/2

3π/2, 5π/3, 7π/4, 11π/6, 2π

Question 6

Integration by u-substitution and recycling

Answer 6

u ≠ x

recycle if u = x and the equation is linear

Question 7

Position and velocity functions

Answer 7

Position is the integral of velocity
Velocity is the derivative of position

Question 8

Setting up and solving FTC

Answer 8

P(b) - P(a) = ∫ᵇₐ P’(t) dt

Question 9

Determining area of shaded region by graph and equation

Answer 9

If scanning horizontally:
∫ upper function - lower function ∆x
- In terms of x

If scanning vertically:
∫ right function - left function ∆y
- In terms of y

Question 10

Disks

Answer 10

∫ π r² h dx
r² should be in terms of x

∫ π r² h dy
r² should be in terms of y

Question 11

Washers

Answer 11

∫ π r² h dx
r² in terms of x, with (outer function)² - (inner function)²

∫ π r² h dy
r² in terms of y, with (outer function)² - (inner function)²

Question 12

Shells

Answer 12

Riemann rectangle going horizontally:
2π ∫ x * (upper function - inner function) dx
- functions should be in terms of x

Riemann rectangle going vertically:
2π ∫ y * (right function - left function) dy
- functions should be in terms of y

Question 13

exponents in the reciprocal

Answer 13

sign changes

Question 14

Displacement vs. Distance

Answer 14

Displacement: x
Distance: |x|