Memorization Quiz - Derivatives and Integrals

∫cosxdx

sinx+c

∫sinxdx

-cosx+c

∫sec²xdx

tanx+c

∫csc² xdx

-cotx+c

∫secxtanxdx

secx+c

∫cscxcotxdx

-cscx+c

∫dx/√(1-x² )

arcsinx+c

∫dx/(1+x² )

arctanx+c

∫dx/(x√(x²-1))

arcsecx+c

∫b^x dx

(b^x/lnb)+c

∫(1/x)dx

ln|x|+c

∫e^x dx

e^x+c

d/dx (sinx)

cosx

d/dx (cosx)

-sinx

d/dx (tanx)

sec² x

d/dx (cotx)

-csc² x

d/dx (secx)

secxtanx

d/dx (cscx)

-cscxcotx

d/dx (arcsinx)

1/√(1-x^2 )

d/dx (arccosx)

-1/√(1-x^2 )

d/dx (arctanx)

1/(1+x^2 )

d/dx (arccotx)

-1/(1+x^2 )

d/dx (arcsecx)

1/(|x| √(x^2-1))

d/dx (arccscx)

-1/(|x| √(x^2-1))

d/dx (b^x )

(b^x)(lnb)

d/dx (lnx)

1/x

d/dx (log₂ x)

1/(xln2)

d/dx (e^x )

e^x

d/dx(g(x)) if g(x) = f⁻¹(x)

1/(f'(g(x))

sin(0,2π)

0

sin(π/6)

1/2

sin(π/4)

√2/2

sin(π/3)

√3/2

sin(π/2)

1

sin(π)

0

sin(3π/2)

-1

cos(0,2π)

1

cos(π/6)

√3/2

cos(π/4)

√2/2

cos(π/3)

1/2

cos(π/2)

0

cos(π)

-1

cos(3π/2)

0

tan(0,2π)

0

tan(π/6)

√3/3

tan(π/4)

1

tan(π/3)

√3

tan(π/2)

∅

tan(π)

0

tan(3π/2)

∅

(lim)(x→∞)⁡(1+(1/x))^(x)

e

Limit Definition of Derivative f'(x) =

(lim)(h→0) (f(x+h) - f(x))/h

Limit Definition of Derivative f'(a) =

(lim)(h→0) (f(a+h) - f(a))/h

Limit Definition of Derivative f'(a) =

(lim)(x→a) (f(x) - f(a))/(x - a)

Speed is increasing when ….

Velocity and Acceleration are the same sign

Speed is decreasing when ….

Velocity and Acceleration are opposite signs

Speed

|v(t)|

Displacement

∫v(t)dt from a to b

Total Distance

∫|v(t)|dt from a to b

Volume (Washer Method)

π∫(R^2 - r^2)dx from a to b

Volume (Cross Section)

∫A(x)dx from a to b

FTC: ∫f'(x)dx from a to b

f(b) - f(a)

Position at b: P(b) =

P(b) = P(a) + ∫v(t)dt from a to b

Value at a Point: f(b) =

f(b) = f(a) + ∫f'(x)dx from a to b

2nd FTC: g(x) = ∫f(t)dt from a to x

g'(x) = f(x)

Average Rate of Change:

(f(b) - f(a))/(b - a)

MVT:

f'(c) = (f(b) - f(a))/(b - a) if continuous and differentiable

Average Value:

(∫f(x)dx from a to b)/(b - a)

Extreme Value Theorem:

If it is a closed interval [a, b], make a table with critical points and end points

Definition of Continuity

(lim)(x→a^+)f(x) = (lim)(x→a^-)f(x) = f(a)

Tangent Line / Linear Approximation

y - y1 = m(x - x1)

Increasing / Decreasing for g given g':

g' is positive → g is increasing

g' is negative → g is decreasing

Concave UP / Concave DOWN for g given g':

g' is increasing → g is CONCAVE UP

g' is decreasing → g is CONCAVE DOWN

Points of Inflection for g given g':

g' is at a min or max (g' goes from increasing to decreasing or decreasing to increasing)