Calculus

continuous functions

100

A discontinuity where the limit exists, and a hole is present. 

What is a removable discontinuity 

200

F(x)=1/x-3 at x=3

What is not continuous 

300

F(x)=8-3x² at X>-2 and F(x)=-4+X at X<-2

limf(x)=-6

limf(x)=-4

what is discontinuous 

400

When a function is continuous 

What is existing 2 sided limits, y values at x=a, and limit=y value

500

F(x)= x²-1/x-1 at X=1 and F(x)=3 X≠1

What is f(1)=3 and lim f(x)=2, so f(x) is not continuous at X=1

Rules

100

Product rule

What is f’(x)=u(x)v’(x) + v(x)u’(x)

200

Quotient rule

f’(x)=v(x)u’(x) - u(x)v’(x)/[v(x)]²

300

Chain rule

h’(x)=f’(g(x)) (g’(x))

400

Power rule (a/x)

What is ax^-1

500

The rule used in related rates

What is product rule 

trig integrals

100

e^xdx 

What is e^x+c

200

Cosxdx

SinX+C

300

Sec²xdx

Tanx+C

400

Secxtanx dx 

Secx+c

500

cscxcotx dx

-cscx+c

Derivative

100

The derivative of f(x) = x²

What is f'(x) = 2x.

200

The derivative of g(x) = 5x³

What is g'(x) = 15x²

300

g(x)=e^x(5x³+2x+6)

(e^x)(15x²+2)+(e^x)(5x³+2x+6)

400

5x²-15x

10x-15

500

f(x)=-4log₂(3x⁵)

—4(1/ln(2)3x⁵)(15x⁴)

Substitution

100

S 20x⁴e^{2x^5+3}dx

2e^{2x^5+3}+c

200

S 12e^3xdx

4e^3x+c

300

S(2x³+1)⁷(x²)dx

1/48(2x³+1) +C

400

S 4Xsin(3x²)dx

-2/3cos(3x²)+C

500

S (2x+5)(x²+5x)⁷dx

(1/8)(x²+5x)⁸+C