Pre Calc BC Honors Final test of the year - what you need to know

Definition of the derivative

graph of f(x) = eˣ

graph of f(x) = lnx

f(x) is continuous at x=a if

f(a) is defined

\
lim f(x) exists

x→a

\
f(a) = lim f(x)

x→a

Types of discontinuity

Removable, Jump, and Infinite

When functions are not differentiable?

Discontinuity, corner, vertical tangent line, or a cusp

d/dx \[c\] =

(c is any real number)

0

d/dx \[mx\] =

\[m is a constant\]

m

d/dx \[xⁿ\] =

nxⁿ⁻¹

d/dx \[cf(x)\] =

(c is any real number)

cf’(x)

d/dx \[f(x)±g(x)\] =

f’(x) ± g’(x)

d/dx \[f(x)g(x)\] =

g(x)f’(x) + f(x)g’(x)

d/dx \[ f(x)/g(x)\] =

\[g(x)f’x)-f(x)g’(x)\]/\[g(x)\]²

d/dx\[f(g(x))\] =

f’(g(x)) ⋅ g’(x)

d/dx \[sin x\] =

cos x

d/dx \[cos x\] =

\-sinx

d/dx \[tan x\] =

sec²x

d/dx \[cot x\] =

\-csc²x

d/dx \[sec x\] =

secx tanx

d/dx \[csc x\] =

\-cscx cotx

d/dx \[eˣ\] =

eˣ

d/dx \[bˣ\] =

bˣ ⋅ lnb

d/dx \[ln x\] =

1/x

d/dx \[log꜀x\] =

1/ x⋅lnc

d/dx \[sin⁻¹\] =

1/  √1-x²

d/dx \[cos⁻¹\] =

\-1/  √1-x²

d/dx \[tan⁻¹\] =

1/1-x²

Trig table

Unit Circle

L’Hopital’s Rule

Mean Value Theorem Definition

If f is continuout on the closed interval \[a,b\] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that

Intermediate Value Theorem

if f is continuous on \[a,b\] and k is any number between f(a) and f(b), inclusive then there is at least one number c in the interval \[a,b\] such that f(c) = k

Extreme Value Theorem

If a function f(x) is continuous on a closed interval \[a,b\] then f(x) has both a maximum and minimum value on \[a,b\]