AP Calc AB - Derivatives

limit definition of a derivative

lim (f(x+h)-f(x))/(h)
h→0

alternate definition of a derivative

f¹(c)=lim x→c (f(x)-f(c))/(x-c)

when is a fcn not differentiable

when the fcn is not continuous

differentiability→

continuous

continuous≠

differentiability

if the fcn is not continuous...

it is not differentiable

equation of tangent line

find the derivative to find the slope then put into an equation

equation of a normal line

perpendicular line→neg reciprocal of tangent slope

when is a tangent line horizontal

m=0

At what values of x is the tangent line horizontal?

take derivative and set it =0

s(t)

position

v(t)

velocity

a(t)

acceleration

v(t)=

s'(t)

a(t)=

v'(t)=sⁿ(t)

sⁿ(t)

second derivative of position

position formula of a free falling object

s(t)=-16t²+v₀t+s₀

velocity formula of a free falling object

v(t)=-32t+v₀

v₀=

initial velocity

s₀=

initial position

average rate of change of a function over an interval...

∆f(x)/∆x

average velocity over an interval...

Vavg=∆s(t)/∆t

average acceleration over an interval...

Aavg=∆v(t)/∆t

instantaneous velocity of an object at a certain point in time...

v(t)=s'(t)

point in time when the object is at rest

v(t)=0

points in time when the object is at ground level...

s(t)=0

"rate of change" implies

derivative

a particle is speeding up when the velocity and acceleration of the particle...

have the same sign

a particle is slowing down when the velocity and acceleration of the particle...

have different signs

s't=v(t)=

lim ∆t→0 v(t+∆t)-v(t) / ∆t

speed=

|v(t)|

velocity HAS

direction and magnitude→is a vector

speed has NO

direction