AP Calc AB UNit 2.1-2.3 Review

derivative of a function

• f’(x)= lim h→0 [f(x+h) - f(x)]/h

• gives you the slope of the tangent line at a specific point when you plug the x value into the derivative

derivative at a point

f’(a)= lim h→0 [f(x) - f(a)]/x-a

graphing f’ from f:

• when f is decreasing

• when f is increasing,

• f has a horizontal tangent

• f’ is negative

• f’ is positive

• changing from increasing to decreasing or vice versa, when f’ is 0

when is y=f(x) differentiable on a closed interval [a,b]

if it has a derivative at every interior point of the interval, and if the limits lim h→0+ [f(a+h) - f(a)]/h and lim h→0- [f(b+h) - f(b)]/h exist at end points

• derivatives at endpoints are one-sided limits

when will f’(a) fail to exist?

where the slopes of the secant lines: [f(x) - f(a)]/x - a fail to approach a limit as x approaches a

1. corner (ex: f(x)= |x|)

2. cusp (ex: f(x)= x²/³)

3. vertical tangent (ex: f(x) = 3rd root of x

4. discontinuity (ex: unit step function)

locally linear

a function that is differentiable at a closely resembles its own tangent line very close to a

• differentiable curves will “straighten out” when we zoom in on them at a point of differentiability

symmetric difference quotient

[f(a + h) - f(a - h)]/2h

• computes the slopes between two nearby points on opposite sides of a

numerical derivative of f at a

[f(a + 0.001) - f(a - 0.001)]/0.002

• NDER(f(x), a)

• number

numerical derivative of f

• [f(x + 0.001) - f(x - 0.001)]/0.002

• NDER(f(x), x)

• function

what’s the relationship between differentiability and continuity

if f has a derivative at x=a, then f is continuous at x=a

the converse of this is not true

intermediate value theorem for derivatives

if a and b are two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b)

derivative of a constant function

df/dx= d/fx (c) = 0

power rule

d/dx(x^n)= nx^n-1

constant multiple rule

d/dx (cu)= c du/dx

sum and difference rules

d/dx (u + or - v)= du/dx + or - dv/dx)

when is there a horizontal tangent on f(x)

when f’(x)=0

product rule

d/dx (uv)= u dv/dx + v du/dx

quotient rule

d/dx(uv)=[v du/dx - u dv/dx]/v²

higher order derivatives notation

y^n = [dy^(n-1)]/dx = d^n y/dx^n

d/dx cos(x)=

-sin (x)

d/dx sin (x)=

cos (x)

ln 1=

0

ln 0=

U.D

e^0=

1

e^ln a =

a

ln e^a =

a

d/dx a^x=

a^x ln a

d/dx e^x=

e^x

d/dx log a (x)=

(1/x) (1/ln a)

d/dx ln(x)=

1/x