Calculus AB Golden Notes

Everything you need to know & understand for the AB calculus exam

Derivative Power Rule

If f

Derivative exponential rule

\

Derivative e Rule

Derivative Ln Rule

Derivative Square Root Rule

Derivative Tangent Rule

Derivative Sine Rule

Derivative Cosine Rule

Derivative Inverse Sine Rule

Derivative Inverse cos rule

Derivative Inverse tan rule

Derivative constant Rule

Derivative Chain Rule

Derivative Product Rule

Derivative Quotient Rule

Derivative Addition Rule

Anti-derivative power rule

Anti-derivative expanded power rule

Anti-derivative exponential rule

Anti-derivative expanded exponential rule

Anti-derivative Ln Rule

Anti-derivative Ln expanded rule

Anti-derivative sine rule

Anti-derivative expanded sin rule

Anti-derivative cos rule

Anti-derivative expanded cos rule

Derivative of inverse f(x)

Displacement

Total Distance

Derivative of an integral

Differentiable if

continuous, no corner or vertical tangent

Continuous if

No removable discontinuity, jumps, or vertical asymptotes.

Limits if x-\>∞ then

1. compare terms that add
2. Factor & divide
3. Left & Right
4. L'hopital's rule

Place in order of growing fastest as x -\>∞:
x^99, e^x, lnx

lnx, x^99, e^x

Find the average value of f(x)

Find the average rate of change

v(t) is the

rate at which x is changing; tangent slope; instantaneous rate of change

Average value of f'(x) is the same as

average rate of change

secant slope is the

average rate of change

Find the secant slope

e^(lnA)

A

lne^A

A

e^(A+B)

e^Ae^B

ln12-ln4

ln(12/4)

f(x) has a critical point when

f'(x)\=0 or f'(x)\=undefined

Min-Max Theorem

The absolute Max/Min of f(x) is at the beginning of f(x) at the end of f(x) or at a critical point on f(x)

f(x) has an inflection point when

f(x) changes concavity, OR f'(x) changes I to D or D to I or when f"(x) changes sign

L'Hopitals Rule

The limit exists if

Area of a semicircle

Solve an Equation

Find value which makes equation true OR graph both halves of equation & find intersection

The particular solution y\=B(t) of a differential equation dB/dt\=1/5(100-B) with initial condition B(0)\=20 what would you use?

Use SACI

SACI

Separate, Anti Differentiate, Constant-tate, Isolate

Speed is increasing when

v(t) and a(t) are the same sign

Approximate the instant rate of change by:

calculating the average rate of change

Approximate the tangent slope by:

calculating the nearest secant slope

When the in rate is E(t) and the out rate is L(t) what is the equation for the rate?

A'(t)\=E(t)-L(t)

Solve an anti-derivative

1. Rule 2. u substitution 3. Algebra trick

Average rate of change of velocity is the same as

average acceleration

average rate of change of position is the same as

average velocity

secant slope is the same as

average rate of change of f(x)

secant slope or average roc or f(x)

average roc of x(t) or average velocity

average roc of v(t) or average acceleration

speed

F'(x)\=

f(x)

anti-derivative of f(x)

F(x)

anti-derivative of f'(x)

f(x)

integral from a to b of a(t) equals

v(b)-v(a)

integral from a to b of v(t) equals

x(b)-x(a)

integral from a to b of f(x) equals

F(b)-F(a)

integral from a to b of f'(x) equals

f(b)-f(a)

integral of a rate equals

change in amount

Mean Value Theorem

If f(x) is continuous and differentiable the "tangent slope at c" \= secant slope

Tangent line formula

If f(x) is concave down the tangent line is

an OVER approximation

If f(x) is concave up the tangent line is

an UNDER approximation

Trapezoidal riemann sum formula

f'(x)\=dy/dx\= Formula to find:

1. Instantaneous rate of change of f(x)
2. Slope of line tangent to f(x)
3. Slope of f(x) at a point
4. Instant rate at which f(x) is changing

f(x) has relative/local max when

f'(x) changes + to - or when f"(x) changes I to D

lne^2

2

lne

1

lne^0

0

ln1

0

ln(1/e)

-1

lne^(-1)

-1

ln(1/e^-2)

-2

rate of change of position

x'(t) or v(t)

rate of change of velocity

v'(t) or a(t)

Vertical Tangent when

number/0

Jump discontinuity when

the left limit is different from the right limit

Removable discontinuity when

the value is different than the limits on the left and right. Limits must be the same on left and right.

Horizontal asymptote

the value of the limit as x-\>infinity

When given a rate and then asked to find the amount use

Fundamental Theorem

When given a rate that includes the output variable and then asked to find the amount use

SACI

f has an inflection point when

f changes concavity

f has a relative or local max when

f changes from increasing to decreasing

f has a relative extrema when

f changes from I to D or D to I or when f' changes + to - or - to +

f has a critical point when

the slope of f is 0 or undefined or when f' has a y-coord. of 0 or und

tangent slope means

instantaneous rate of change