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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
compare terms that add
Factor & divide
Left & Right
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
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:
Instantaneous rate of change of f(x)
Slope of line tangent to f(x)
Slope of f(x) at a point
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
Note 
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