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AP Calculus AB Cram Sheet
Definition of the Derivative Function:
f ' (x) = limh 0 ccccccccccccf^ +x^ h/ hcc^ ccccccc^ f^ + cccc^ xcc^ /c
Definition of Derivative at a Point:
f ' (a) = limh 0 ccccccccccccccccf^ +ahh/cccccccc f^ +ccccca/ (note: the first definition results in a function, the second definition results in a number. Also note that the difference quotient, ccccccccccccccccf^ +ahh/cccccccc f^ +ccccca/, by itself, represents the average rate of change of f from x = a to x = a + h)
Interpretations of the Derivative: f ' (a) represents the instantaneous rate of change of f at x = a, the slope of the tangent line to the graph of f at x = a, and the slope of the curve at x = a.
Derivative Formulas: (note:a and k are constants)
ccccccc^ d dx +k/^0
ccccccc^ d dx (k·f(x))= k·f ' (x)
ccccccc^ d dx +^ f^ +x//
n (^) n+ f +x//n (^1) f ' +x/
ccccccc^ d dx [f(x) ± g(x)] = f ' (x) ± g ' (x)
ccccccc^ d dx [f(x)·g(x)] = f(x)·g ' (x) + g(x) · f ' (x)
ccccccc^ d dx ,^ cccccccccccf^ +x/ g+x/ 0 ccccccccccccccccccccccccccccccccg+x/^ f^ '^ +x/^ ^ f^ +ccccccccx/g^ cccccccccc'^ +x/ +g+x//^2
ccccccc^ d dx sin(f(x)) = cos (f(x)) ·f ' (x)
ccccccc^ d dx cos(f(x)) = -sin(f(x))·f ' (x)
ccccccc^ d dx tan(f(x)) = sec (^2) + f +x// º f ' +x/
ccccccc^ d dx ln(f(x)) =^ ccccccccccc^1 f +x/ º^ f^ '^ +x/
ccccccc^ d dx e^ f +x/ (^) e f +x/ (^) º f ' +x/
ccccccc^ d dx a^ f +x/ (^) a f +x/ (^) º ln a º f ' +x/
ccccccc^ d dx sin
(^1) f +x/ ccccccccccccccccrf ' (^) +xcccccccc/cccc 1 + f +x//^2
ccccccc^ d dx cos
(^1) f +x/ ccccccccccccccccrf^ '^ +xcccccccc/cccc 1 + f +x//^2
cccccccd dx tan (^1) f +x/ ccccccccccccccccf^ '^ +x/cccccc 1 + f +x//^2
ccccccc^ d dx +^ f^ (^1) +x// at x f +a/ equals cccccccc^1 cccccc f '+x/ at^ x^ a
L'Hopitals's Rule:
If limxa cccccccccccgf^ ++xx// cccc^00 or cccccc and if limxa ccccccccgf^ '' ++ccccccxx// exists then
limxa cccccccccccfg^ ++xx// limxa ccccccccgf^ ''^ ++ccccccxx//
The same rule applies if you get an indeterminate form ( cccc^00 or cccccc ) for limx cccccccccccfg^ ++xx// as well.
Slope; Critical Points: Any c in the domain of f such that either f ' (c) = 0 or f ' (c) is undefined is called a critical point or critical value of f.
Tangents and Normals The equation of the tangent line to the curve y = f(x) at x = a is
y - f(a) = f ' (a) (x - a)
The tangent line to a graph can be used to approximate a function value at points very near the point of tangency. This is known as local linear approximations. Make sure you use instead of = when you approximate a function.
The equation of the line normal(perpendicular) to the curve y = f(x) at x = a is
y - f(a) = ccccccccf '^1 +cccccca/ +x a/
Increasing and Decreasing Functions A function y = f(x) is said to be increasing/decreasing on an interval if its deriva- tive is positive/negative on the interval.
Maximum, Minimum, and Inflection Points The curve y = f(x) has a local (relative) minimum at a point where x = c if the first derivative changes signs from negative to positive at c.
The curve y = f(x) has a local maximum at a point where x = c if the first deivative changes signs from positive to negative.
The curve y = f(x) is said to be concave upward on an interval if the second derivative is positive on that interval. Note that this would mean that the first derivative is increasing on that interval.
The curve y = f(x) is siad to be concave downward on an interval if the second derivative is negative on that interval. Note that this would mean that the first derivative is decreasing on that interval.
The point where the concavity of y = f(x) changes is called a point of inflection.
The curve y = f(x) has a global (absolute) minimum value at x = c on [a, b] if f(c) is less than all y values on the interval.
¼ tan^ uÅ^ u^ ln^ «^ sec^ u^ «^ C
¼ sec^2 uÅ^ u^ tan^ u^ ^ C
¼ euÅ^ u^ eu^ ^ C
¼ auÅ^ u^ a
u cccccccccln a C
¼ ccccccccr acccccccc (^21) uccccc 2 Å^ u^ sin^1 ^ ccccua ^ C
¼ cccccccca (^2) ^1 cccccccu 2 Å^ u^ cccc^1 a tan^1 ^ ccccua ^ C
The Fundamental Theorems
The First Fundamental Theroem of Calculus states
If f is continuous on the closed interval [a, b] and F ' = f, then,
¼a b (^) f +x/Å x F+b/ F+a/
The Second Fundamental Theorem of Calculus States
If f is continuous on [a, b], then the function
F(x) = (^) ¼a^ x^ f +t/Å t
has a derivative at every point in [a, b]. and
F ' (x) = cccccccdxd ¼a^ x^ f +t/Å t f +x/
Definite Integral Properties (in addition to the indefinite integral properties)
- (^) ¼a a f +x/Å x 0
- (^) ¼a b f +x/Å x ¼b a f +x/Å x
- (^) ¼a^ b^ f +x/Å x (^) ¼a^ c^ f +x/Å x (^) ¼c^ b^ f +x/Å x
Areas: If f(x) is positive for some values of x on [a, b] and negative for others, then
¼a
b (^) f +x/Å x
represents the cumulative sum of the signed areas between the graph of y = f(x) and the x-axis (where the areas above the x- axis are counted positively and the areas below the x-axis are counted negatively)
Thus,
¼a
b (^) « f +x/ « Å x
represents the actual area between the curve and the x-axis.
The area between the graphs of f(x) and g(x) where f(x) g(x) on [a, b] is given by
¼a
b# f +x/ g+x/'Å x
Volumes:
The volume of a solid of revolution (consisting of disks) is given by
Volume = S¼left endright enpdointpoint+radius/^2 Å x+or dy/
The volume of a solid of revolution (consisting of washers) is given by
Volume = S¼left endpoint right endpoint #+outside radius/^2 +inside radius /^2 'Å x+or dy/
The volume of a solid of known cross-sectional areas is given by
Volume = (^) ¼left endright enpdointpoint+cross sectional area/Å x+or dy/
Arc Length:
If the derivative of a function y = f(x) is continuous on the interval [a, b], then the length s of the arc of the curve of y = f(x) from the point where x = a to the point where x = b is given by
s = (^) Ä a
b 1 , cccccccdydx 0 ^2 Å x or (^) Ã a
b 1 + f '+x//^2 Å x
