AP Calculus Formulas and Theorems
Implicit Differentiation
Differentiating both sides of the equation with respect to x and then solving the resulting equation for y' (dy/dx)
Squeeze Theorem
If f(x) ≤ g(x) ≤ h(x) and as x→a, f(x)→L and h(x)→L then g(x)→L
EVT
If function f(x) is continuous on the closed interval [a, b], then f must attain a maximum and a minimum, each at least once.
MVT
If function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].
IVT
If function f(x) is continuous on the closed interval [a,b] and that k is between f(a) and f(b). Then, there exists some value c on the open interval (a,b) such that f(c) =k.
Definition of Concavity
Let f be a function differentiable on an open interval I. The graph of f is said to be concave up on I if f' increases on I it is said to be concave down on I if f' decreases on I.
Rolle's Theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
Alternate Form of Derivative
[f(x)-f(c)]÷[x-c]
Definition of an Inflection Point
a point on the curve where the concavity changes MAY occur when... y"=0 or y" DNE
Chain Rule
d/dx [f(g(x)] = f'(g(x))*g'(x)
Product Rule
d/dx [f(x)*g(x)] = f(x)g'(x) + f'(x)g(x)
Quotient Rule
d/dx [f(x)/g(x)] = [g(x)f'(x) - g'(x)f(x)]/[g(x)²]
Power Rule
d/dx[xⁿ]=nxⁿ⁻¹
synonym for antiderivative
definite integral
2nd derivative test
f(x) is... cc down where y" < 0 cc up where y" > 0
reciprocal parent function
f(x)= 1/x
Cosine Parent Function
f(x)=cosx
Logarithmic Parent Function
f(x)=lnx
sine parent function
f(x)=sinx
Tangent Parent Function
f(x)=tanx
Cubic Parent Function
f(x)=x³
Absolute Parent Function
f(x)=|x|
Square Root Parent Function
f(x)=√x
Formal definition of derivative
lim [f(x+h)-f(x)]÷h h→0
Special Trig Limits
lim sinx / x = 1 x→0 lim (1-cosx) / x = 0 x→0
synonyms for derivative
line tangent to f(x) rate of change
1st Derivative Test
set f'(x)=0 then find the critical numbers determine the sign of f'(x) at the test values of each interval f' > 0 → increasing f' < 0 → decreasing
Critical Number
x values where f'(x) is zero or undefined.
Exponential Parent Function
y=a∧x
Linear parent function
y=x
Quadratic Parent Function
y=x²
average rate of change
ƒ(b) - ƒ(a) / b-a
Anti-Power Rule
∫(xⁿ)dx= [xⁿ⁺¹/n+1]+c
