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Level 593

Calculus BC


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The limit as x approaches a of ƒ(x) exists if and only if
limit as x approaches a from the right of ƒ(x) is equal to the limit as x approaches a from the left of ƒ(x)
ƒ(x) is continuous at x=a if
limit as x approaches a of ƒ(x) is equal to ƒ(a)
Intermediate Value Theorem
If f(a) > 0 and f(b) < 0 then there is a real zero between a and b
Derivative
limit as h approaches 0 of {ƒ(x+h) - ƒ(x)}/h
A function is differentiable at x=a if
The function is continuous at x=a and the limit as h approaches 0 of {ƒ(x+h) - ƒ(x)}/h
limit as x approaches ±∞ of ƒ(x)=b
A function has a horizontal asymptote at y = b if
#/0
Vertical asymptotes occur at values of x that produce
0
Limit as x→∞ of (ax^m)/(bx^n), n>m
a/b
Limit as x→∞ of (ax^m)/(bx^n), n=m
increasing
ƒ'(x)>0
decreasing
ƒ'(x)<0
Relative (or Local) Maximum
ƒ'(x) changes from positive to negative
Relative (or Local) Minimum
ƒ'(x) changes from negative to positive
ƒ''(x)>0
Concave Up
ƒ''(x)<0
Concave Down
point of inflection
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
critical point
a point in the domain of the function at which the derivative is zero or undefined.
ln(ab)
ln(a)+ln(b)
ln(a/b)
ln(a) - ln(b)
aln(b)
ln(b^a)
0
Derivative: c
cnxⁿ⁻¹
Derivative: (cxⁿ)
1/(2√x)
Derivative: √2
Derivative: (ƒ(x)g(x))
ƒ(x)g'(x) + ƒ'(x)g(x)
(g(x)ƒ'(x)-g'(x)ƒ(x))/g(x)²
Derivative: (ƒ(x))/(g(x))
ƒ'(g(x))g'(x)
Derivative: (ƒ(g(x)))
ƒ'(x)e^ƒ(x)
Derivative: e^ƒ(x)
ƒ'(x)/ƒ(x)
Derivative: ln(ƒ(x))