Section 1
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f(x) has a local max
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Last updated
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Date created
Mar 1, 2020
Cards (51)
Section 1
(50 cards)
f(x) has a local max
f'(x) changes from + to -
Chain Rule
d/dx f(g(x)) = f'(g(x)) g'(x)
Rolle's Theorem
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c)= 0
f(x) has a local min
f'(x) changes from - to +
derivative of tanx
sec^2x
Derivative of cosine
-sinx
f(x) is concave down
f''(x) is negative
Mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
f(x) is decreasing when
f'(x) is negative
ln(x) =
1/x
The General Power Rule
d/dx(u^n)=nu^(n-1)*u'
e^lnx = (inverse relationship)
x
Derivative of cscx
-cscxcotx
Derivative of secx
secxtanx
sin^-1(x) =
1/√(1-x^2)
sec^-1(x) =
1/|x|sqrt(x^2-1)
cos^-1(x) =
-1/√(1-x^2)
cot^-1(x) =
-1/(1+x^2)
log_a(x)
1/lna lnx
ln(1) =
0
Quotient Rule
(vu'-uv')/v²
ln(a/b) =
ln(a)-ln(b)
Difference Rule for derivatives
d/dx [f(x)-g(x)] = f'(x)-g'(x)
Constant Multiple Rule
d/dx [cf(x)] = cf'(x)
derivative of sine
cosx
f(x) has a point of inflection
f''(x) changes signs
f(x) is increasing
f''(x) is positive
Critical Points of f
points in the domain of f - that are endpoints of the domain - where f' doesn't exist - where f'=0
ln(a^n) =
n ln(a)
Constant Rule for Derivatives
d/dx (c)= 0
ln(e^x) = (inverse relationship)
x
a^x =
e^(lna)x
csc^-1(x)
-1/|x|sqrt(x^2-1)
e^x =
e^x
f(x) is concave down
f'(x) is decreasing
ln(ab) =
ln(a) + ln(b)
f(x) is concave up
f'(x) is increasing
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.
definition of continuity
1. lim x→c f(x) exists. 2. f(c) exists. 3. lim x→c f(x) = f(c)
a^x =
a^x lna
sum rule of derivatives
d/dx [f(x)+g(x)] = f'(x)+g'(x)
Second Derivative test
if f'(c) = 0 and f''(c) > 0 then minimum; if f'(c) = 0 and f''(c) < 0 then maximum
f(x) has a point of inflection
f'(x) changes from inc. to dec. or dec. to inc.
f(x) is increasing
f'(x) is positive
Derivative of cotx
-csc^2x
First Deviantart Test
Used to determine where a function's graph has a min/max and is inc. or dec. (if a derivative goes + to -, it's a relative max. - to + is a relative min)
tan^-1(x) =
1/(1+x^2)
log_a(x) =
1/xln(a)
Product Rule
vu' + uv'
Section 2
(1 card)