Section 1
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Definition of Derivative
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Last updated
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Date created
Mar 1, 2020
Cards (53)
Section 1
(50 cards)
Definition of Derivative
lim [f(x) - f(a)] / [x-a] x->a
tanx
sec^2x = (secx)^2
sinx
cosx
tangent line
share the same slope and point
if f is continuous from [a,b], and N is a number between f(a) and f(b), f(a)/= f(b), then there is a number c in (a,b) where f(c) = N
On a continuous graph, you can't get from one point to another without going over every x value and y value between those two points
cosx
-sinx
lim f/g = lim f(x)/lim g(x) x->a x->a x->a
division
notation for derivative
f'(x)
lim f(x) = L x-> a
the limit as x approaches a of f(x) is L
notation for derivative on a point
d(f(a))/dx
1 +- oo
undefined
notation for derivative
Df(x)
derivative
rate
lim cf(x) = c lim f(x) x->a x->a
constant
Power Rule
If f(x) = x^n, then f'(x) = nx^n-1
lim (f(x) + g(x)) = lim f(x) + lim g(x) x->a x->a x->a
addition
0/0
undefined
lim (f(x) - g(x)) = lim f(x) - lim g(x) x->a x->a x->a
subtraction
derivative
rate of change
if h(x) = f(x) +- g(x)
then h'(x) = f'(x) +- g'(x)
notation for derivative on a point
dy/dx | | a
derivative of a function
lim [f(x + h)−f(x)] / [h] h->0
notation for derivative
d(f(x))/dx
lim (1-cos x) / x x->0
0
derivative of a point
lim [f(a + h)−f(a)] / [h] h->0
lim f(x) = L x->a+
the limit as x approaches a from the right of f(x) is L
oo - oo
undefined
lim f(x) = L iff lim f(x) = L AND lim f(x) = L x->a x->a+ x->a-
the limit as x approaches a of f(x) is L if and only if the limit as x approaches a from the left of f(x) is L AND the limit as x approaches a from the right of f(x) is L
notation for derivative on a point
f'(a)
0 * +-oo
undefined
derivative
velocity
lim [f(x)]^n = [lim f(x)]^n x->a x->a
to the power n
-oo + oo
undefined
notation for derivative
y'
lim f(x) = L x->a-
the limit as x approaches a from the left of f(x) is L
lim (f(x)g(x)) = lim f(x) * lim g(x) x->a x->a x->a
multiplication
derivative
slope
derivative
speed
lim (sin x) / x x->a
1
notation for derivative
dy/dx
notation for derivative
Dxf(x)
notation for derivative
df/dx
Quotient Rule
If h(x) = f(x)g(x), then h'(x) = f'(x)g(x)-g'(x)f(x)/(g(x))^2
if h(x) = cf(x),
then h'(x) = cf'(x)
notation for derivative
dy/dt
lim f(x) = lim f(x) = f(a) x->a+ x->a-
Definition of continuous
oo^0
undefined
0^0
undefined
+- oo/ +- oo
undefined
lim c = c x->a
constant
Section 2
(3 cards)