Unit 1 Flashcards
1/x
x^-1
1/x^p
x^-p
√x
x^1/2
m√x^n
x^n/m
√(x^2)
|x|
(√x)^2
x
The instantaneous rate of change is the __________ of the function.
derivative
Average rate of change of f(x) on [a,b]
f’(x)= (f(b) - f(a))/ b - a
lim x->∞ (1/x)
0
What does it mean when a problem states that “f(x) is differentiable?”
f’x exists
ln(a*b)=
ln(a) + ln(b)
ln(a/b)=
ln(a) - ln(b)
lnx^p
plnx
What is lim x->c- f(x)
The limit of f(x) as x approaches c from the left
What is lim x->c+ f(x)
The limit of f(x) as x approaches from the right
What is lim x->c f(x)
The limit of f(x) as x approaches from both sides
When does the derivative of the function not exist?
-When the function is not continuous
-At a corner, cusp, sharp turn
-Vertical tangent line
State the relationship between continuity and differentiability?
-If f(x) is continuous f(x) may or many not be differentiable
-If f(x) is differentiable f(x) is continuous
The Intermediate Value Theorem (IVT):
If f(x) is continuous on [a,b], the f(a)<f(c)<f(b) for some c between a and b. (a<c<b)
The Mean Value Theorem:
If f(x) is continuous and differentiable on [a,b] then f’(c)= (f(b)-f(a))/b-a. For some c between a and b.
Equation of a tangent line
y-y1=m(x-x1)
Different of a square: a^2-b^2=
(a+b)(a-b)
Definition of derivative
f’(x)= lim h->0 f(x+h)-f(x)/h
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