CALCULUS TRUE or FALSE?
if f(x)=g(x)+c, then f ' (x) = g ' (x).
and
if f(x) < g(x) for all numbers where x does not equal a, then the limit as x approaches a of f(x) < the limit as x approaches a of g(x).
PLEASE EXPLAIN HOW...
if f(x)=g(x)+c, then f ' (x) = g ' (x).
and
if f(x) < g(x) for all numbers where x does not equal a, then the limit as x approaches a of f(x) < the limit as x approaches a of g(x).
PLEASE EXPLAIN HOW...
Anonymous
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First one is true.
Proof:
LEFT SIDE:
Derivative of f(x) = f'(x)
RIGHTSIDE
Derivative of g(x) + c
Consider Sum rule of Derivation:
Derivative of (g + c) = g' + c'
Therefore, Derivative of g(x) = g'(x)
Derivative of c, a constant, is always 0
Therefore, Derivative of g'(x) = g'(x)
Which proves the statement.
The 2nd statement is also true.
If the graph of f(x) is continually smaller then the graph of g(x), then no matter where you take the limit, the y value of g(x) will always be greater than f(x)