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Unformatted text preview: + √x is not defined for x < 0 and hence f(x) is not continuous
continuous
contin
for x < 0. So we may straight away say that f(x) is not differentiable at x = 0.
differentiable
Another way to think about this is : since f(x) is not defined for x < 0 then certainly
f’(x) is not defined for x ≤ 0.
Let us make f(x) continuous at x = 0 by defining f(x) = 0 for x < 0. Now :
continuous
+ √x for x ≥ 0
0, a constant for x < 0
irst
We may now apply the method of finding the derivative from fir st principles as we
have been doing so far. Alternatively, we may mechanically apply the formula to find
the derivative of xn when n is rational. Either way we get :
1/
deriv tiv
for x ≥ 0 is the derivative from the right
deri
right
f’(x) = { 2√ x
0 for x < 0 is the derivative from the left
derivative
left
At x = 0 : the right derivative f’(x) = 1/ 2√ x is not defined. So at x = 0 :
right deriv tiv
f(x) = { left derivative ≠ right derivative
right We say : f(x) is not differentiable at x = 0.
differentia
dif entiab
When finding the derivatives of functions with rational exponents, we must be
rational
cautious and check to see if f(x) and f’(x) are defined at x = 0 and x < 0.
92 [meters] Exer cise
ercis
Ex er cise : A car weighing one ton and star ting fr om r est (0 kmph)
accelerates steadily due EAST (along the xaxis) for 10 secs and levels
off at 90 kmph (25 m/sec). Its position function x(t) and the graph is shown
below.
x(t) = { 2.5t2/ over [0, 10]
2
25t − 125 for t ≥ 10 375
250
125
0 10 20 [secs] t What is the speed x’(t) over speed over [0, 10] ?
What is the speed x’(t) over speed over t ≥ 10 ?
Is x’(t) DIFFERENTIABLE at t = 10 secs ?
Draw the speed graph x’(t) over [0, 25].
What is the acceleration x”(t) over [0, 10] ?
What is the acceleration x”(t) over t ≥ 10 ?
Is x”(t) DIFFERENTIABLE at t = 10 secs ?
Draw the acceleration graph x”(t) over [0, 25].
( Hint : see example 3 page 89 and example 2 page 181 ) . 93 Before finding the INSTANTANEOUS RATE OF CHANGE of a function the
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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