Sample Questions with Solutions for Math 53 DLRC Review Jan 2011

1. Questions:
1. If f  x  = 〚 x〛 x then what is f '  x at x = a∈ℤ ?
2. If ∣ f  x −1∣  x 2 e x for all x ∈ℝ then lim f  x as x 0 is ?
sinx
3. Use the Intermediate Value Theorem to show that f  x = has at least one root on the interval [2,5]
x 5−7
Solutions:
Solution 1:
We use the definition of the derivative.
CASE 1: as x  a
f  x − f a  〚 x〛x−〚a〛−a x −a
lim = lim = lim =1
x−a x−a x −a
CASE 2 :as x  a−
f  x − f a  〚 x〛x−〚a〛−a a−1a−a−a
lim = lim  =∞
x−a x−a 0-
Thus , the derivative of f does not exist at x=a ∈ℤ
Solution 2 :
We use the SqueezeTheorem.
∣ f  x −1∣  x 2 e x for all x∈ℝ
- x 2 e x  f  x −1x 2 e x for all x∈ℝ
as x  0, lim - x 2 e x   lim  f  x−1  lim  x 2 e x  for all x∈ℝ
as x  0, lim  f  x −1 = 0 for all x∈ℝ
as x  0, lim f  x = 1 for all x ∈ℝ
Solution 3:
Note that onthe interval 0,  , sinx0
Since 2∈[0,] , f 20
Note that onthe interval  , 2  , sinx0
Since5∈ , 2 , f 50
Thus ,it follows from the Intermediate Value Theorem that
∃ c∈[2,5] such that f  c=0, ie. f has a zero on the interval [2,5]