Chapter 2 | Limits
191
2.4 EXERCISES For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. 131. f ( x ) = 1 x
⎧ ⎩ ⎨
π 2
sin θ , 0≤ θ <
146. f ( θ ) =
cos( θ + k ), π
θ ≤ π
2 ≤
⎧ ⎩ ⎨ x 2 +3 x +2 x +2 ,
x ≠ −2 k , x =−2
147. f ( x ) =
132. f ( x ) = 2
x 2 +1
⎧ ⎩ ⎧ ⎩ ⎨ e kx , 0≤ x <4 x +3, 4≤ x ≤8 ⎨ kx , 0≤ x ≤3 x +1, 3< x ≤10
148. f ( x ) =
133. f ( x ) = x x 2 − x 134. g ( t ) = t −1 +1 e x −2 136. f ( x ) = | x −2| x −2 137. H ( x ) = tan2 x 135. f ( x ) = 5
149. f ( x ) =
In the following exercises, use the Intermediate Value Theorem (IVT).
⎧ ⎩
⎨ 3 x 2 −4, x ≤2 5+4 x , x >2
150. Let h ( x ) =
Over the interval
[0, 4], there is no value of x such that h ( x ) =10, although h (0) <10 and h (4) >10. Explain why this does not contradict the IVT. 151. A particle moving along a line has at each time t a position function s ( t ), which is continuous. Assume s (2) =5 and s (5) =2. Another particle moves such that its position is given by h ( t ) = s ( t )− t . Explain why there must be a value c for 2< c <5 such that h ( c ) =0. 152. [T] Use the statement “The cosine of t is equal to t cubed.” a. Write a mathematical equation of the statement. b. Prove that the equation in part a. has at least one real solution. c. Use a calculator to find an interval of length 0.01 that contains a solution. 153. Apply the IVT to determine whether 2 x = x 3 has a solution in one of the intervals ⎡ ⎣ 1.25, 1.375 ⎤ ⎦ or ⎡ ⎣ 1.375, 1.5 ⎤ ⎦ . Briefly explain your response for each interval.
t +3 t 2 +5 t +6
138. f ( t ) =
For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? 139. f ( x ) 2 x 2 −5 x +3 x −1 at x =1 140. h ( θ ) = sin θ −cos θ tan θ at θ = π
⎧ ⎩ ⎨ ⎪ ⎪
6 u 2 + u −2
u ≠ 1 2 u = 1 2
2 u −1 if
141. g ( u ) =
, at u = 1 2
7 2
if
πy ) tan( πy ) ,
142. f ( y ) = sin(
at y =1
⎧ ⎩ ⎧ ⎩
⎨ x 2 − e x if x <0 x −1 if x ≥0
143. f ( x ) =
, at x =0
⎨ x sin( x ) if x ≤ π x tan( x ) if x > π
144. f ( x ) =
, at x = π
In the following exercises, find the value(s) of k thatmakes each function continuous over the given interval.
⎧ ⎩ ⎨ 3 x +2, x < k 2 x −3, k ≤ x ≤8
145. f ( x ) =
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