题目
题目

MATH 2A LEC A: CALCULU... Midterm

多项选择题

Consider the function  𝑓 ( 𝑥 ) = { 1 − 𝑥 2 if  𝑥 ≤ 1 ln ⁡ ( 𝑥 ) if  𝑥 > 1 . Determine which of the following is correct. Select all that apply. 

选项
A.𝑓 is discontinuous at 𝑥 = 1 since 𝑓 ( 1 ) does not exist
B.𝑓 is discontinuous at 𝑥 = 1 since lim 𝑥 → 1 𝑓 ( 𝑥 ) does not exist
C.𝑓 is discontinuous at 𝑥 = 1 since lim 𝑥 → 1 𝑓 ( 𝑥 ) ≠ 𝑓 ( 1 )
D.𝑓 is continuous at 𝑥 = 1
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标准答案
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思路分析
To approach this question, I’ll examine the behavior of f around x = 1 from both sides and compare it to the value at x = 1. Option 1: 'f is discontinuous at x=1 since f(1) does not exist' — This is incorrect. Since x ≤ 1 at x = 1, f(1) = 1 − 1^2 = 0, so f(1) exists. Therefore, the premise of ......Login to view full explanation

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类似问题

Determine the following statements are True or False. 1) There is no solution of 𝑒 𝑥 + 𝑒 − 𝑥 2 = 2 on [-2,2]. [ Select ] True False 2) Let f be a continuous function on [0,1] such that 0 < 𝑓 ( 𝑥 ) < 1 for all 𝑥 ∈ [ 0 , 1 ] . We can conclude that there exists a point 𝑎 ∈ [ 0 , 1 ] such that 𝑓 ( 𝑎 ) = 𝑎 . [ Select ] True False 3) If a function f is continuous on [a,b], then there is a c in [a,b] with 𝑓 ( 𝑐 ) = 𝑓 ( 𝑎 ) + 𝑓 ( 𝑏 ) 2 . [ Select ] True False 4) The function 𝑓 ( 𝑥 ) = 1 + 𝑥 2 𝑥 2 − 4 has a maximum on [-3,3]. [ Select ] True False

Find numbers a and b, or k, so that f is continuous at every point.

The function [math: f(x)={−2x+4if x<0,−4x−6if x>0]f(x)=\left \{\begin {array}{ll}-2x+4&\text {if }x<0,\\-4x-6&\text {if }x>0\end {array}\right . is continuous.

Suppose we know the following information about the function  𝑓 ( 𝑥 ) :  𝑓 ( − 1 ) = − 4 , 𝑓 ( 2.5 ) = 3 , 𝑓 ( 𝜋 ) = 2.4 and 𝑓 ( 1 ) does not exist lim 𝑥 ⟶ − 1 − 𝑓 ( 𝑥 ) = − 4 lim 𝑥 ⟶ − 1 + 𝑓 ( 𝑥 ) = − 4 lim 𝑥 ⟶ 2.5 + 𝑓 ( 𝑥 ) = − ∞ lim 𝑥 ⟶ 𝜋 𝑓 ( 𝑥 ) = 0 lim 𝑥 ⟶ 8 − 𝑓 ( 𝑥 ) = 3 lim 𝑥 ⟶ 8 + 𝑓 ( 𝑥 ) = 3.01   What does this information tell us about the continuity of 𝑓 ( 𝑥 ) ? At 𝑥 = − 1 , 𝑓 ( 𝑥 ) is/has a [ Select ] jump discontinuity infinite discontinuity continuous discontinuous, but there is not enough information to tell which type there is not enough information to tell anything removable discontinuity . At 𝑥 = 1 , 𝑓 ( 𝑥 )  is/has a [ Select ] infinite discontinuity continuous there is not enough information to tell anything jump discontinuity removable discontinuity discontinuous, but there is not enough information to tell which type . At 𝑥 = 2.5 , 𝑓 ( 𝑥 )  is/has a [ Select ] continuous discontinuous, but there is not enough information to tell which type removable discontinuity jump discontinuity there is not enough information to tell anything infinite discontinuity . At 𝑥 = 𝜋 , 𝑓 ( 𝑥 )  is/has a [ Select ] there is not enough information to tell anything discontinuous, but there is not enough information to tell which type jump discontinuity continuous infinite discontinuity removable discontinuity . At 𝑥 = 8 , 𝑓 ( 𝑥 )  is/has a [ Select ] jump discontinuity there is not enough information to tell anything continuous removable discontinuity infinite discontinuity discontinuous, but there is not enough information to tell which type .

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