Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #2 : Limits And Continuity

Screen shot 2015 08 17 at 6.27.41 pm

Given the above graph of \(\displaystyle f(x)\), over which of the following intervals is \(\displaystyle f(x)\) continuous?

Possible Answers:

None of the above

\(\displaystyle (-\infty,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,\infty)\)

Correct answer:

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

Explanation:

For a function \(\displaystyle f(x)\) to be continuous at a given point \(\displaystyle (a, f(a))\), it must meet the following two conditions:

1.) The point \(\displaystyle (a, f(a))\) must exist, and

2.) \(\displaystyle \lim_{x\rightarrow a}f(x)=f(a)\).

 

Examining the above graph, \(\displaystyle f(x)\) is continuous at every possible value of \(\displaystyle x\) except for \(\displaystyle x=0\). Thus, \(\displaystyle f(x)\) is continuous on the interval \(\displaystyle (-\infty,0)\cup(0,\infty)\).

Example Question #8 : Limits And Continuity

Screen shot 2015 08 17 at 6.36.23 pm

Given the above graph of \(\displaystyle f(x)\), over which of the following intervals is \(\displaystyle f(x)\) continuous?

 

Possible Answers:

\(\displaystyle (-\infty,-1)\cup(-1,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,\infty)\)

Correct answer:

\(\displaystyle (-\infty,-1)\cup(-1,1)\cup(1,\infty)\)

Explanation:

For a function \(\displaystyle f(x)\) to be continuous at a given point \(\displaystyle (a, f(a))\), it must meet the following two conditions:

1.) The point \(\displaystyle (a, f(a))\) must exist, and

2.) \(\displaystyle \lim_{x\rightarrow a}f(x)=f(a)\).

 

Examining the above graph, \(\displaystyle f(x)\) is continuous at every possible value of \(\displaystyle x\) except for \(\displaystyle x=-1\) and \(\displaystyle x=1\). Thus, \(\displaystyle f(x)\) is continuous on the interval \(\displaystyle (-\infty,-1)\cup(-1,1)\cup(1,\infty)\).

Example Question #1 : Limits And Continuity

Screen shot 2015 08 17 at 6.45.07 pm

Given the above graph of \(\displaystyle f(x)\), over which of the following intervals is \(\displaystyle f(x)\) continuous?

Possible Answers:

\(\displaystyle (-\infty,-1)\cup(-1,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\)

None of the above

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,0)\cup(0,\infty)\)

Correct answer:

\(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\)

Explanation:

For a function \(\displaystyle f(x)\) to be continuous at a given point \(\displaystyle (a, f(a))\), it must meet the following two conditions:

1.) The point \(\displaystyle (a, f(a))\) must exist, and

2.) \(\displaystyle \lim_{x\rightarrow a}f(x)=f(a)\).

 

Examining the above graph, \(\displaystyle f(x)\) is continuous at every possible value of \(\displaystyle x\) except for \(\displaystyle x=0\) and \(\displaystyle x=1\). Thus, \(\displaystyle f(x)\) is continuous on the interval \(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\).

Example Question #2 : Limits And Continuity

Screen shot 2015 08 18 at 9.51.12 am

Given the above graph of , over which of the following intervals is continuous?

 

Possible Answers:

\(\displaystyle (-\infty,-\frac{1}{2})\cup(\frac{1}{2},\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,\infty)\)

\(\displaystyle (-\infty,1)\cup(1,\infty)\)

None of the above

Correct answer:

Explanation:

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

 

Examining the above graph, is continuous at every possible value of except for . Thus, is continuous on the interval .

Example Question #501 : Limits

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Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

\(\displaystyle (-\infty,-2)\cup(-2,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,-2)\cup(-2,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,2)\cup(2,\infty)\)

\(\displaystyle (-\infty,-2)\cup(-2,2)\cup(2,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\)

Correct answer:

\(\displaystyle (-\infty,0)\cup(0,2)\cup(2,\infty)\)

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for  and \(\displaystyle x=2\). Thus,  is continuous on the interval \(\displaystyle (-\infty,0)\cup(0,2)\cup(2,\infty)\).

Example Question #11 : Limits And Continuity

Screen shot 2015 08 18 at 11.04.49 am

Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,2)\cup(2,\infty)\)

\(\displaystyle (-\infty,-\frac{1}2{})\cup(-\frac{1}{2},\infty)\)

\(\displaystyle (-\infty,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-1)\cup(-1,\infty)\)

Correct answer:

\(\displaystyle (-\infty,-1)\cup(-1,\infty)\)

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for \(\displaystyle x=-1\). Thus,  is continuous on the interval \(\displaystyle (-\infty,-1)\cup(-1,\infty)\).

Example Question #501 : Limits

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Given the above graph of , over which of the following intervals is continuous?

Possible Answers:

\(\displaystyle (-\infty,-11)\cup(-11,11)\cup(11,\infty)\)

\(\displaystyle (-\infty,11)\cup(11,\infty)\)

\(\displaystyle (-\infty,-11)\cup(-11,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,11)\cup(11,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

Correct answer:

\(\displaystyle (-\infty,11)\cup(11,\infty)\)

Explanation:

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

 

Examining the above graph, is continuous at every possible value of except for \(\displaystyle x=11\). Thus, is continuous on the interval \(\displaystyle (-\infty,11)\cup(11,\infty)\).

Example Question #504 : Limits

Screen shot 2015 08 19 at 2.31.48 pm

Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

\(\displaystyle (-\infty,-2)\cup(-2,2)\cup(2,\infty)\)

\(\displaystyle (-\infty,2)\cup(2,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-2)\cup(-2,\infty)\)

Correct answer:

\(\displaystyle (-\infty,2)\cup(2,\infty)\)

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for \(\displaystyle x=2\). Thus,  is continuous on the interval \(\displaystyle (-\infty,2)\cup(2,\infty)\).

Example Question #505 : Limits

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Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

\(\displaystyle (-\infty,-1)\cup(-1,0)\cup(0,\infty)\)

\(\displaystyle (-\infty,5)\cup(5,\infty)\)

\(\displaystyle (-\infty,-5)\cup(-5,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,5)\cup(5,\infty)\)

\(\displaystyle (-\infty,-5)\cup(-5,5)\cup(5,\infty)\)

Correct answer:

\(\displaystyle (-\infty,5)\cup(5,\infty)\)

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for \(\displaystyle x=5\). Thus,  is continuous on the interval \(\displaystyle (-\infty,5)\cup(5,\infty)\).

Example Question #506 : Limits

Screen shot 2015 08 19 at 4.51.39 pm

Given the above graph of , over which of the following intervals is continuous?

Possible Answers:

\(\displaystyle (-\infty,-1)\cup\(-1,0)\cup(0,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-1)\cup\(-1,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,0)\cup(0,1)\cup(1,\infty)\)

\(\displaystyle (-\infty,-\frac{1}{2})\cup\(-\frac{1}{2},0)\cup(0,\frac{1}{2})\cup(\frac{1}{2},\infty)\)

\(\displaystyle (-\infty,-2)\cup\(-2,0)\cup(0,2)\cup(2,\infty)\)

Correct answer:

\(\displaystyle (-\infty,-1)\cup\(-1,0)\cup(0,1)\cup(1,\infty)\)

Explanation:

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

 

Examining the above graph, is continuous at every possible value of except for \(\displaystyle x=-1\), and \(\displaystyle x=1\). Thus, is continuous on the interval \(\displaystyle (-\infty,-1)\cup\(-1,0)\cup(0,1)\cup(1,\infty)\).

 

 

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