High School Math : Calculus II — Integrals

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #1 : Using Limits With Continuity

Function

 

The above graph depicts a function . Does  exist, and why or why not?

Possible Answers:

 does not exist because 

 exists because 

 exists because 

 does not exist because 

 does not exist because .

Correct answer:

 does not exist because .

Explanation:

 exists if and only if . As can be seen from the diagram, , but . Since ,    does not exist.

Example Question #1 : Using Limits With Continuity

Function

The above graph depicts a function . Does  exist, and why or why not?

Possible Answers:

 exists because 

 does not exist because 

 does not exist because  is not continuaous at .

 does not exist because 

 does not exist because 

Correct answer:

 exists because 

Explanation:

 exists if and only if ;

the actual value of  is irrelevant, as is whether  is continuous there.

As can be seen,

 and ;

therefore, ,

and  exists.

Example Question #1 : Using Limits With Continuity

A function is defined by the following piecewise equation:

At , the function is:

Possible Answers:

continuous

discontinuous

Correct answer:

continuous

Explanation:

The first step to determine continuity at a point is to determine if the function is defined at that point. When we substitute in 3 for , we get 18 as our -value. is thus defined for this function.

The next step is determine if the limit of the function is defined at that point. This means that the left-hand limit must be equal to the right-hand limit at . Substitution reveals the following:

Both sides of the function, therefore, approach a -value of 18.

 

Finally, we must ensure that the curve is smooth by checking the limit of the derivative of both sides.

 

Since the function passes all three tests, it is continuous.

Example Question #1 : Calculus Ii — Integrals

Function

The graph depicts a function . Does  exist?

Possible Answers:

 exists because .

 does not exist because  is undefined.

 does not exist because .

 does not exist because  is not continuous at .

 exists because  is constant on .

Correct answer:

 exists because .

Explanation:

 exists if and only if ; the actual value of  is irrelevant.

As can be seen,  and ; therefore, , and   exists.

Example Question #1 : Calculus Ii — Integrals

The polar coordinates of a point are . Give its -coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

Correct answer:

Explanation:

Given the polar coordinates , the  -coordinate is  .  We can find this coordinate by substituting :

Example Question #1 : Polar

The polar coordinates of a point are . Give its -coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

Correct answer:

Explanation:

Given the polar coordinates , the  -coordinate is  . We can find this coordinate by substituting :

Example Question #1 : Parametric, Polar, And Vector

The polar coordinates of a point are . Give its -coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

Correct answer:

Explanation:

Given the polar coordinates , the  -coordinate is  . We can find this coordinate by substituting :

Example Question #1 : Polar

The polar coordinates of a point are . Give its -coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

Correct answer:

Explanation:

Given the polar coordinates , the  -coordinate is  . We can find this coordinate by substituting :

Example Question #2 : Parametric, Polar, And Vector

Find the vector where its initial point is and its terminal point is .

Possible Answers:

Correct answer:

Explanation:

We need to subtract the -coordinate and the -coordinates to solve for a vector when given its initial and terminal coordinates:

 

Initial pt:

Terminal pt:

Vector:

Vector:

Example Question #1 : Understanding Vector Coordinates

Find the vector where its initial point is  and its terminal point is .

Possible Answers:

Correct answer:

Explanation:

 

We need to subtract the -coordinate and the -coordinate to solve for a vector when given its initial and terminal coordinates:

 

Initial pt:

Terminal pt:

Vector: 

Vector:

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