# High School Math : Calculus II — Integrals

## Example Questions

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### Example Question #1 : Using Limits With Continuity

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

exists because

exists because

does not exist because

does not exist because .

does not exist because

does not exist because .

Explanation:

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

### Example Question #2 : Using Limits With Continuity

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

exists because

does not exist because  is not continuaous at .

does not exist because

does not exist because

does not exist because

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 #2 : Using Limits With Continuity

A function is defined by the following piecewise equation:

At , the function is:

continuous

discontinuous

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 #4 : Using Limits With Continuity

The graph depicts a function . Does  exist?

exists because  is constant on .

does not exist because .

exists because .

does not exist because  is undefined.

does not exist because  is not continuous at .

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).

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).

Explanation:

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

### Example Question #3 : Calculus Ii — Integrals

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

Explanation:

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

### Example Question #4 : Calculus Ii — Integrals

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

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 .

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 #6 : Calculus Ii — Integrals

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

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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