### All High School Math Resources

## Example Questions

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

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

**Possible Answers:**

exists because

exists because

does not exist because

does not exist because .

does not exist because

**Correct answer:**

does not exist because .

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?

**Possible Answers:**

exists because

does not exist because is not continuaous at .

does not exist because

does not exist because

does not exist because

**Correct answer:**

exists because

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:

**Possible Answers:**

continuous

discontinuous

**Correct answer:**

continuous

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?

**Possible Answers:**

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 .

**Correct answer:**

exists because .

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:**

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:**

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).

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

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:**

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 .

**Possible Answers:**

**Correct answer:**

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: **