### All GRE Subject Test: Math Resources

## Example Questions

### Example Question #1 : Numerical Approximation

For which values of p is

convergent?

**Possible Answers:**

All positive values of

only

it doesn't converge for any values of

**Correct answer:**

only

We can solve this problem quite simply with the integral test. We know that if

converges, then our series converges.

We can rewrite the integral as

and then use our formula for the antiderivative of power functions to get that the integral equals

.

We know that this only goes to zero if . Subtracting p from both sides, we get

.

### Example Question #1 : Simpson's Rule

Solve the integral

using Simpson's rule with subintervals.

**Possible Answers:**

**Correct answer:**

Simpson's rule is solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is therefore

### Example Question #1 : Numerical Approximation

Solve the integral

using Simpson's rule with subintervals.

**Possible Answers:**

**Correct answer:**

Simpson's rule is solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is therefore

### Example Question #2 : Numerical Approximation

Solve the integral

using Simpson's rule with subintervals.

**Possible Answers:**

**Correct answer:**

Simpson's rule is solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is therefore

### Example Question #2 : Simpson's Rule

Solve the integral

using Simpson's rule with subintervals.

**Possible Answers:**

**Correct answer:**

Simpson's rule is solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is therefore

### Example Question #1 : Numerical Integration

Solve the integral

using the trapezoidal approximation with subintervals.

**Possible Answers:**

**Correct answer:**

Trapezoidal approximations are solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #2 : Trapezoidal Rule

Solve the integral

using the trapezoidal approximation with subintervals.

**Possible Answers:**

**Correct answer:**

Trapezoidal approximations are solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #1 : Numerical Integration

Solve the integral

using the trapezoidal approximation with subintervals.

**Possible Answers:**

**Correct answer:**

Trapezoidal approximations are solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #3 : Trapezoidal Rule

Solve the integral

using the trapezoidal approximation with subintervals.

**Possible Answers:**

**Correct answer:**

Trapezoidal approximations are solved using the formula

where is the number of subintervals and is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #2 : Trapezoidal Rule

Evaluate using the Trapezoidal Rule, with n = 2.

**Possible Answers:**

**Correct answer:**

1) n = 2 indicates 2 equal subdivisions. In this case, they are from 0 to 1, and from 1 to 2.

2) Trapezoidal Rule is:

3) For n = 2:

4) Simplifying:

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