GRE Subject Test: Math : Numerical Approximation

Example Questions

Example Question #1 : Numerical Approximation

For which values of p is

convergent?

All positive values of

only

it doesn't converge for any values of

only

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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: