All GRE Subject Test: Math Resources
Example Questions
Example Question #11 : Comparing Series
For which values of p is
convergent?
All positive values of
only
it doesn't converge for any values of
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 : Numerical Approximation
Solve the integral
using Simpson's rule with subintervals.
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 : Simpson's Rule
Solve the integral
using Simpson's rule with subintervals.
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 #11 : Midpoint Riemann Sums
Solve the integral
using Simpson's rule with subintervals.
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 #4 : Simpson's Rule
Solve the integral
using Simpson's rule with subintervals.
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 : Trapezoidal Rule
Solve the integral
using the trapezoidal approximation with subintervals.
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 : Numerical Integration
Solve the integral
using the trapezoidal approximation with subintervals.
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 : Numerical Integration
Solve the integral
using the trapezoidal approximation with subintervals.
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 #4 : Numerical Integration
Solve the integral
using the trapezoidal approximation with subintervals.
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
Evaluate using the Trapezoidal Rule, with n = 2.
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:
All GRE Subject Test: Math Resources
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