All AP Calculus AB Resources
Example Questions
Example Question #31 : Numerical Approximations To Definite Integrals
Example Question #31 : Numerical Approximations To Definite Integrals
Example Question #33 : Numerical Approximations To Definite Integrals
Example Question #31 : Numerical Approximations To Definite Integrals
Example Question #35 : Numerical Approximations To Definite Integrals
Example Question #36 : Numerical Approximations To Definite Integrals
Example Question #37 : Numerical Approximations To Definite Integrals
Example Question #38 : Numerical Approximations To Definite Integrals
Example Question #39 : Numerical Approximations To Definite Integrals
Let .
A relative maximum of the graph of can be located at:
The graph of has no relative maximum.
At a relative minimum of the graph , it will hold that and .
First, find . Using the sum rule,
Differentiate the individual terms using the Constant Multiple and Power Rules:
Set this equal to 0:
Either:
, in which case, ; this equation has no real solutions.
has two real solutions, and .
Now take the second derivative, again using the sum rule:
Differentiate the individual terms using the Constant Multiple and Power Rules:
Substitute for :
Therefore, has a relative minimum at .
Now. substitute for :
Therefore, has a relative maximum at .
Example Question #40 : Numerical Approximations To Definite Integrals
Estimate the integral of from 0 to 3 using left-Riemann sum and 6 rectangles. Use
Because our is constant, the left Riemann sum will be