Orals Questions for Exam 3
APPM 1340
INTEGRALS
1. If you integrate y = f(x) on the interval [a,b], what will that tell you? (Give a
geometric explanation….have them draw the picture and explain. Be sure their
graph has some negative values.)
2. What is the connection between the indefinite integral ! f (t )dt and the definite
integral
!
b
a
f (t )dt
3. Suppose that you know that
! (ax
3
"3
2
)
+ bx + c dx = 0 , does that tell you that the
function is odd? Draw a graph to support you claim.
4. Given the function f (t ) = 6t ! t 2 what should the limits of
!
b
a
f (t )dt be in order
to maximize the integral?
5. Use the geometry you know to calculate
!
2
0
(2 x + 4 " x 2 )dx
RIEMANN SUMS
1. a) Draw the function y = 4 x ! x 2 on [-2,4] and then draw the rectangles that
would correspond to n = 3 using right hand method.
b) How would you improve your approximation of the integral?
c) Graph the same function and draw the rectangles that correspond to midpoint
method.
d) If I were to take the limit as n approaches infinity of the sum in part c, what
would I find?
e) Letting n approach infinity is the same as letting delta x approach what?
f) Why does letting the norm of the partition approach zero achieve the same thing
as letting delta x approach zero where the delta x’s are equal? Draw graphs to
demonstrate
2. Draw a rough graph of y = 1 + x 3 on the interval [2,6] .
a) Using n = 4, draw the rectangles that correspond to Riemann sums if you are
using midpoint method.
b) Using n = 4, draw trapezoids you would have if you wished to approximate the
integral using trapezoidal method.
c) From your graphs, which method seems to give you the best approximation?
FUNDAMENTAL THEOREM OF CALCULUS
1. a) State both parts of the Fundamental Theorem of Calculus and be sure to
include all conditions.
b) State the MVT for integrals
2. What does the FTC Part 1 really say?
TRUE OR FALSE
b
a) If f(x) is continuous on the interval [a,b], then 5 ! f (t )dt =
a
a
b
b
! f (t )dt = ! f (t )dt
then ! tf (t )dt = t ! f (t )dt
b) If f(x) is continuous on the interval [a,b], then
c) If f(x) is continuous on the interval [a,b],
b
! 5 f (t )dt
a
a
b
b
a
a
d) If f(x) and g(x) are differentiable on the interval (a,b) and f(x) ! g(x) on the
interval, then f’(x) ! g’(x) on (a,b).
e)
! (ax
3
"3
2
)
3
(
)
+ bx + c dx = 2 2! ax 2 + bx + c dx
0
1
!3
dx =
4
8
x
g) All continuous functions have antiderivatives
h) All continuous functions are integrable
i) If I were to integrate that function from 1 to 5, I would be calculating the area
between the curve, f(t), and the x-axis.
f)
1
!
"2
LINEARIZATION
x
1. Linearize the function f(x) = 4x -
(1 + t 2 )dt
! 6 "t4
" 2
at the point where x = ! 2
TRAPEZOIDAL RULE
1. What is the smallest value of n that would assure that using the trapezoidal rule
you would have and error less than 10 !3 if you were estimating the value of the
2
(1 + t 2 )
! t 4 dt given that the error
"1
ET #
b"a
(!x )2 M
12
AVERAGE VALUE
4
1. Find the average value of
!
(1 + x )
dt . Will that value of the function occur
x
anywhere inside [1,4]? How do you know?
1
CALCULATING INTEGRALS AND DERIVATIVES
1. y =
3
( x 2 ! 3) 4
(2 x + 5) 7
x3
b) y =
(1 + t 2 )
! t 4 dt
"1
2
2. Evaluate:
a)
!
"1
" 3t 3
(t + 5)
4
4
dt
b)
!
1
(1 + x )
x
dt