Name: __________________
Class:
1 By reading values from the given graph of f, use five rectangles to
find a lower estimate for the area from x = 0 to x = 10 under
the given graph of f. Round your answer to the nearest tenth.
Date: _____________
4
4
x , 1 x 4, approximate the area under
If f (x) =
the curve using ten approximating rectangles of equal widths and
left endpoints.
Select the correct answer. The choices are rounded to the nearest
hundredth.
a.
2.88
c.
2.70
e.
4.68
b.
3.66
d.
3.40
f.
2.34
5 The speed of a runner increased steadily during the first three
seconds of a race. Her speed at half second intervals is given in
the table. Find a lower estimate for the distance that she traveled
during these three seconds.
________
2 Estimate to the hundredth the area from 1 to 5 under the graph of
f (x) = 4 using four approximating rectangles and right
x
endpoints.
Select the correct answer. The choices are rounded to the nearest
hundredth.
a.
b.
3
4.17
5.75
c.
d.
5.13
3.35
e.
f.
0
0.5
1.0
1.5
2.0
2.5
3.0
v (ft/s)
0
4.5
8.1
11.4
12.9
13.4
14
a.
23.65
c.
25.35
e.
25.95
b.
24.35
d.
25.25
f.
25.15
6 When we estimate distances from velocity data, it is sometimes
necessary to use times t0 , t1 , t2 , ... that are not equally spaced. We
can still estimate distances using the time periods t = ti ti 1
. For example, on May 7, 1992, the space shuttle Endeavor was
launched on mission STS 49, the purpose of which was to
install a new perigee kick motor in an Intelsat communications
satellite. The table, provided by NASA, gives the velocity data for
the shuttle between liftoff and the jettisoning of the solid rocket
boosters. Use this data to estimate an upper bound for the space
shuttle Endeavor's height above Earth's surface 60 seconds after
liftoff.
Event
Time (s)
Velocity (ft/s)
5.39
Launch
0
0
6.85
Begin roll maneuver
12
180
End roll maneuver
13
318
Throttle to 89%
20
454
Throttle to 67%
34
744
Approximate the area under the curve y =
5 from 1 to 2 using
2
x
ten approximating rectangles of equal widths and right endpoints.
Select the correct answer. The choices are rounded to the nearest
hundredth.
a.
2.32
c.
2.92
e.
2.62
b.
2.80
d.
0.34
f.
1.60
PAGE 1
t (s)
Throttle to 104%
56
1321
Maximum dynamic pressure
60
1436
Solid rocket booster separation
121
4208
a.
51024
c.
50406
e.
49921
b.
51072
d.
49949
f.
50878
Name: __________________
Class:
7 The velocity graph of a car accelerating from rest to a speed of 7
km/h over a period of 10 seconds is shown. Estimate to the
nearest integer the distance traveled during this period. Use a right
sum with n = 10.
Date: _____________
9 Find an expression for the area from 1 to 7 under the curve y =
3
x as a limit.
a.
n
lim
n
b.
i = 1
n
lim
n
c.
i = 1
n
lim
n
d.
i = 1
lim
n
e.
i = 1
lim
n
f.
n
n
i = 1
lim
n
n
i = 1
1 + 8i
n
3
5
n
1 + 7i
n
3
7
n
1 + 7i
n
38
1 + 9i
n
3
1 + 8i
n
38
1 + 6i
n
36
n
7
n
n
n
10 Evaluate the Riemann sum for
a.
19
c.
13
e.
15
b.
23
d.
17
f.
20
8 Determine a region whose area is equal to
lim
n
n
i = 1
tan i .
5n
5n
y = tan x, 0 x b.
y = tan x, 0 x c.
y = tan x, 0 x e.
f.
y = tan x, 0 x y = tan x, 0 x y = tan x, 0 z 2, with four subintervals,
taking the sample points to be right endpoints.
4.5
a.
x lim
6
3
8
11
n b.
122.5
c.
12.25
n
7 x sin x x ,
i
i=1
[8,16]
i
12 Express the limit as a definite integral on the given interval.
lim
n n
5x
i = 1
2
i
12x
i
x,
[6,11]
13 Express the integral as a limit of sums. Then evaluate the limit.
9
5
a.
PAGE 2
2
z , 0 11 Express the limit as a definite integral on the given interval.
a.
d.
f(z) = 8 9
b.
1
9
sin 9x dx
0
c.
2
9
Name: __________________
Class:
14 Evaluate the integral by interpreting it in terms of areas.
Date: _____________
19 Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
4
(3
+ 5x ) dx =
5
________
g(x) = 5tan(t) d t
2
x
15 Evaluate the integral by interpreting it in terms of areas.
3
(2
1
8
a.
16
b.
5
c.
14
If
f (x)dx = 3.2 and
dg(x) = 5tan(x)
dx
b.
dg(x) = 5tan(5)
dx
x ) dx
f (x)dx = 1 find
9
12
If
9
f (x)dx .
1
g(x) =
12
f (x) dx = 3.8 and
4
f (x) dx = 1, find
f (x) dx .
b.
4.8
c.
2.8
18 Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
g(x) =
x
3 cos(t) d t
t
2
a.
dg(x) = 1.5 cos( x )
dx
x
b.
dg(x) = 1.5 cos( x )
dx
x
c.
dg(x) = 3cos( x )
dx
4
2.8
5tan(x)
6
6
a.
dg(x) = dx
20 Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
________
17
c.
4
14
1
a.
x
8 + 6t d t
21
22
Find the most general antiderivative of f (x) = 6 x
6 x + 5.
Find the most general antiderivative of f (x) =
1
2
6 , x > 0
5
x
.
a.
dg(x) =
6
dx
2 8 + 6x
b.
dg(x) =
dx
8 + 6x
c.
dg(x) =
dx
7 + 6x
23 Find the most general antiderivative of the function on the
interval
2
,
.
2
x
2
f ( x ) = 4e + 5sec x
24 Find f, given that
t
f' ' ( t ) = 9e + 7sint ,
f ( 0 ) = 0, f ( ) = 0
25 Given that the graph of f passes through the point (2, 32) and
that the slope of its tangent line at (x, f (x)) is 10 x + 2,
find f (4).
f (4) =
PAGE 3
________
Name: __________________
Class:
Date: _____________
30 What constant acceleration is required to increase the speed of a
car from 30 mi/h to 50 mi/h in 5 s?
26 The graph of a function f is shown. Which graph is an
antiderivative of f ?
Round your answer to the nearest hundredth.
a ________
27 A particle has velocity v(t) = 15 t and its position at t =
1 is s(1) = 15 . Find the position function of the particle.
28 A stone was dropped off a cliff and hit the ground with a speed
of 112 ft/s. What is the height of the cliff?
h =
________ ft
29 If a diver of mass m stands at the end of a diving board with
length L and linear density then the board takes on the shape
of a curve y = f (x), where
EIy' ' = mg(L x) + 1 g(L 2
x)
2
E and I are positive constants that depend on the material of the
board and g(> 0) is the acceleration due to gravity.
(a) Find an expression for the shape of the curve.
(b) Use f (L) to estimate the distance below the horizontal at
the end of the board.
PAGE 4
________
ft/s
2