Name Pre Calc HW – Distance as Area 1)

Name
Pre Calc HW – Distance as Area
1)
What is the total distance traveled by a tour bus, which travels at the rate of 55 mph
for 2.5 hours, 40 mph for 30 minutes, and 25 mph for 15 minutes?
2)
Use summation notation to express the total distance traveled by an object whose
rate-time graph is given at the right.
3)
If a space probe travels in a straight line with an initial speed of 120 m/sec and a
constant acceleration of 9.8 m/sec2, then its velocity at time t seconds is given by
120 + 9.8t. Find the distance it will have traveled in 8 seconds.
4)
The rate-time graph below shows the speed of a truck during a trucker’s 8-hour work
day. Estimate the distance the trucker traveled.
5)
The rate-time graph below depicts a cyclist riding from home to work. From the
graph estimate the distance to work.
→
→
6)
Given u = (3,−5) , find u .
7)
Let g be the function with g(x) = 2 x 5 − 10 x 2 + 3 . What is the slope of the tangent line
to the graph of g at x = 3?
8)
Evaluate the limit: lim
9)
Label the functions with f(x), fI(x), and fII(x).
4x
x → −4 2 x 2 + 4
25
20
15
10
5
-3
-2
-1
1
-5
-10
-15
2
3
Name
1)
Pre Calc HW – Riemann Sum and Integral Notation
The function f(x) = − .1(x − 7)2 + 8 is graphed.
f(x) = -0.1⋅(x-7)2 +8
8
a)
Use the upper Riemann sum, with
interval lengths of 2, to estimate the
area under the curve from 0 to 6
(the shaded area).
6
4
2
2
4
6
b)
Use the lower Riemann sum ,with interval lengths of 2, to estimate the area
under the curve from 0 to 6 (the shaded area)
c)
Use the upper Riemann sum, with
interval lengths of 1, to estimate the
area under the curve from 0 to 6
(the shaded area)
8
f(x) = -0.1⋅(x-7)2 +8
6
4
2
2
d)
4
6
Use the lower Riemann sum ,with interval lengths of 1, to estimate the area
under the curve from 0 to 6 (the shaded area)
Directions: For each of the pictures below express the area of the shaded region as a
definite integral. You do not have to find the value of the integral.
2)
3)
______________________
4)
5)
_______________________
6)
_________________________
_________________________
As a freight train travels eastward through Colorado, an instrument in the
locomotive keeps a record of the train’s velocity at any time on a paper scroll. The
velocity time record for the trip is pictured below.
Find how far east of the starting point the train is after 2 hours . (distance traveled)
**You do not have to estimate. This picture can be broken up into several pieces of which
you know how to find the area of like rectangles, trapezoids, and right triangles.
Name
Pre Calc HW – Integrating Using Area Formulas
Directions for 1 – 3:
a)
Express the area of the shaded region in integral notation.
b)
Tell whether the value of the integral appears to be positive or negative.
c)
Use area formulas to find the value of the integral in part a.
1)
2)
3)
Directions for 4 – 7:
a)
draw a picture to represent the integral as a shaded area.
b)
Evaluate the integral using area formulas.
5
4)
∫ 7dx
4
5)
−3
0
7
8
6)
∫ − 2x + 1 dx
2
1
∫ 2 x + 6 dx
*7)
∫
5
49 − x 2 dx
Name
Pre Calc HW – Integrating
7
1)
Evaluate:
 1

∫  − 2 x + 4 dx
3
4
2)
Evaluate:
∫ (15x
4
)
+ 2 x dx
0
5
3)
Evaluate:
∫ (2 x
2
)
+ 3 x dx
1
3
4)
Evaluate ∫ (25x 4 − 28 x 3 + 6x 2 )dx
2
5)
Express the area of the shaded region as a definite integral. (you do not need to
solve)
10
8
6
4
2
-4
-2
f(x) = -2⋅x2+x+3
2
4
-2
-4
-6
Find the value of the definite integral using area formulas.
9
6)
∫
81− x 2 dx
7
5
7)
∫ (2x + 3) dx
0
Name
Pre Calc HW – More Integrating
In #1 - 2, use properties of integrals to write the expression as a single integral
and then evaluate.
25
1.
∫ 3x
25
2
dx -
0
2.
∫ 3x
2
dx
15
2
5
0
2
3
3
∫ (8 x − 5)dx + ∫ (8 x − 5)dx
In #3-5, evaluate the integrals
3
3.
∫ (2x
2
− 3x + 5)dx
0
2
4.
∫ (x
3
+ 4x)dx
−3
6
5.
∫
5
36 − x 2 dx
In 6 and 7, express the area of each shaded region using integral notation and find its
value:
8.
Suppose a car accelerates from 0 to 80 ft/sec in 6 seconds so that its velocity v(t) in
this time interval in ft/sec after t seconds is given by v(t) = -2(t – 6)2 + 80. What is the
total distance traveled in these 6 seconds?
9.
Suppose a car accelerates from 0 to 90 ft/sec in 8 seconds so that its velocity v(t) in
this time interval in ft/sec after t seconds is given by v(t) = -2(t – 8)2 + 90. What is the
total distance traveled in these 8 seconds?