section 5.1 hw

5/2/14, 10:34 PMSection 5.1 HW

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Current Score : 82 / 83 Due : Thursday, April 24 2014 11:59 PM EDT

Section 5.1 HW (Homework)Frances CoronelMAT 151 Calculus I, Spring 2014, section 01, Spring 2014Instructor: Ira Walker

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5/2/14, 10:34 PMSection 5.1 HW


1. 6/6 points | Previous Answers SCalcET7 5.1.002.

Consider the following.

(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24.

(i) Sample points are left endpoints.L6 = 343.2

(ii) Sample points are right endpoints.R6 = 279.2

(iii) Sample points are midpoints.M6 = 314

(b) Is L6 an underestimate or overestimate of the true area?

(c) Is R6 an underestimate or overestimate of the true area?

(d) Which of the numbers gives the best estimate?

overestimate

underestimate

overestimate

underestimate

M6

L6

R6

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5/2/14, 10:34 PMSection 5.1 HW


2. 6/6 points | Previous Answers SCalcET7 5.1.003.MI.

(a) Estimate the area under the graph of f(x) = 2 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints.(Round your answers to four decimal places.)R4 = 1.5815

Sketch the graph and the rectangles.

5/2/14, 10:34 PMSection 5.1 HW


Is your estimate an underestimate or an overestimate?

Master It

Estimate the area under the graph of f(x) = cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. Isyour estimate an underestimate or an overestimate?Part 1 of 5

Rectangle areas are found by calculating

The width of each rectangle equals and the height of each rectangle is given by the function value at the right-hand side of therectangle.

So we must calculate where represent the right-hand endpoints of

four equal sub-intervals of

Since we wish to estimate the area over the interval using 4 rectangles of equal widths, then each rectangle will have width

(b) Repeat part (a) using left endpoints.L4 = 2.3669

Sketch the graph and the rectangles.

underestimate

overestimate

height × width.

Δx

R4 = ,4

f(xi)Δx = [f(x1) + f(x2) + f(x3) + f(x4)] Δxi = 1

x1, x2, x3, x4

0, .π2

0, π2

Δx = (No Response) .

5/2/14, 10:34 PMSection 5.1 HW


Is your estimate an underestimate or an overestimate?

5/2/14, 10:34 PMSection 5.1 HW


Master It

Estimate the area under the graph of f(x) = cos(x) from x = 0 to x = π/2 using four approximating rectangles and left endpoints. Is yourestimate an underestimate or an overestimate?Part 1 of 5

We must calculate where represent the left-hand endpoints of

four equal sub-intervals of

Since we wish to estimate the area over the interval using 4 rectangles of equal widths, then each rectangle will have width

3. 16/16 points | Previous Answers SCalcET7 5.1.003.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points forthe skipped part, and you will not be able to come back to the skipped part.

Exercise (a)

Estimate the area under the graph of f(x) = cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints.Is your estimate an underestimate or an overestimate?

Part 1 of 5

Rectangle areas are found by calculating

The width of each rectangle equals and the height of each rectangle is given by the function value at the right-hand side ofthe rectangle.

So we must calculate where represent the right-hand

endpoints of four equal sub-intervals of

Since we wish to estimate the area over the interval using 4 rectangles of equal widths, then each rectangle will have

width

Part 2 of 5

We wish to find

underestimate

overestimate

L4 = ,4

f(xi − 1) Δx = [f(x0) + f(x1) + f(x2) + f(x3)]Δxi = 1

x0, x1, x2, x3

0, .π2

0, π2

Δx = (No Response) .

height × width.

Δx

R4 = ,4

f(xi)Δx = [f(x1) + f(x2) + f(x3) + f(x4)] Δxi = 1

x1, x2, x3, x4

0, .π2

0, π2

Δx = .

R4 = [f(x1) + f(x2) + f(x3) + f(x4)] .π8


5/2/14, 10:34 PMSection 5.1 HW


Since represent the right-hand endpoints of the four sub-intervals of then we must have the following.

Part 3 of 5Round your answer to four decimal places.

= .790766

Part 4 of 5We can use the above information to sketch the graph of the function and the rectangles.

x1, x2, x3, x4 0, ,π2

x1 =

x2 =

x3 =

x4 =

R4 = cos + cos + cos + cosπ8

π4

3π8

π2

π8

5/2/14, 10:34 PMSection 5.1 HW


Part 5 of 5Is your estimate an underestimate or an overestimate?

Exercise (b)

Estimate the area under the graph of f(x) = cos(x) from x = 0 to x = π/2 using four approximating rectangles and left endpoints.Is your estimate an underestimate or an overestimate?

Part 1 of 5

We must calculate where represent the left-hand

endpoints of four equal sub-intervals of

Since we wish to estimate the area over the interval using 4 rectangles of equal widths, then each rectangle will have

underestimate

overestimate

L4 = ,4

f(xi − 1) Δx = [f(x0) + f(x1) + f(x2) + f(x3)]Δxi = 1

x0, x1, x2, x3

0, .π2

0, π2

5/2/14, 10:34 PMSection 5.1 HW


width

Part 2 of 5

We wish to find .

Since represent the left-hand endpoints of the four sub-intervals of then we must have the following.

Part 3 of 5Round your answer to four decimal places.

= 1.183465

Part 4 of 5We can use the above information to sketch the graph of the function and the rectangles.

Δx = .

L4 = [f(x0) + f(x1) + f(x2) + f(x3)] π8

x0, x1, x2, x3 0, ,π2

x0 =

x1 =

x2 =

x3 =

L4 = cos(0) + cos + cos + cosπ8

π4

3π8

π8

5/2/14, 10:34 PMSection 5.1 HW


Part 5 of 5Is your estimate an underestimate or an overestimate?

You have now completed the Master It.


(a) Estimate the area under the graph of using three rectangles and right endpoints.R3 = 13

Then improve your estimate by using six rectangles.R6 = 10.75

Sketch the curve and the approximating rectangles for R3.

underestimate

overestimate

f(x) = 1 + 2x2 from x = −1 to x = 2


5/2/14, 10:34 PMSection 5.1 HW


Sketch the curve and the approximating rectangles for R6.

5/2/14, 10:34 PMSection 5.1 HW


(b) Repeat part (a) using left endpoints.

L3 = 7

L6 = 7.75

Sketch the curve and the approximating rectangles for L3.

5/2/14, 10:34 PMSection 5.1 HW


5/2/14, 10:34 PMSection 5.1 HW


Sketch the curve and the approximating rectangles for L6.

(c) Repeat part (a) using midpoints.

M3 = 8.5

M6 = 8.875

Sketch the curve and the approximating rectangles for M3.

5/2/14, 10:34 PMSection 5.1 HW


Sketch the curve and the approximating rectangles for M6.

5/2/14, 10:34 PMSection 5.1 HW


5/2/14, 10:34 PMSection 5.1 HW


(d) From your sketches in parts (a)-(c), which appears to be the best estimate?


Evaluate the upper and lower sums for with Illustrate with diagrams like the figure shownbelow. (Round your answers to two decimal places.)

n = 2:

upper sum 9.424

lower sum 6.283

Illustrate.

L6

R6

M6

f(x) = 2 + sin x, 0 ≤ x ≤ π, n = 2, 4, and 8.


5/2/14, 10:34 PMSection 5.1 HW


n = 4:

upper sum 8.9645

lower sum 7.3937

Illustrate.

5/2/14, 10:34 PMSection 5.1 HW


5/2/14, 10:34 PMSection 5.1 HW


n = 8:

upper sum 8.65

lower sum 6.29

Illustrate.

5/2/14, 10:34 PMSection 5.1 HW



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 lower and upper estimates for the distance that she traveled during these three seconds.

33.65 ft (smaller value)

43.75 ft (larger value)

t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0

v (ft/s) 0 6.7 9.2 14.1 17.5 19.8 20.2

Master ItThe 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 lower and upper estimates for the distance that she traveled during these three seconds.

t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0

v (ft/s) 0 6.2 10.8 14.9 18.1 19.4 20.2

Part 1 of 5We will use either L6 or R6 for the upper and lower estimates.

Since the runner's speed is an increasing function, then (No Response) will give the lower estimate, and (No Response) will give theupper estimate.



5/2/14, 10:34 PMSection 5.1 HW


7. 17/17 points | Previous Answers SCalcET7 5.1.013.MI.SA.


Tutorial ExerciseThe 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 lower and upper estimates for the distance that she traveled during these three seconds.

t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0

v (ft/s) 0 6.2 10.8 14.9 18.1 19.4 20.2

Part 1 of 5We will use either L6 or R6 for the upper and lower estimates.

Since the runner's speed is an increasing function, then will give the lower estimate, and willgive the upper estimate.

Part 2 of 5The sub-interval widths for this situation are

Part 3 of 5The first two sub-intervals in the table are and

When calculating L6, we should use the function values and for these two sub-intervals, respectively.

Part 4 of 5Enter your answers from smallest to largest. In the last step, round your answer to one decimal place. For the lower estimate, we have thefollowing.

Part 5 of 5Enter your answers from smallest to largest. In the last step, round your answer to one decimal place. For the upper estimate, we have thefollowing.


Δt Δt = .5 .

[0, 0.5] [0.5, 1.0].

v = 0 v = 6.2

L6 = (0.5)(0 + 6.2 + 10.8 + 14.9 + 18.1 + 19.4 )

= 34.7 ft

R6 = (0.5)( 6.2 + 10.8 + 14.9 + 18.1 + 19.4 + 20.2 )

= 44.8 ft


5/2/14, 10:34 PMSection 5.1 HW



Speedometer readings for a motorcycle at 12-second intervals are given in the table.

t (s) 0 12 24 36 48 60

v (ft/s) 30 28 25 21 25 28

(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the timeintervals.1548 ft

(b) Give another estimate using the velocities at the end of the time periods.1524 ft

(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.

(b) is a lower estimate and (a) is an upper estimate since v is a decreasing function of t.

(a) is a lower estimate and (b) is an upper estimate since v is an increasing function of t.

(a) and (b) are neither lower nor upper estimates since v is neither an increasing nor decreasing function of t.


5/2/14, 10:34 PMSection 5.1 HW



When we estimate distances from velocity data, it is sometimes necessary to use times that are not equally spaced. Wecan still estimate distances using the time periods For example, a space shuttle was launched on a mission, the purposeof which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and thejettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 secondsafter liftoff. (Give the upper approximation available from the data.)h = 51782 ft

Event Time (s) Velocity (ft/s)

Launch 0 0

Begin roll maneuver 10 180

End roll maneuver 15 319

Throttle to 89% 20 453



Maximum dynamic pressure 62 1453

Solid rocket booster separation 125 4052

Master ItWhen we estimate distances from velocity data, it is sometimes necessary to use times that are not equally spaced. Wecan still estimate distances using the time periods For example, a space shuttle was launched on a mission, the purposeof which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and thejettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 secondsafter liftoff.


Launch 0 0








Part 1 of 4We can obtain an upper estimate for the height by using the final velocity for each time interval.

The first interval begins at t = 0 and ends at t = 10, so its width is

t0, t1, t2, t3, . . .Δti = ti − ti − 1.

t0, t1, t2, t3, . . .Δti = ti − ti − 1.

Δt1 = (No Response) .


5/2/14, 10:34 PMSection 5.1 HW


10.9/9 points | Previous Answers SCalcET7 5.1.016.MI.SA.


Tutorial ExerciseWhen we estimate distances from velocity data, it is sometimes necessary to use times that are not equally spaced. Wecan still estimate distances using the time periods For example, a space shuttle was launched on a mission, the purposeof which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and thejettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 secondsafter liftoff.


Launch 0 0








Part 1 of 4We can obtain an upper estimate for the height by using the final velocity for each time interval.

The first interval begins at t = 0 and ends at t = 10, so its width is

Part 2 of 4The second interval begins at t = 10 and ends at t = 15, so its width is

Part 3 of 4The final velocities for the first two intervals are, respectively,

and

Part 4 of 4Now, we can calculate an upper estimate for the height.


t0, t1, t2, t3, . . .Δti = ti − ti − 1.

Δt1 = 10 .

Δt2 = 5 .

v1 = 185 v2 = 319 .

h ≈ (185 × 10) + (319 × 5) + ( 447 × 5) + (742 × 12 ) + ( 1325 × 27) + (1445 × 3 )

= 54694 ft


5/2/14, 10:34 PMSection 5.1 HW


11.1/1 points | Previous Answers SCalcET7 5.1.017.

The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied. (Use M6 to

get the most precise estimate.)120 ft


section 5.1 hw

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