Section 5.1 QUIZ

Current Score : 41 / 45 Due : Thursday, April 24 2014 11:59 PM EDT

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 = 36.

(i) Sample points are left endpoints.L6 = 780

(ii) Sample points are right endpoints.R6 = 636

(iii) Sample points are midpoints.M6 = 705

(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?

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

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overestimate

underestimate

overestimate

underestimate


(a) Estimate the area under the graph of using four approximating rectangles and right

endpoints. (Round your answers to four decimal places.)R4 = 30.7313

Sketch the graph and the rectangles.

L6

R6

M6

f(x) = 5 from x = 0 to x = 4x

http://www.webassign.net/web/Student/Assignment-Responses/last?dep=8337885#

Is your estimate an underestimate or an overestimate?

(b) Repeat part (a) using left endpoints.L4 = 20.7313

Sketch the graph and the rectangles.

overestimate

underestimate

Is your estimate an underestimate or an overestimate?

Enhanced Feedback

Please try again. For estimating the area under an interval of a graph using four rectangles, divide the domain of the interval intofour segments of equal length. Next, find the y value on the graph that corresponds to the right (or left) endpoint of eachsegment. This y value will be the height of the rectangle, while the width of each rectangle will be the length of its segment.Lastly, add together the areas of the rectangles.


(a) Estimate the area under the graph of using three rectangles and right endpoints.R3 = 34

Then improve your estimate by using six rectangles.R6 = 31.75

overestimate

underestimate

f(x) = 8 + 2x2 from x = −1 to x = 2


Sketch the curve and the approximating rectangles for R3.

Sketch the curve and the approximating rectangles for R6.

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

L3 = 28

L6 = 28.75

Sketch the curve and the approximating rectangles for L3.

Sketch the curve and the approximating rectangles for L6.

(c) Repeat part (a) using midpoints.

M3 = 29.5

M6 = 29.875

Sketch the curve and the approximating rectangles for M3.

Sketch the curve and the approximating rectangles for M6.

(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 thefigure shown below. (Round your answers to two decimal places.)

n = 2:

upper sum 9.424

lower sum 6.283

Illustrate.

R6

M6

L6

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


n = 4:

upper sum 8.9645

lower sum 7.3937

Illustrate.

n = 8:

upper sum 8.65

lower sum 6.30

Illustrate.


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 27 25 21 25 27

(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of thetime intervals.1536 ft

(b) Give another estimate using the velocities at the end of the time periods.1500 ft

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

(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.

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




Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and values of the rate at two hour timeintervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.

V ≈ 63.2 L (lower estimate)

V ≈ 70.2 L (upper estimate)

t (h) 0 2 4 6 8 10

r(t) (L/h) 8.6 7.9 6.7 6.2 5.7 5.1


The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied. (UseM6 to get the most precise estimate.)

130 ft



8. –/2 points SCalcET7 5.1.030.

(a) Let An be the area of a polygon with n equal sides inscribed in a circle with radius r.

By dividing the polygon into n congruent triangles with central angle show that the following is true.

(No Response)

This answer has not been graded yet.

(b) Show that [Hint: Use the equation.]

(No Response)

This answer has not been graded yet.

2π/n,

An = nr2 sin12

2πn

An = πr2.lim n → ∞

http://www.webassign.net/scalcet7/3-3-equation-02.gif


9. 2/2 points | Previous Answers SCalcET7 5.1.013.MI.

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in thetable. Find lower and upper estimates for the distance that she traveled during these three seconds.

35.5 ft (smaller value)

45.6 ft (larger value)

t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0

v (ft/s) 0 5.7 11.2 15.5 18.8 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 thetable. 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 givethe upper estimate.

10.1/1 points | Previous Answers SCalcET7 5.1.023.

Determine a region whose area is equal to the given limit. Do not evaluate the limit.

lim n → ∞

n tan

i = 1

π3n

iπ3n

tan(x) on [0, 3π]

tan(x) on − , π3π3

x tan(x) on − , π3π3

tan(x) on 0, π3

x tan(x) on 0, π3



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Section 5.1 QUIZ