Consider f(x) = 1/xon the interval [1,2].Estimate

| August 30, 2017

Question
1. Consider f(x) = 1/xon the interval [1,2].Estimate the area under the graph offby

subdividing the interval into three equal subintervals and using the right-hand endpoints.Explain whythe estimate is an overestimate or an underestimate

2. Find a plane region whose area is equal to

lim n→∞ n, i=1 [2 +(5i/n)^3] (5/n)

Do not evaluate the limit! (Hint: Compare with lim

n→∞ ni=1 f(a+i∆x)∆x which gives the “area” under

the graph of f over [a, b] where ∆x = b−a

n , to find a, b, and f)

MAT 109
Quiz 8, Due in class on Monday, Dec 7

Name:

Please provide the details of your solutions. Full credit will not be given to incomplete answers.
Make sure to staple pages together before bringing your papers to class.

1.

(3 points) Find the shortest distance from the point (4, 0) to a point on the curve y 2 = 2x.

2.

(2 points) Find an antiderivative of the following functions

(a) f (x) =


x4 +x3 + x
x4

(b) g(x) = sec2 (x) + 3 sin(x)

3.

(2 points) Find f (x) if f (x) =

2

x

and f (4) = 1.

4.

1
(2 points) Consider f (x) = x on the interval [1, 2]. Estimate the area under the graph of f by
subdividing the interval into three equal subintervals and using the right-hand endpoints. Explain why
the estimate is an overestimate or an underestimate.

5.

(1 point) Find a plane region whose area is equal to
n

lim

n→∞

2+
i=1

Do not evaluate the limit! (Hint: Compare with lim

n→∞

the graph of f over [a, b] where ∆x =

b−a
n ,

5i
n

3

5
.
n

n
i=1 f (a+i∆x)∆x

to find a, b, and f ).

which gives the “area” under

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