# 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:

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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