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Practice Test # 1, Math 1550, Spring 2012

In general in your test will be used the problems similar to your WebAssign and home-

works.

1. Given a graph of a function fsketch its derivative f′below.

-2

-1

1

2

3

4

5

-10

10

20

30

40

50

-2

-1

1

2

3

4

5

-40

-20

20

40

2. Determine the type of each given limit and ﬁnd evaluate them, without approxima-

tions or numerical estimations.

a) lim

x→0(4√x3+ 2 −5) = 4√2−5, the type is a “number” (or #).

b) lim

x→−10

x+ 10

x2+ 4x−60 =−1

16 , the type is 0

0.

c) lim

x→1+

ex2−1

sin x=1

sin 1, the type is a “number” (or #).

d) lim

x→5−

x−5

|5−x|=−1, the type is 0

0.

1

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2e) lim

h→−∞

tan−1(x3−x) = −π

2, the type is f(−∞).

f) lim

t→−3

√1−t−2

t+ 3 =−1

4, the type is 0

0.

g) lim

t→01

3t−1

t(t+ 3)=1

9, the type is ∞ − ∞.

h) lim

s→9

s−6

√s−3DNE, the type is #

0

3. Use the Squeeze Theorem to ﬁnd lim

x→0(x3cos 1

x).

Answer: For the Squeeze Theorem we need a double inequation, which bounds the

function.

−1≤cos 1

x≤1 multiply everything by |x3|

−|x3| ≤ |x|3cos 1

x≤ |x3|.

Then using that −|a| ≤ a≤ |a|:

−|x3| ≤ x3cos 1

x≤ |x3|

Now we take limits on the left and on the right, they should agree for the Squeeze

Theorem to work.

lim

x→0(−|x3|) = 0 = lim

x→0|x3|.

Hence lim

x→0x3cos 1

x= 0.

4. Use the Intermediate Value Theorem to show that the equation 2x5+x+ 1 = 0 has

at least one root in the interval (−1,0).

Answer: Deﬁne f(x) = 2x5+x+ 1. This is a polynomial function, thus it’s continuous.

Calculating f(−1) = −2 and f(0) = 1 we can see that 0 lies between -2 and 1. Then

by Intermediate Value Theorem there is such x=cin the interval (−1,0) so f(c) =

2c5+c+ 1 = 0.

5. a ) A ball is dropped from a state of rest at time t= 0. The distance traveled after

tseconds is s(t) = 10 −16t2ft. Compute the average velocity over time interval [0.6,0.7].

Round your answer to two decimal places.

Answer: vav =−20.8

b) ﬁnd a slope of the secant line of the graph below over the interval [-1.8, -0.8]. What

can you say about slopes of tangent lines at x=-2, -0.8, -0.5, 0.2?

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