MA 125-5B, Spring 2004

TEST # 1

February 6, 2004 (70 minutes)

Name: SSN:

Max. Points: 100 + 5 Bonus Points: Test Grade:

Turn in all the work which you did to solve the problems, not just the ﬁnal answer.

In particular, include intermediate steps in calculations wherever they are needed.

You may write on the back of a page if you need extra space.

To receive credit, all solutions have to be based on the methods from Chapter 2 of

Stewart’s book.

No book, no notes, and no calculator are to be used!

1. (a) Deﬁne what it means that a function fis continuous at a number a. (3P)

(b) Deﬁne what it means that a function fis diﬀerentiable at a number a. (3P)

2. State the intermediate value theorem. (6P)

3. Find the following limits: (4P+4P+4P+4P)

(a) lim

x→−2

x+ 2

x2−4

(b) lim

x→0

1

cos x

(c) lim

x→∞

2x3−x

1−3x3

(d) lim

x→0

1

x3

4. From the graph of fprovided on the right ﬁnd all numbers asuch that (3P+3P+3P+3P)

(a) lim

x→a+f(x) does not exist

(b) lim

x→a

f(x) does not exist

(c) fis not continuous at a

(d) fis not diﬀerentiable at a

5. For f(x) = 1

xﬁnd the equation for the tangent line to the graph at the point

P(2,1

2). (12P)

6. The position of a vehicle after tseconds is s(t) = t3meters. (a) Find the velocity

and the acceleration of the vehicle after 2 seconds. (12P)

(b) At which time does the vehicle have a velocity of 27 m/s? (4P)

7. For the given graph of f, sketch the graphs of f0and f00. Make sure that these

graphs reﬂect where fis increasing or decreasing and where fis concave upwards

or concave downwards. (14P)

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