# Continuous Function - Calculus I - Exam, Exams for Calculus I. Birla Institute of Technology and Science

## Calculus I

Description: Continuous Function, Differentiable Function, Intermediate Value Theorem, Find Following Limits, Equation for Tangent Line, Position of Vehicle, Acceleration of Vehicle, Concave Downwards are some points from this exam paper of Calculus I.
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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
x24
(b) lim
x0
1
cos x
(c) lim
x→∞
2x3x
13x3
(d) lim
x0
1
x3
4. From the graph of fprovided on the right ﬁnd all numbers asuch that (3P+3P+3P+3P)
(a) lim
xa+f(x) does not exist
(b) lim
xa
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)