Introduction to Calculus

Course Head: Robin Gottlieb
Office: Science Center 430
Email: gottlieb@math.harvard.edu

Exams

Here are some previous exams and solutions that should provide a good review of the material covered to date. Please note that not all exam problems apply. Solutions for the Fall 2000 and Fall 1999 Exams have been posted. Solutions for the Fall 2002 and Spring 1998 Exams will be posted by October 18. Students are encouraged to attempt the practice exams before checking the solutions.

Midterm 1

Additional corrections to Fall 2002:
#1(a) lim x to 2 should be written two more times.
#1(c) Unlike the damped oscillations in part (d), these are getting wilder as x increases.

#2: if the explanation given in the corrections is not adequate, please see Ex 6 p. 137 in your text.
#3c arctan x would have two horizontal asymptotes.
#4. When you give a linearization, make sure that what you have is the equation of a line! L (x) = f(a) + f'(a) (x-a) or in this case
L(x) = f(25) + f'(25) (x-25). It is not necessary to use this "formula" in order to get the linear approximation. The correct answer is 5.1 as stated in the corrections.
#6: I would say that the velocity graph does NOT have a vertical asymptote at x=3, but rather a jump discontinuity.

Corrections to spring 1998:
# 1(d) False: let f(x) = 1/x and g(x) = 5/x.
The limit rule quoted is only true if both limits independently exist.
1(e) The statement is false but the reason given doesn't make sense. If the function is continuous from the left and right with the definitions stated then the function must be continous. However, the left and right hand limits could be different, or we could have the left and right hand limits equal but the function defined at c to be something entirely different.

#5: f'(2) is not 1. This says that as h approaches 0 the difference between f at 2+h and at 2 approaches -1. Try f(x) = 4 for x <2 and f(x) = 3 for x>2 for example.

Exam	Solution
Fall 2002 page1 page2 page3 page4 page5 page6 page7 page8 page9 page10	*Solution (additional corrections above the box)* <Solution Correction> page1 page2 page3 page4 page5 page6 page7 page8 page9 page9 page10 page11 page12 page13
Fall 2000 Note: Omit #4, 5, 7 and 8	*Solution*
Fall 1999 Note: Omit #5, 6 and 7 AND On Question 3b use the limit definition of a derivative	*Solution*
Spring 1998 page1 page2 page3 page4 page5 Note: Omit #1b and 4	*Solution (see corrections above the box).* page1 page2 page3 page4 page5 page6 page7 page8

Mathematics 1a

Introduction to Calculus

Exams