FALL 2007: [Final Exam] [Solutions]

Monday, Dec 17: to take a look at your exam and find out your final grade, please stop by Krieger 301 as follows

• Monday, Dec 17, 12 noon-1pm: Sections 1, 3
• Monday, Dec 15, 2 pm-3pm: Sections 2,4

Please note that I will hold on to your papers after you examine them.

If you cannot make it on Monday, you can get the grade from the Registrar's office. Your instructor or TA will not email you the grade.

MIDTERM II: Wednesday, November 14, in class.

Exam questions. Solutions.

Try to arrive at least 5 minutes before the exam starts.

Midterm topics

• Local/global extremum. Critical points. Optimization.
• Rolle's Theorem. Mean Value Theorem. (Proofs and applications.)
• Monotonicity: first derivative. Concavity: second derivative.
• Second Derivative test: to determine local max/min.
• Definite Integral. Riemann sums. Area under the graph of a function.
• Antiderivative (indefinite integral).
• Evaluation Theorem
• Substitution Rule (to be covered on Monday).

Old exams (Fall 06): [Second Midterm] [Solutions]

MIDTERM I: Wednesday, October 10, in class. Try to arrive at least 5 minutes in advance.

Exam topics: sections 1.1.-2.5. of the textbook, plus the part of 2.8 covered in lecture (linear approximation).

Also review section material and homework exercises. The textbook has a good review of chapter 2 on page 139.

Topics covered in lecture [exam material]:

Week 1

• Functions: definition (domain, range).
• Graphs of real-valued functions.
• Basic functions (and their graphs): linear functions, quadratic functions, square-root.
• Monotone functions.
• sin and cos: definition from scratch and basic properties.

• Operations on functions (algebraic, composition).
• Limit of a real-valued function.
• Basic limits (subject to substitution).
• Operations with limits (sum, product, quotient). More complicated limits.
• Squueze theorem. Relevant examples.

• lim_{x->0}sin(x)/x=1. Proof and variations.
• Continuous functions and operations: addition, multiplication, composition.
• Intermediate value theorem
• Horizontal and vertical asymptotes (limits involving infinity)
• Tricks with limits involving powers of x: at 0 and at infinity.

• Differentiation: definition [a limit] and geometric meaning [slope of tangent line].
• Differentiation of basic functions from scratch: mx+n, sin(x).
• Rules of differentiation: sum, product, quotient. Diff. of more complicated functions.
• Linear approximation: f(x)~f(a)+f'(a)(x-a). Example: sqrt{1.04}~1.02.
• Theorem: differentiability=>continuity. Proof.
• Graphs of non-differentiable functions [we discussed 3 types].
• Equation of the tangent line to the graph of a function.
• Chain Rule. Proof. Examples.
• A harder example: x^2*sin(1/x).

Practice exams

Fall 06. [First Midterm] [Solutions]