Calculus 1 -- Fall 2017

The second exam is scheduled for Friday, 11/3.   Remember to bring your portfolio for review during the exam, and to take it with you when you leave.

The exam will cover sections 3.1 - 3.7, 3.9, 3.10,  4.1, 4.2, plus the material on optimization from Stewart's Calculus text, plus the additional material I presented on differential equations (including sections 11.1-11.3).  On Wednesday, November 1, class will be devoted to reviewing for the exam.  Here is a  sample exam

The material for this exam has less of a conceptual and theoretical emphasis than we saw in Chapter 2.  As a result, the exam will focus more on basic knowledge and procedures.  You should be able to find correct formulas for derivatives (as well as higher derivatives) for just about any sort of function.  That means you have to be able to decide which rules of differentiation to apply, and to apply them correctly.  To review for this material, I recommend that you work on a random selection of Review Exercises 1-73 at end of chapter 3.  You may skip any question that mentions sinh, cosh, or tanh, which are all from section 3.8.    When you are asked to find a derivative, any correct answer (with appropriate work and/or justification) will earn full credit. Algebraic simplification is not required. But if you do simplify, be careful to do so correctly. Simplification errors may result in a partial loss of credit.

Finding Derivatives.  At the exam I will pass out this sheet showing the basic rules for derivatives, as well as basic geometric and trigonometric facts.  In part, this reflects my belief that memorizing all of the basic derivative formlas is not essential for a general understanding of this material for most calculus 1 students.  However, anyone who will go on to calculus 2 and more advanced courses SHOULD know all the basic derivative rules and formulas, as readily as they know basic rules of algebra and arithmetic.  Without this knowledge you will be at a disadvantage in learning and demonstrating your command of later material.  So, if you are going on in math, make an effort to learn (which is not exactly the same as memorizing) the derivative rules and formulas.  Similarly for the basic trigonometry.  The geometric facts are provided as a resource for max/min problems.  This information is general math knowledge that math majors should have, but is not specific to calculus and will not be as critical for future math courses.

In some cases, for problems that apply derivatives, you may be given equations for the derivative and/or the second derivative of a function.  Such problems may be included because errors in finding derivatives can make it much more difficult (or impossible) to carry out the steps that use those derivatives.  Be on the look out for this kind of problem.  If at any point you decide you need to take the derivative of a function, check whether that derivative has already been given as part of the problem.  Problem 7 on the sample exam is a problem of this type.

Differential Equations.  For this material the emphasis will be a bit more conceptual.  You should understand the basic ideas from the powerpoint presentation: what differential equations are, how they can arise in applications, the idea of making highly simplified assumptions that are reasonable over very short intervals of time, and by using limits obtain an equation about an unknown function and its derivative, how differential equations can be used to make predictions about future evolution of a model, including the role of initial conditions, and the cyclic nature of model development.  You should be able to give examples of systems for which differential equation methods produce highly accurate results, but should also be able to give an example of a system where differential equations cannot give us accurate predictions (chaos).  Be prepared for either essay questions or multiple choice questions on this material.  We have seen a method to solve ONE particular type of differential equation, by reducing it to something of the form
 (1) (ln f (x))' = constant. (1)
You should be prepared to use this method.  However, you should also know that the solution to this special differential equation is of the form
 (2) f (x) = Aekx, (2)
where A and k are constants, A is the value of  f (0), and k is the constant on the right side of  equation (1).  (This k is the relative growth rate, given by  f  '(x)/f(x) for any x.)    For sections 11.2, you should be able to explain what a slope field is and how it is created, and should be able to use a slope field to estimate solutions to a differential equation.  For section 11.3, know how to use Euler's method to approximate points on a solution curve to a differential equation, and understand how Euler's method is related to the idea of linear approximations.

Theorems and Definitions.  You should know the statement's of the theorems in section 3.10, and be able to use these theorems to draw conclusions about functions in particular situations.  For example, you should be able to use the Mean Value Theorem to prove that if f  is a function whose derivative is zero everywhere, then f  must be a constant function.  Similarly, you should be able to show that if f is a function whose derivative is positive at every point of an interval (a,b), then f is an increasing function on (a,b).  You should know definitions inctroduced in chapter 4, such as local maxima and minima, critical points, critical numbers, inflection points, global (aka absolute) maxima and minima.  Know the theorems pertaining to local and global maxima and minima, including the first and second derivative tests, the Extreme Value Theorem, the First Derivative Test For Absolute Extreme Values (from the handout posted on blackboard).    Be able to describe, justify, and use the procedure for finding global maxima and minima for a continous function defined on a closed interval.