Chapter 3 Study Guide I have listed each section of chapter 3 below and given the main points from each. That being said, there may be information I have missed so it is still a good idea to look at the book and your professor or TA s study guide. 3.1 The limit of the difference quotient 3.2 can be interpreted as: the slope of the graph of at the slope of the tangent line to the graph at the (instantaneous) rate of change at the derivative at a point You will need to know that all of these mean the same thing as derivative. So when you see one of these in a question it means to take the derivative. Also, you will need to know the limit definition of derivative for the midterm. A derivative can be expressed as a function instead of at a single point. So A function is differentiable on an interval if and only if for every x in the interval. In other words, must have a derivative everywhere in. Differentiability implies Continuity If is differentiable at, then it is also continuous at. But if we only know a function is continuous we don t know that it is differentiable. For example, is continuous everywhere but does not have a derivative at, because the limits of the derivatives from the left and right are not equal.
So a function is not differentiable when there is a corner or sharp point (cusp), a vertical tangent, or a discontinuity. 3.3 Derivative Rules Know all of these by heart and be prepared to use all of them!! [ ] 3.4 Derivative as a Rate of Change There are a lot of different possibilities for simple rate of change problems. Some may involve shapes. For instance: What is the rate of change of the area of a circle with respect to its radius? Notice it says rate of change, which cues us to take the derivative using the formula for the area of a circle. So Many problems will involve moving objects and their position, velocity, and acceleration. A good thing to know here for an object is the following:
Also, a moving object will change direction (or switch from moving in positive direction to negative direction and vice versa) when. 3.5 Trig Derivatives Memorize these. (Notice a pattern with the functions on the left and their co-functions on the right.) 3.6 Chain Rule This rule is very important and is also something to know by heart and be comfortable using several times in a problem ( Think of this as working from outside to inside, but keeping the inside untouched until you get to it. You can also think of it as or (the derivative of with respect to ) times (the derivative of with respect to ). Example: 3.7 Implicit Differentiation This method avoids the trouble of solving for and getting several functions and several derivatives. 1. Differentiate both sides with respect to. Treat as a function of.
2. Move every term with a to one side of the equation and everything else to the other side. 3. Factor out and divide out the remaining terms to the other side. Example: [( ) ] 3.8 Natural Log and Logarithmic Differentiation Natural Log Rules (pg. 44) (Be very comfortable using these rules, they are a life saver!!!) Logarithmic Differentiation This is a method of differentiation that makes difficult functions easier to differentiate. 1. Take the natural log of both sides. 2. Break down the expression as much as possible using the natural log rules. 3. Differentiate both sides. 4. Multiple by on both sides and substitute the original expression for back into the equation.
Example: ( ) ( ) [ ] Extra Comments: Always look for ways not to do quotient rule. Quotient rule is an ugly process, which can often times be avoided all together. This can be done be changing a power in the denominator to a negative and using chain rule or product rule, canceling something out, changing your trig functions, and logarithmic differentiation, just to name a few. Look up the Trig Handout on Eric s SI webpage. It has everything you will ever need to know about trig functions on the handout. Study it and be familiar with much it, many of the information can be used on exams in this class and down the road. There is also a derivative handout that might be useful as well.