Example Of Report On Also, What Do You Use To Help Remember These? ***
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***There are a lot of Rules and Procedures to find derivatives -- rank these in order from easiest to most difficult
a) Difference Quotient b) Power Rule c) Quotient Rule d) Chain Rule e) General Power Rule
In my opinion, the ranking would be as follows: 1.Power Rule, 2.Quotient Rule, 3. General power rule, 4.Chain Rule, 5.Difference Quotient. The Power Rule goes to the first place as it only involves changing the exponent. Quotient Rule needs derivatives of two different functions and the 2nd place suits it. General Power rule as a special case of Chain Rule involves derivating of a function consisting of another function, and it’s one level more complicated than the Quotient Rule and one level less than the Chain Rule. Since Difference Quotient needs an extra skill of calculating Limits which can be very tricky and difficult, it is placed last.
Practice is the key to perfection. To my way of thinking, one can make a list of these rules and specify what combination of functions suit best with them. Then he or she may try this list in exercises until he or she masters and keep it in mind for a long time.
***Imagine you had an equation that described you driving a car from home to the store. Tell everyone about the increasing/decreasing, critical points, concave up/concave down and inflection points of the curve that would describe your driving. ***
Imagine that we are driving up a sloped road or a hill. If it feels like we are going higher as we are driving the road, it means that the road at that specific interval has a positive slope. In the other words, the first derivative of the function describing you driving is positive and this function is increasing. In contrast, if it feels like we are falling then the function has a negative first derivative and it is decreasing. When we reach the top or bottom of a hill or a sloped road, for a moment we feel no sloping (no going down or up); it means that the first derivative of the function describing you driving is zero and this specific point is critical and an extremum of this function. If it feels like we are driving inside a circle, the road is concaved up in that specific interval and the second derivative is positive. Similarly, if it feels we are driving on a circle then the road is concaved down and the second derivative is negative. The point we feel a change of concavity after driving past it, is an inflection Point and has a second derivative equaling to zero.
***In just about every exponential growth/decay problem you end up with an equation that looks like thisA = A0 e kt
Describe what each term means and give an example of it's use***
In such equations, A0 is the initial value, k the relative growth rate and t the variable. If k has a positive/negative value, then it is a growth/decaying problem. Let’s consider the population growth as an example for its application, then t would be the time, k the relative population growth rate, A0 the initial population and A the population after a time t has passed.
***We know that an indefinite integral leaves us with a + C at the end. Discuss what this is all about and give an example of a real life situation where the + C is important. ***
C is an arbitrary constant, which expresses that each function has an infinite number of distinct antiderivatives. As an example of a real life situation, suppose a car driving a distance of ∆x=x-x0 with a constant velocity of V. If we calculate car location x based on a givenV, the C left from indefinite integral would be the initial position of the carx0. Therefore, we cannot neglect C or consider it zero, otherwise we would not know where the car is now!
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