e substitute the given point in for a in the above formula rather than work

# E substitute the given point in for a in the above

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above…i.e. substitute the given point in for “ a ” in the above formula), rather than work with the definition in 3.2 which has “ x ” in it. Also, know how to calculate the equation of a tangent line to a curve at a given point…this involves finding the slope (as described above), and then using the given point on the curve to find the equation of the line [you can use b x m y , or ) ( 1 1 x x m y y ]. You should also be able to go “backwards”…given a limit, guess what derivative it represents. Finally, know that, in general, the instantaneous rate of change of something is the limit of the average rate of change between two points as the distance between the two points gets very small. In the context of physicals applications, for velocity, you should know that you’re supposed to differentiate the position function. Also, if you’re asked when a particle is at rest, this means you’re looking for times when the velocity is zero. 3.2: The Derivative as a Function You should be able to state the domain of the derivative (you already know how to find domains from 1.1, but also recall that the derivative can’t exist at a place where the function is not continuous). Also, know how to interpret graphs and signs of derivatives…i.e. match graphs of functions to graphs of their derivatives. 3.3: Differentiation Rules and 3.9: Derivatives of Exponential Functions Know the various rules to help you calculate derivatives, e.g. power rule, product rule, and quotient rule, and how to apply them to find derivatives. These are listed below: 0 ) ( c dx d 1 ) ( n n x n x dx d x x e e dx d f c f c ) ( g f g f ) ( g f g f ) ( f g g f g f ) ( 2 g g f f g g f Oh, and as far as second and higher order derivatives are concerned, well, if you can differentiate once, differentiating a second or third time really shouldn’t be any trouble…you just have to differentiate again. 3.4: Derivatives as Rates of Change Basically, you’re just finding derivatives in this section using the rules from the previous section. The “new” part here is just in understanding what the application is, and what you’re being asked to do. Applications include biology (rate of change of population), chemistry (rate of reaction is rate of change of chemical species), physics (velocity is rate of change of position), etc.
3.5: Derivatives of Trigonometric Functions Know the derivatives of sin( x ) and cos( x ). The derivative formulas will be provided for the rest. However, you are expected to remember that: ) sin( 1 ) csc( , ) cos( 1 ) sec( , ) sin( ) cos( ) cot( , ) cos( ) sin( ) tan( x x x x x x x x x x . You should be able to use the rules from 3.3, as well as the chain rule, to differentiate any trig function. 3.6: The Chain Rule This is very important, since it is needed in many differentiation questions. Basically, you differentiate your function of x using one of the rules you’ve already learned, and then you multiply by the derivative of the inner function. I.e.

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