Chapter 2

I. The Derivative and the Tangent Line Problema. A tangent line is a line that touches only one point on a graphb. Finding a tangent line basically involves finding it’s slopec. The slope of a tangent line can be found through differentiation

i. The process of finding the derivative of a functiond. A derivative can be used to find the instantaneous rate of changee. A function is differentiable at x when its derivative exits at x and is differentiable on

an open intervali. Differentiable at every point in an interval

f. Limit notation for derivatives isi. h being the change in x

g. A line must be continuous for it to be differentiablei. The graph cannot contain a corner or cusp either

II. Basic Differentiation Rules and Rates of Changea. The constant rule states that the derivative of a constant is 0

i. The derivative of any real number by itselfb. The most basic way to find a derivative is the product rulec. Used in polynomials that are being added or subtractedd. To use the power the power is brought in front of the variable and then the power is

decreased by onei. the derivative of x by itself is always 1

e. If a polynomial is being divided by a monomial then each piece can be separated to use the power rule

f. The derivative of sin x is cos x and the derivative of cos x is –sin xi. If there is a constant with the sin or cos it stays the same

g. Derivatives can be applied to position functionsi. The first derivative of a position function tells the average velocity at a

given heightIII. Product and Quotient Rules and Higher-Order Derivatives

a. The product rule is used when two functions being multiplied togetherb. To use it the first function is multiplied by the derivative of the second plus the

second function multiplied by the derivative of the firsti. f(x)g’(x) + g(x)f’(x)

c. The quotient rule is used when one function is dividing another d. To use it the bottom function is multiplied by the derivative of the top function

subtracted by the top function multiplied by the derivative of the bottomi. All of this is then divided by the bottom function squared

ii. g(x)f’(x) – f(x)g’(x) [g(x)]^2

e. “Low D high minus high D low, draw a line and square belowf. More can be found out from position functions using derivatives

i. The second derivative of a position function tells the acceleration of the object

1. The acceleration will be the same for the entire function