Math 151 Spring 2010 c Benjamin Aurispa 3 1 Derivatives Definition of the Derivative at a point The derivative of a function f at a number a denoted f a is f a lim h 0 f a h f a h or equivalently f a lim x a f x f a x a We have already seen all of this in the previous section The derivative of a function at a has the following interpretations f a is the slope of the tangent line to the graph of f at x a f a is the instantaneous rate of change of the function f at x a If f t is a position function then f a is the instantaneous velocity at time t a Either of the above formulas can be used to find the derivative of a function at a specific value x a We did many examples of this in the previous section We can also find a function that will give the derivative at all values of x for which it is defined Definition of the Derivative as a function If f is a function the derivative of f denoted f is f x h f x h 0 h f x lim Example Find the derivative of f x x 1 2x 1 1 Math 151 Spring 2010 c Benjamin Aurispa Example Find f if f x 1 x 1 There are many different notations for the derivative of a function If y f x the following notations denote the derivative of f d dy df f x f x y dx dx dx Taking the derivative of a function is also referred to as differentiation A function is differentiable at a if f a exists A function is not differentiable at Discontinuities Corners Vertical tangent lines 2 Math 151 Spring 2010 c Benjamin Aurispa Where is the following function not differentiable 6 4 2 6 4 2 2 4 6 8 10 14 16 2 4 6 Example Show why the function f x x 2 is not differentiable at x 2 3 Math 151 Spring 2010 c Benjamin Aurispa Given each graph of f below sketch a graph of f 6 4 4 2 2 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 4 4 2 2 2 4 6 8 10 12 14 2 2 4 4
View Full Document
Unlocking...