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Derivatives Overriding principal is the concept of change (speed, weather, populations,..) Average speed An object is dropped from a 500 ft building. From physics we know that the distance above the ground is d (t ) 500 16 t 2 . Sketch a graph of distance versus time. Find time until object hits the ground. Compute average speed on the interval [a, b] using d (b) d (a) v ave ba Geometrically the average speed is the slope of the secant line. I. Instantaneous velocity Greeks realized that at any given instant even a moving object is frozen. Newton and his contemporaries gave up on the notion of speed at an instant and replaced it with average speeds over shorter and shorter intervals. Compute instantaneous speed at the time t a using d (t ) d (a) vinst lim . t a ta Geometrically the instantaneous speed is the slope of the tangent line. II. The derivative at a point Can apply the same process to any function. The instantaneous rate of change of y f (x) at the point x a is defined by f ( x) f ( a ) f ' (a) lim or letting h x a x a xa f ( a h) f ( a ) f ' (a) lim h 0 h III. The derivative as a function We can apply the process of finding the derivative to many different x values and in fact for every x value there is a corresponding value for the derivative. For any function y f (x) we define the derivative function f ( x h) f ( x ) f ' ( x) lim h 0 h There are several equivalent notations for the derivative function including dy df f ' ( x), y' ( x), , , Dx f ( x), Dx y dx dx IV. Interpretations of the derivative As a slope and as a rate of change. dC Ex. Suppose C is cost and t is time. Then is the rate of change of cost. dt V. Graphical derivatives Function Derivative Increasing Positive Decreasing Negative Constant Zero Concave up Increasing Concave down Decreasing Inflection point Max/Min Smooth max/min Zero HA (x=a) HA (x=a) VA (y=a) VA (y=0) Sharp corner undefined VI. Algebraic shortcuts for computing derivatives Table of known derivatives Power rule d n x nx n 1 dx Trigonometric functions d sin x cos x dx d cos x sin x dx d tan x sec2 x dx d cot x csc2 x dx d sec x sec x tan x dx d csc x csc x cot x dx Combinations Sums, Differences, Products, Quotients d f ( x) s ( x) d f ( x) d s ( x) dx dx dx d f ( x) s ( x) d f ( x) d s ( x) dx dx dx d f ( x) s ( x) f ( x) d s ( x) s ( x) d f ( x) d dx dx dx d d d ( x ) n( x ) n( x ) d ( x ) d n( x ) dx dx d ( x) dx d ( x) 2 Chain rule d f ( g ( x)) f ' ( g ( x)) g ' ( x) dx Implicit differentiation VII. Focus on theory Limits Intuitive definition lim f ( x) L means that as x gets close to a , f (x) gets close to L . x a Existence of limits (limits can fail to exist if there are small jumps, unbounded behavior or excessive oscillations) Rigorous definition lim f ( x) L means that for every 0 there exists a 0 such that x a f ( x) L whenever x a . Continuity Intuitively a function is continuous if you can sketch its graph without lifting your pencil. From a rigorous perspective a function is continuous at x a if (i) f (a) is defined (ii) lim f ( x) exists xa (iii) lim f ( x) f (a) x a Differentiability A function is differentiability if it possesses the property of local linearity. Differentiability continuity Continuity differentiability f (b) f (a) Differentiable functions have the Mean value property f ' (c) ba for some value of c (a, b) . This means there is a point where the slope of the tangent line is equal to the slope of the secant line or that there is point where the instantaneous rate of change is equal to the average rate of change. VIII. Homework assignment Calculus home page

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posted: | 1/1/2011 |

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