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Understanding Derivatives: Instantaneous Rates of Change and Continuity

This chapter delves into the concept of derivatives, highlighting their significance in calculating instantaneous rates of change. It defines the derivative of a function ( f ) at a specific point ( a ), using limits. The chapter includes examples demonstrating how to derive functions and find tangent lines, as well as discusses differentiability and continuity. It emphasizes that a function can be continuous but not differentiable at specific points, illustrated through practical examples and the exploration of higher-order derivatives.

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Understanding Derivatives: Instantaneous Rates of Change and Continuity

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  1. Derivatives CHAPTER 2 Instantaneous rates of change appear so often in applications that they deserve a special name. 2.4 Continuity DefinitionThe derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a)) / h if this limit exists. f’(a)= lim x  a ( f(x) – f(a)) / (x – a).

  2. Example Find the derivative of the function f(x) = 1+ x – 2x2 at the number a.

  3. The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope isequal to f’(a), the derivative of f at a. The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

  4. Example If g(x) = 1 – x3, find g’(0) and use it to find an equation of the tangent line to the curve y = 1 – x3 at the point (0,1).

  5. Example State fand a in the following limit which represents the derivative of some function f at some number a. lim h --> 0 ( (2 + h)3 – 8 ) / h .

  6. The Derivative as a Function CHAPTER 2 2.4 Continuity The derivative at a number a is a number. If we let a vary, x=a, then the derivate will be a function as well. f’(x) = lim h --> 0 ( f(x + h) – f(x)) / h

  7. ____ Example Find the derivative of the function f(x) = 1+ x and find the domain of f’.

  8. Definition A function is differentiable at aif f’(a) exists. It is differentiable on an open interval (a,b) or (a, ) or (-, ) if it is differentiable at every number in the interval. Theorem Iff is differentiable at a, then f is continuous at a. Can a function be continuous but not differentiable at some number?

  9. Example Show that f(x) = | x – 6 | is not differentiable at 6. Find the formula for f’ and sketch its graph.

  10. The Second Derivative If f is differentiable function, then its derivative f’ is also a function, so f’ may have a derivative of its own, denoted by ( f’)’ = f”. This new function f” is called the second derivative of f.

  11. Example Find f” for f(x) = 3x3 – 4x2 +7 .

  12. Various notations for nth derivative of the function y=f(x): y n = f n (x) = (dn y) / (d xn ) =Dn x

  13. Example If f(x) = x 3 – 2x2 + 9, find 3rd and 4th derivatives of f.

  14. Example Based on the following animation, state whether the function is differentiable at x=0. animation animation

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