Grade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability

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1 Grade 2 (MCV4UE) AP Calculus Page of 5 The Derivative at a Point f ( a h) f ( a) Recall, lim provides the slope of h0 h the tangent to the graph y f ( at the point, f ( a), and the instantaneous rate of change of y f ( with respect to at = a. Since this limit plays a central role in calculus, it is given a name and a concise notation. It is called the derivative of f( at = a. It is denoted by f (a) and is read as f prime of a. Other Notation for Derivatives If y f (, then y ' or dy are used instead of f '( ). d Leibniz Notation dy y lim derivative of y with respect to d 0 First Principles Definition of the Derivative The derivative of f at the number a is given by f '( a) lim h 0 f ( a h) h f ( a), provided that this limit eists. Eample : Determine a Derivative using the First Principles Definition 2 a) Use the first principles definition to determine the derivative of f (. b) What is the domains of f ( and f '(? c) What do you notice about the nature of the derivative? Describe the relationship between the function and its derivative. d) Determine the value of i) f '( 3), ii) f '( 0) and iii) f '( 2) and what are the values represent? e) Determine the equation of the tangent of the f ( at = 2.

2 Grade 2 (MCV4UE) AP Calculus Page 2 of 5 Eample 2: Determine a Derivative using the First Principles Definition 3 a) Use the first principles definition to determine the derivative of f (. b) Determine the equation of the tangent line at = - and show the line on the given cubic graph. Eample 3: Determine a Derivative using the First Principles Definition Determine the derivative f '( of the function f (, 0. Eample 4: Graphing f from f Graph the derivative of the function f whose graph is given. Working diagrams

3 Grade 2 (MCV4UE) AP Calculus Page 3 of 5 Eample 5: Determine a Derivative using the First Principles Definition a) Determine the derivative f '( of the function f (. b) Determine and sketch the equation of the tangent of f( at = 2. c) Determine and sketch the equation of the line that is perpendicular to the tangent (normal) to f ( at = 2and that intersects it at the point of tangency. f ( The Eistence of Derivatives (Differentiability) A function f is said to be differentiable at a if f '( a) eists. At points where f is not differentiable, we say that the derivative does not eist. Three common ways for a derivative to fail to eist are shown. Cusp The slopes of the secant lines approach from one side and from the other. 2 / 3 Eg) f Vertical Tangent The slopes of the secant lines approach either or from both sides. Eg) f 3 Discontinuity Will cause one or both of the one-sided derivatives to be noneistent)

4 Grade 2 (MCV4UE) AP Calculus Page 4 of 5 Eample 6: Recognize and Verify where a Function is Non-differentiable 5, if 2 A piecewise function f is defined by f (. The graph 0.5 2, if 2 of f consists of two line segments that form a verte, or corner, at (2, 3). a) Use the first principles to prove that the derivative f (2) does not eist. b) Graph the slope of the tangent for each on the function. How does this graph support your results in part a)?

5 Grade 2 (MCV4UE) AP Calculus Page 5 of 5 Derivatives on a Calculator (nderiv) m f h f h 2h tangent line The numerical derivative of f as a function is denoted by NDER f( & nderiv (Ti Calculators) which is similar to the concept of a lim a h or h0 Ti procedures: MATH;nDeriv(epression,variable,value) 0.00 f 0.00 f NDERf Let h 0.00is more than adequate. a-h a a+h Eample 7: Computing the Numerical derivatives Compute the numerical derivatives 3,3, the numerical derivative of a) NDER b) NDER,0, the numerical derivative of at = 0 3 at = 3 Theorems If f has derivative at = a, then f is continuous at = a. If a and b are any two points in an interval on which f is differentiable, then f, takes on every value between a f ' b. f ' and Homework: P. 05 #,3,5-3-6,8-20, 22, 24, 3,32 P. 4 #-26, 3-35, 39 Optional (Cal & Vectors) P. 73 #, 5, 6, 7, 8, 0,, 5, 9