PPT-Derivative of a Function

Chapter 31 Definition of the Derivative In the previous chapter we defined the slope of the tangent line to a curve at a point as When this limit exists it is called the derivative of . We have that AA 1 that is that the product of AA is the sum of the outer products of the columns of To see this consider that AA ij 1 pi pj because the ij element is the th row of which is the vector a a ni dotted with the th column of which is Relating f, f’, and f” . Problem A. Problem B. Conceptual Problems. Inability to see derivative as a function, only a value. Derivative is object but not as an operation. Derivative vs. Differentiation vs. “Finding the derivative”. Section 3.1b. Remember, that in . graphical terms. , the derivative of a. function at a given point can be thought of as the . slope. of the curve at that point…. Therefore, we can get a good idea of what the graph of. Points of Inflection. Section 4.3a. Writing: True . or . False – A . critical point . of. a function always signifies . an . extreme. value . of the . function. Explain.. FALSE!!! – Counterexample???. Derivative. A . derivative. of a function is the instantaneous rate of change of the function at any point in its domain.. We say this is the derivative of . f. with respect. to the variable . x. .. Example. For. . find the derivative of . f. and state the domain of . f’. . The derivative can be regarded as a new function. Example. Given the graph of the function, . f. derivative. Lecture. . 5. Handling. a . changing. . world. x. 2. -x. 1. y. 2. -y. 1. The. . derivative. x. 2. -x. 1. y. 2. -y. 1. x. 1. x. 2. y. 1. y. 2. The. . derivative. . describes. . the. VALUE THEOREMS. Derivability of a function :. A function . f . defined on [. a, b. ] is said to be derivable or differentiable at if exists. This limit is called derivative of . Section 3.1a. Answers to the “Do Now” – Quick Review, p.101. 1.. 2.. 3.. 5. Slope:. 6.. 4.. 7.. 8.. 9. No, the one-sided limits. at . x. = 1 are different. 10. No, . f. is discontinuous. at . -More Effort Needed!. -Wording of Problems (derivative, slope at a point, slope of tangent line…). -Product / Quotient Rules!!!. -Quiz . I:g. and . II:a. -Weekly 7 , 8 , 10 . The Chain Rule. 4.1.1. Slope of the Tangent Line. If . f. is defined on an open interval containing . c. and the limit exists, then . . and the line through (. c. , . f. (. c. )) with slope . m. is the line tangent to the graph of . The Second Derivative and the Function. The first derivative tells us where a function is increasing or decreasing. But how can we tell the manner in which a function is increasing or decreasing?. For example, if . NOW: . Replace: . Graph of . , with words:. Graph: (. , the . slope of the tangent line to the . function . . at that . point). .  . CALCULUS problem:. Graph: (. , the slope of the tangent line to the function . T. he line y=L is a horizontal asymptote of the graph of f if lim f(x)=L or limf(x)=L. Horizontal Asymptotes. X->. 8. X-> -. 8. Finding a horizontal asymptote (when looking at exponential degree):.