PPT-Differentiable functions

are Continuous Connecting Differentiability and Continuity Differentiability and Continuity Continuous functions are not necessarily differentiable For instance start with . 1 The Differential and Partial Derivatives Let be a function of the three variables In this chapter we shall explore how to evaluate the change in near a point and make use of that evaluation For functions of one variable this led to the derivative Differentiability. A function is differentiable at point . c . if and only if. the derivative from the left of . c. equals the derivative from the right of . c. .. AND. if . c. is in the domain of . Chapter 3.2. How . Might Fail to Exist.  . A function will not have a derivative at a point . where the slopes of the secant lines . fail to approach a limit as . Some of the common ways where a function fails to have a derivative:. AKA “Shortcuts”. Review from 3.2. 4 places derivatives do not exist:. Corner. Cusp. Vertical tangent (where derivative is undefined). Discontinuity (jump, hole, vertical asymptote, infinite oscillation). 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. Differentiability. Local Linearity. Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look like a straight line. Thus the slope of the curve at that point is the same as the slope of the tangent line at that point. c x = , the function must be continuous, and we will then see if it is differentiable. Let 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 3.6c. Suppose that functions . f. and . g. and their derivatives have the. f. ollowing values at . x. = 2 and . x. = 3.. 2. 3. 8. 3. 2. –4. 1/3. –3. 5. Evaluate the derivatives with respect to . -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. Differentiability. Local Linearity. Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look like a straight line. Thus the slope of the curve at that point is the same as the slope of the tangent line at that point. 1980 . AB Free Response 3. Continuity and Differentiability of Inverses. If . f. . is continuous in its domain, then its inverse is continuous on its domain. . If . f. . is increasing on its domain, then its inverse is increasing on its domain . Theorem and the Mean Value Theorem.  .  . Mean Value Theorem. The Mean Value Theorem can be interpreted geometrically as follows:. Is the slope of the line segment joining the points where . x. =. Nicholas . Ruozzi. University of Texas at Dallas. Where We’re Going. Multivariable calculus tells us where to look for global optima, but our goal is to design algorithms that can actually find one!.