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 : . The differential equation . . (1). . where n is a parameter, is called . Bessel’s Equation of Order n. .. Any solution of . Bessel’s Equation of Order n . is called a . Bessel Function of Order n. Objective: Understand the relationship between differentiability and continuity. Miss . Battaglia. BC Calculus. Differentiability & Continuity. Alternative limit form of the derivative:. provided this limit exists. Note the limit in this alternative form requires that the one-sided limits. 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:. Exponential Functions & Their Graphs. Logarithmic Functions & Their Graphs. Properties of Logarithms . Exponential and Logarithmic Equations. Exponential and Logarithmic Models. a. b.. iiPrefaceThepurposeofthesenotesistointroduceandstudydierentiablemani-folds.Dierentiablemanifoldsarethecentralobjectsindierentialgeometry,andtheygeneralizetohigherdimensionsthecurvesandsurfacesknown Rolle’s. theorem. Exploration:. Sketch a rectangular coordinate plane on a piece of paper.. Label the points (1, 3) and (5, 3).. Draw the graph of a differentiable function that starts at (1, 3) and ends at (5, 3).. Objective: Find the derivative using the Constant Rule, Power Rule, Constant Multiple Rule, and Sum and Difference Rules. Find the derivatives of the sine function and of the cosine function.. Ms. . Battaglia. Section 3.2a. A function will not have a derivative at a point . P . (. a. , . f. (. a. )) where. the slopes of the secant lines,. How . f. (. a. ) Might Fail to Exist. f. ail to approach a limit as . 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 . 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 . What is a piecewise function?. A piecewise function . (passes vertical line test) . is a function that is made up of different graphs for different intervals.. “Frankenstein Functions”. Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal.. 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. =. . . IN C LANGUAGE. In c, we can divide a large program into the basic building blocks known as function. . The . function contains the set of programming statements enclosed by . {}.. . A function can be called multiple times to provide reusability and modularity to the C program.