We will learn later what these higher order derivatives are used for. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line. 3) Plug x value into f(x) to find the y coordinate of the tangent point. (y double prime) is the third derivative. 2) Plug x value of the indicated point into f '(x) to find the slope at x. Is the first derivative of y with respect to x. Example: f(x) = | x |ĥ To be differentiable, a function must be continuous and smooth.ĭerivatives will fail to exist at: corner cusp discontinuity vertical tangent Note: The converse is false: there are functions that are continuous but not differentiable. Theorem: If f is differentiable at a, then f is continuous at a. It is differentiable on an open interval (a,b) if it is differentiable at every number in the interval. EX: Find the derivatives of each of the functions using different derivative notations. dy / dx should not be regarded as a ratio.Ģ The derivative is the slope of the original function.Ī function f is differentiable at a if f ′(a) exists. Misconception 2: A tangent line to a curve must cross. If we replace number a by a variable x, then the derivative can be interpreted as a function of x : Alternative notations for the derivative: D and d / dx are called differentiation operators. 2.2 tangent lines and the derivative homework answers Among all functions, linear functions are simpler. \): The area of the region under the curve is approximated by summing the areas of thin rectangles.In Section 2.1 we considered the derivative at a fixed number a.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |